src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
author paulson <lp15@cam.ac.uk>
Tue Apr 28 16:23:38 2015 +0100 (2015-04-28)
changeset 60150 bd773c47ad0b
parent 60141 833adf7db7d8
child 60162 645058aa9d6f
permissions -rw-r--r--
New material about complex transcendental functions (especially Ln, Arg) and polynomials
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(*  Author: John Harrison
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    Ported from "hol_light/Multivariate/transcendentals.ml" by L C Paulson (2015)
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*)
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section {* Complex Transcendental Functions *}
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theory Complex_Transcendental
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imports  "~~/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics"
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begin
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lemma cmod_add_real_less:
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  assumes "Im z \<noteq> 0" "r\<noteq>0"
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    shows "cmod (z + r) < cmod z + abs r"
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proof (cases z)
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  case (Complex x y)
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  have "r * x / \<bar>r\<bar> < sqrt (x*x + y*y)"
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    apply (rule real_less_rsqrt)
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    using assms
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    apply (simp add: Complex power2_eq_square)
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    using not_real_square_gt_zero by blast
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  then show ?thesis using assms Complex
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    apply (auto simp: cmod_def)
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    apply (rule power2_less_imp_less, auto)
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    apply (simp add: power2_eq_square field_simps)
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    done
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qed
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lemma cmod_diff_real_less: "Im z \<noteq> 0 \<Longrightarrow> x\<noteq>0 \<Longrightarrow> cmod (z - x) < cmod z + abs x"
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  using cmod_add_real_less [of z "-x"]
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  by simp
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lemma cmod_square_less_1_plus:
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  assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1"
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    shows "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
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  using assms
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  apply (cases "Im z = 0 \<or> Re z = 0")
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  using abs_square_less_1
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    apply (force simp add: Re_power2 Im_power2 cmod_def)
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  using cmod_diff_real_less [of "1 - z\<^sup>2" "1"]
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  apply (simp add: norm_power Im_power2)
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  done
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subsection{*The Exponential Function is Differentiable and Continuous*}
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lemma complex_differentiable_within_exp: "exp complex_differentiable (at z within s)"
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  using DERIV_exp complex_differentiable_at_within complex_differentiable_def by blast
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lemma continuous_within_exp:
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  fixes z::"'a::{real_normed_field,banach}"
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  shows "continuous (at z within s) exp"
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by (simp add: continuous_at_imp_continuous_within)
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lemma continuous_on_exp:
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  fixes s::"'a::{real_normed_field,banach} set"
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  shows "continuous_on s exp"
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by (simp add: continuous_on_exp continuous_on_id)
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lemma holomorphic_on_exp: "exp holomorphic_on s"
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  by (simp add: complex_differentiable_within_exp holomorphic_on_def)
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subsection{*Euler and de Moivre formulas.*}
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text{*The sine series times @{term i}*}
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lemma sin_ii_eq: "(\<lambda>n. (ii * sin_coeff n) * z^n) sums (ii * sin z)"
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proof -
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  have "(\<lambda>n. ii * sin_coeff n *\<^sub>R z^n) sums (ii * sin z)"
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    using sin_converges sums_mult by blast
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  then show ?thesis
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    by (simp add: scaleR_conv_of_real field_simps)
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qed
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theorem exp_Euler: "exp(ii * z) = cos(z) + ii * sin(z)"
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proof -
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  have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n)
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        = (\<lambda>n. (ii * z) ^ n /\<^sub>R (fact n))"
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  proof
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    fix n
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    show "(cos_coeff n + ii * sin_coeff n) * z^n = (ii * z) ^ n /\<^sub>R (fact n)"
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      by (auto simp: cos_coeff_def sin_coeff_def scaleR_conv_of_real field_simps elim!: evenE oddE)
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  qed
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  also have "... sums (exp (ii * z))"
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    by (rule exp_converges)
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  finally have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n) sums (exp (ii * z))" .
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  moreover have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n) sums (cos z + ii * sin z)"
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    using sums_add [OF cos_converges [of z] sin_ii_eq [of z]]
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    by (simp add: field_simps scaleR_conv_of_real)
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  ultimately show ?thesis
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    using sums_unique2 by blast
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qed
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corollary exp_minus_Euler: "exp(-(ii * z)) = cos(z) - ii * sin(z)"
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  using exp_Euler [of "-z"]
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  by simp
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lemma sin_exp_eq: "sin z = (exp(ii * z) - exp(-(ii * z))) / (2*ii)"
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  by (simp add: exp_Euler exp_minus_Euler)
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lemma sin_exp_eq': "sin z = ii * (exp(-(ii * z)) - exp(ii * z)) / 2"
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  by (simp add: exp_Euler exp_minus_Euler)
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lemma cos_exp_eq:  "cos z = (exp(ii * z) + exp(-(ii * z))) / 2"
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  by (simp add: exp_Euler exp_minus_Euler)
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subsection{*Relationships between real and complex trig functions*}
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lemma real_sin_eq [simp]:
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  fixes x::real
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  shows "Re(sin(of_real x)) = sin x"
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  by (simp add: sin_of_real)
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lemma real_cos_eq [simp]:
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  fixes x::real
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  shows "Re(cos(of_real x)) = cos x"
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  by (simp add: cos_of_real)
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lemma DeMoivre: "(cos z + ii * sin z) ^ n = cos(n * z) + ii * sin(n * z)"
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  apply (simp add: exp_Euler [symmetric])
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  by (metis exp_of_nat_mult mult.left_commute)
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lemma exp_cnj:
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  fixes z::complex
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  shows "cnj (exp z) = exp (cnj z)"
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proof -
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  have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) = (\<lambda>n. (cnj z)^n /\<^sub>R (fact n))"
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    by auto
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  also have "... sums (exp (cnj z))"
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    by (rule exp_converges)
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  finally have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (exp (cnj z))" .
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  moreover have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (cnj (exp z))"
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    by (metis exp_converges sums_cnj)
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  ultimately show ?thesis
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    using sums_unique2
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    by blast
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qed
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lemma cnj_sin: "cnj(sin z) = sin(cnj z)"
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  by (simp add: sin_exp_eq exp_cnj field_simps)
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lemma cnj_cos: "cnj(cos z) = cos(cnj z)"
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  by (simp add: cos_exp_eq exp_cnj field_simps)
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lemma complex_differentiable_at_sin: "sin complex_differentiable at z"
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  using DERIV_sin complex_differentiable_def by blast
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lemma complex_differentiable_within_sin: "sin complex_differentiable (at z within s)"
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  by (simp add: complex_differentiable_at_sin complex_differentiable_at_within)
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lemma complex_differentiable_at_cos: "cos complex_differentiable at z"
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  using DERIV_cos complex_differentiable_def by blast
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lemma complex_differentiable_within_cos: "cos complex_differentiable (at z within s)"
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  by (simp add: complex_differentiable_at_cos complex_differentiable_at_within)
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lemma holomorphic_on_sin: "sin holomorphic_on s"
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  by (simp add: complex_differentiable_within_sin holomorphic_on_def)
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lemma holomorphic_on_cos: "cos holomorphic_on s"
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  by (simp add: complex_differentiable_within_cos holomorphic_on_def)
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subsection{* Get a nice real/imaginary separation in Euler's formula.*}
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lemma Euler: "exp(z) = of_real(exp(Re z)) *
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              (of_real(cos(Im z)) + ii * of_real(sin(Im z)))"
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by (cases z) (simp add: exp_add exp_Euler cos_of_real exp_of_real sin_of_real)
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lemma Re_sin: "Re(sin z) = sin(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
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  by (simp add: sin_exp_eq field_simps Re_divide Im_exp)
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lemma Im_sin: "Im(sin z) = cos(Re z) * (exp(Im z) - exp(-(Im z))) / 2"
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  by (simp add: sin_exp_eq field_simps Im_divide Re_exp)
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lemma Re_cos: "Re(cos z) = cos(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
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  by (simp add: cos_exp_eq field_simps Re_divide Re_exp)
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lemma Im_cos: "Im(cos z) = sin(Re z) * (exp(-(Im z)) - exp(Im z)) / 2"
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  by (simp add: cos_exp_eq field_simps Im_divide Im_exp)
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lemma Re_sin_pos: "0 < Re z \<Longrightarrow> Re z < pi \<Longrightarrow> Re (sin z) > 0"
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  by (auto simp: Re_sin Im_sin add_pos_pos sin_gt_zero)
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lemma Im_sin_nonneg: "Re z = 0 \<Longrightarrow> 0 \<le> Im z \<Longrightarrow> 0 \<le> Im (sin z)"
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  by (simp add: Re_sin Im_sin algebra_simps)
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lemma Im_sin_nonneg2: "Re z = pi \<Longrightarrow> Im z \<le> 0 \<Longrightarrow> 0 \<le> Im (sin z)"
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  by (simp add: Re_sin Im_sin algebra_simps)
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subsection{*More on the Polar Representation of Complex Numbers*}
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lemma exp_Complex: "exp(Complex r t) = of_real(exp r) * Complex (cos t) (sin t)"
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  by (simp add: exp_add exp_Euler exp_of_real sin_of_real cos_of_real)
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lemma exp_eq_1: "exp z = 1 \<longleftrightarrow> Re(z) = 0 \<and> (\<exists>n::int. Im(z) = of_int (2 * n) * pi)"
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apply auto
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apply (metis exp_eq_one_iff norm_exp_eq_Re norm_one)
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apply (metis Re_exp cos_one_2pi_int mult.commute mult.left_neutral norm_exp_eq_Re norm_one one_complex.simps(1) real_of_int_def)
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by (metis Im_exp Re_exp complex_Re_Im_cancel_iff cos_one_2pi_int sin_double Re_complex_of_real complex_Re_numeral exp_zero mult.assoc mult.left_commute mult_eq_0_iff mult_numeral_1 numeral_One of_real_0 real_of_int_def sin_zero_iff_int2)
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lemma exp_eq: "exp w = exp z \<longleftrightarrow> (\<exists>n::int. w = z + (of_int (2 * n) * pi) * ii)"
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                (is "?lhs = ?rhs")
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proof -
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  have "exp w = exp z \<longleftrightarrow> exp (w-z) = 1"
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    by (simp add: exp_diff)
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  also have "... \<longleftrightarrow> (Re w = Re z \<and> (\<exists>n::int. Im w - Im z = of_int (2 * n) * pi))"
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    by (simp add: exp_eq_1)
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  also have "... \<longleftrightarrow> ?rhs"
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    by (auto simp: algebra_simps intro!: complex_eqI)
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  finally show ?thesis .
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qed
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lemma exp_complex_eqI: "abs(Im w - Im z) < 2*pi \<Longrightarrow> exp w = exp z \<Longrightarrow> w = z"
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  by (auto simp: exp_eq abs_mult)
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lemma exp_integer_2pi:
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  assumes "n \<in> Ints"
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  shows "exp((2 * n * pi) * ii) = 1"
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proof -
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  have "exp((2 * n * pi) * ii) = exp 0"
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    using assms
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    by (simp only: Ints_def exp_eq) auto
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  also have "... = 1"
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    by simp
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  finally show ?thesis .
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qed
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lemma sin_cos_eq_iff: "sin y = sin x \<and> cos y = cos x \<longleftrightarrow> (\<exists>n::int. y = x + 2 * n * pi)"
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proof -
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  { assume "sin y = sin x" "cos y = cos x"
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    then have "cos (y-x) = 1"
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      using cos_add [of y "-x"] by simp
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    then have "\<exists>n::int. y-x = real n * 2 * pi"
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      using cos_one_2pi_int by blast }
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  then show ?thesis
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  apply (auto simp: sin_add cos_add)
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  apply (metis add.commute diff_add_cancel mult.commute)
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  done
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qed
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lemma exp_i_ne_1:
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  assumes "0 < x" "x < 2*pi"
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  shows "exp(\<i> * of_real x) \<noteq> 1"
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proof
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  assume "exp (\<i> * of_real x) = 1"
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  then have "exp (\<i> * of_real x) = exp 0"
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    by simp
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  then obtain n where "\<i> * of_real x = (of_int (2 * n) * pi) * \<i>"
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    by (simp only: Ints_def exp_eq) auto
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  then have  "of_real x = (of_int (2 * n) * pi)"
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    by (metis complex_i_not_zero mult.commute mult_cancel_left of_real_eq_iff real_scaleR_def scaleR_conv_of_real)
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  then have  "x = (of_int (2 * n) * pi)"
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    by simp
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  then show False using assms
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    by (cases n) (auto simp: zero_less_mult_iff mult_less_0_iff)
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qed
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lemma sin_eq_0:
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  fixes z::complex
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  shows "sin z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi))"
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  by (simp add: sin_exp_eq exp_eq of_real_numeral)
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lemma cos_eq_0:
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  fixes z::complex
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  shows "cos z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi) + of_real pi/2)"
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  using sin_eq_0 [of "z - of_real pi/2"]
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  by (simp add: sin_diff algebra_simps)
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lemma cos_eq_1:
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  fixes z::complex
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  shows "cos z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi))"
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proof -
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  have "cos z = cos (2*(z/2))"
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    by simp
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  also have "... = 1 - 2 * sin (z/2) ^ 2"
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    by (simp only: cos_double_sin)
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  finally have [simp]: "cos z = 1 \<longleftrightarrow> sin (z/2) = 0"
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    by simp
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  show ?thesis
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    by (auto simp: sin_eq_0 of_real_numeral)
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qed
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lemma csin_eq_1:
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  fixes z::complex
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  shows "sin z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + of_real pi/2)"
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  using cos_eq_1 [of "z - of_real pi/2"]
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  by (simp add: cos_diff algebra_simps)
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lemma csin_eq_minus1:
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  fixes z::complex
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  shows "sin z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + 3/2*pi)"
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        (is "_ = ?rhs")
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proof -
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  have "sin z = -1 \<longleftrightarrow> sin (-z) = 1"
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    by (simp add: equation_minus_iff)
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  also have "...  \<longleftrightarrow> (\<exists>n::int. -z = of_real(2 * n * pi) + of_real pi/2)"
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    by (simp only: csin_eq_1)
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  also have "...  \<longleftrightarrow> (\<exists>n::int. z = - of_real(2 * n * pi) - of_real pi/2)"
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    apply (rule iff_exI)
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    by (metis (no_types)  is_num_normalize(8) minus_minus of_real_def real_vector.scale_minus_left uminus_add_conv_diff)
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  also have "... = ?rhs"
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    apply (auto simp: of_real_numeral)
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    apply (rule_tac [2] x="-(x+1)" in exI)
lp15@59746
   301
    apply (rule_tac x="-(x+1)" in exI)
lp15@59746
   302
    apply (simp_all add: algebra_simps)
lp15@59746
   303
    done
lp15@59746
   304
  finally show ?thesis .
lp15@59862
   305
qed
lp15@59746
   306
lp15@59862
   307
lemma ccos_eq_minus1:
lp15@59746
   308
  fixes z::complex
lp15@59746
   309
  shows "cos z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + pi)"
lp15@59746
   310
  using csin_eq_1 [of "z - of_real pi/2"]
lp15@59746
   311
  apply (simp add: sin_diff)
lp15@59746
   312
  apply (simp add: algebra_simps of_real_numeral equation_minus_iff)
lp15@59862
   313
  done
lp15@59746
   314
lp15@59746
   315
lemma sin_eq_1: "sin x = 1 \<longleftrightarrow> (\<exists>n::int. x = (2 * n + 1 / 2) * pi)"
lp15@59746
   316
                (is "_ = ?rhs")
lp15@59746
   317
proof -
lp15@59746
   318
  have "sin x = 1 \<longleftrightarrow> sin (complex_of_real x) = 1"
lp15@59746
   319
    by (metis of_real_1 one_complex.simps(1) real_sin_eq sin_of_real)
lp15@59746
   320
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + of_real pi/2)"
lp15@59746
   321
    by (simp only: csin_eq_1)
lp15@59746
   322
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + of_real pi/2)"
lp15@59746
   323
    apply (rule iff_exI)
lp15@59746
   324
    apply (auto simp: algebra_simps of_real_numeral)
lp15@59746
   325
    apply (rule injD [OF inj_of_real [where 'a = complex]])
lp15@59746
   326
    apply (auto simp: of_real_numeral)
lp15@59746
   327
    done
lp15@59746
   328
  also have "... = ?rhs"
lp15@59746
   329
    by (auto simp: algebra_simps)
lp15@59746
   330
  finally show ?thesis .
lp15@59862
   331
qed
lp15@59746
   332
lp15@59746
   333
lemma sin_eq_minus1: "sin x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 3/2) * pi)"  (is "_ = ?rhs")
lp15@59746
   334
proof -
lp15@59746
   335
  have "sin x = -1 \<longleftrightarrow> sin (complex_of_real x) = -1"
lp15@59746
   336
    by (metis Re_complex_of_real of_real_def scaleR_minus1_left sin_of_real)
lp15@59746
   337
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + 3/2*pi)"
lp15@59746
   338
    by (simp only: csin_eq_minus1)
lp15@59746
   339
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + 3/2*pi)"
lp15@59746
   340
    apply (rule iff_exI)
lp15@59746
   341
    apply (auto simp: algebra_simps)
lp15@59746
   342
    apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
lp15@59746
   343
    done
lp15@59746
   344
  also have "... = ?rhs"
lp15@59746
   345
    by (auto simp: algebra_simps)
lp15@59746
   346
  finally show ?thesis .
lp15@59862
   347
qed
lp15@59746
   348
lp15@59746
   349
lemma cos_eq_minus1: "cos x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 1) * pi)"
lp15@59746
   350
                      (is "_ = ?rhs")
lp15@59746
   351
proof -
lp15@59746
   352
  have "cos x = -1 \<longleftrightarrow> cos (complex_of_real x) = -1"
lp15@59746
   353
    by (metis Re_complex_of_real of_real_def scaleR_minus1_left cos_of_real)
lp15@59746
   354
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + pi)"
lp15@59746
   355
    by (simp only: ccos_eq_minus1)
lp15@59746
   356
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + pi)"
lp15@59746
   357
    apply (rule iff_exI)
lp15@59746
   358
    apply (auto simp: algebra_simps)
lp15@59746
   359
    apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
lp15@59746
   360
    done
lp15@59746
   361
  also have "... = ?rhs"
lp15@59746
   362
    by (auto simp: algebra_simps)
lp15@59746
   363
  finally show ?thesis .
lp15@59862
   364
qed
lp15@59746
   365
lp15@59746
   366
lemma dist_exp_ii_1: "norm(exp(ii * of_real t) - 1) = 2 * abs(sin(t / 2))"
lp15@59862
   367
  apply (simp add: exp_Euler cmod_def power2_diff sin_of_real cos_of_real algebra_simps)
lp15@59746
   368
  using cos_double_sin [of "t/2"]
lp15@59746
   369
  apply (simp add: real_sqrt_mult)
lp15@59746
   370
  done
lp15@59746
   371
lp15@59746
   372
lemma sinh_complex:
lp15@59746
   373
  fixes z :: complex
lp15@59746
   374
  shows "(exp z - inverse (exp z)) / 2 = -ii * sin(ii * z)"
lp15@59746
   375
  by (simp add: sin_exp_eq divide_simps exp_minus of_real_numeral)
lp15@59746
   376
lp15@59746
   377
lemma sin_ii_times:
lp15@59746
   378
  fixes z :: complex
lp15@59746
   379
  shows "sin(ii * z) = ii * ((exp z - inverse (exp z)) / 2)"
lp15@59746
   380
  using sinh_complex by auto
lp15@59746
   381
lp15@59746
   382
lemma sinh_real:
lp15@59746
   383
  fixes x :: real
lp15@59746
   384
  shows "of_real((exp x - inverse (exp x)) / 2) = -ii * sin(ii * of_real x)"
lp15@59746
   385
  by (simp add: exp_of_real sin_ii_times of_real_numeral)
lp15@59746
   386
lp15@59746
   387
lemma cosh_complex:
lp15@59746
   388
  fixes z :: complex
lp15@59746
   389
  shows "(exp z + inverse (exp z)) / 2 = cos(ii * z)"
lp15@59746
   390
  by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lp15@59746
   391
lp15@59746
   392
lemma cosh_real:
lp15@59746
   393
  fixes x :: real
lp15@59746
   394
  shows "of_real((exp x + inverse (exp x)) / 2) = cos(ii * of_real x)"
lp15@59746
   395
  by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lp15@59746
   396
lp15@59746
   397
lemmas cos_ii_times = cosh_complex [symmetric]
lp15@59746
   398
lp15@59862
   399
lemma norm_cos_squared:
lp15@59746
   400
    "norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
lp15@59746
   401
  apply (cases z)
lp15@59746
   402
  apply (simp add: cos_add cmod_power2 cos_of_real sin_of_real)
lp15@59746
   403
  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
lp15@59746
   404
  apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
lp15@59746
   405
  apply (simp add: sin_squared_eq)
lp15@59746
   406
  apply (simp add: power2_eq_square algebra_simps divide_simps)
lp15@59746
   407
  done
lp15@59746
   408
lp15@59746
   409
lemma norm_sin_squared:
lp15@59746
   410
    "norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
lp15@59746
   411
  apply (cases z)
lp15@59746
   412
  apply (simp add: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double)
lp15@59746
   413
  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
lp15@59746
   414
  apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
lp15@59746
   415
  apply (simp add: cos_squared_eq)
lp15@59746
   416
  apply (simp add: power2_eq_square algebra_simps divide_simps)
lp15@59862
   417
  done
lp15@59746
   418
lp15@59746
   419
lemma exp_uminus_Im: "exp (- Im z) \<le> exp (cmod z)"
lp15@59746
   420
  using abs_Im_le_cmod linear order_trans by fastforce
lp15@59746
   421
lp15@59862
   422
lemma norm_cos_le:
lp15@59746
   423
  fixes z::complex
lp15@59746
   424
  shows "norm(cos z) \<le> exp(norm z)"
lp15@59746
   425
proof -
lp15@59746
   426
  have "Im z \<le> cmod z"
lp15@59746
   427
    using abs_Im_le_cmod abs_le_D1 by auto
lp15@59746
   428
  with exp_uminus_Im show ?thesis
lp15@59746
   429
    apply (simp add: cos_exp_eq norm_divide)
lp15@59746
   430
    apply (rule order_trans [OF norm_triangle_ineq], simp)
lp15@59746
   431
    apply (metis add_mono exp_le_cancel_iff mult_2_right)
lp15@59746
   432
    done
lp15@59746
   433
qed
lp15@59746
   434
lp15@59862
   435
lemma norm_cos_plus1_le:
lp15@59746
   436
  fixes z::complex
lp15@59746
   437
  shows "norm(1 + cos z) \<le> 2 * exp(norm z)"
lp15@59746
   438
proof -
lp15@59746
   439
  have mono: "\<And>u w z::real. (1 \<le> w | 1 \<le> z) \<Longrightarrow> (w \<le> u & z \<le> u) \<Longrightarrow> 2 + w + z \<le> 4 * u"
lp15@59746
   440
      by arith
lp15@59746
   441
  have *: "Im z \<le> cmod z"
lp15@59746
   442
    using abs_Im_le_cmod abs_le_D1 by auto
lp15@59746
   443
  have triangle3: "\<And>x y z. norm(x + y + z) \<le> norm(x) + norm(y) + norm(z)"
lp15@59746
   444
    by (simp add: norm_add_rule_thm)
lp15@59746
   445
  have "norm(1 + cos z) = cmod (1 + (exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
lp15@59746
   446
    by (simp add: cos_exp_eq)
lp15@59746
   447
  also have "... = cmod ((2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
lp15@59746
   448
    by (simp add: field_simps)
lp15@59746
   449
  also have "... = cmod (2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2"
lp15@59746
   450
    by (simp add: norm_divide)
lp15@59746
   451
  finally show ?thesis
lp15@59746
   452
    apply (rule ssubst, simp)
lp15@59746
   453
    apply (rule order_trans [OF triangle3], simp)
lp15@59746
   454
    using exp_uminus_Im *
lp15@59746
   455
    apply (auto intro: mono)
lp15@59746
   456
    done
lp15@59746
   457
qed
lp15@59746
   458
lp15@59746
   459
subsection{* Taylor series for complex exponential, sine and cosine.*}
lp15@59746
   460
lp15@59862
   461
context
lp15@59746
   462
begin
lp15@59746
   463
lp15@59746
   464
declare power_Suc [simp del]
lp15@59746
   465
lp15@59862
   466
lemma Taylor_exp:
lp15@59746
   467
  "norm(exp z - (\<Sum>k\<le>n. z ^ k / (fact k))) \<le> exp\<bar>Re z\<bar> * (norm z) ^ (Suc n) / (fact n)"
lp15@59746
   468
proof (rule complex_taylor [of _ n "\<lambda>k. exp" "exp\<bar>Re z\<bar>" 0 z, simplified])
lp15@59746
   469
  show "convex (closed_segment 0 z)"
lp15@59746
   470
    by (rule convex_segment [of 0 z])
lp15@59746
   471
next
lp15@59746
   472
  fix k x
lp15@59746
   473
  assume "x \<in> closed_segment 0 z" "k \<le> n"
lp15@59746
   474
  show "(exp has_field_derivative exp x) (at x within closed_segment 0 z)"
lp15@59746
   475
    using DERIV_exp DERIV_subset by blast
lp15@59746
   476
next
lp15@59746
   477
  fix x
lp15@59746
   478
  assume "x \<in> closed_segment 0 z"
lp15@59746
   479
  then show "Re x \<le> \<bar>Re z\<bar>"
lp15@59746
   480
    apply (auto simp: closed_segment_def scaleR_conv_of_real)
lp15@59746
   481
    by (meson abs_ge_self abs_ge_zero linear mult_left_le_one_le mult_nonneg_nonpos order_trans)
lp15@59746
   482
next
lp15@59746
   483
  show "0 \<in> closed_segment 0 z"
lp15@59746
   484
    by (auto simp: closed_segment_def)
lp15@59746
   485
next
lp15@59746
   486
  show "z \<in> closed_segment 0 z"
lp15@59746
   487
    apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   488
    using of_real_1 zero_le_one by blast
lp15@59862
   489
qed
lp15@59746
   490
lp15@59862
   491
lemma
lp15@59746
   492
  assumes "0 \<le> u" "u \<le> 1"
lp15@59862
   493
  shows cmod_sin_le_exp: "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   494
    and cmod_cos_le_exp: "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   495
proof -
lp15@59746
   496
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   497
    by arith
lp15@59746
   498
  show "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
lp15@59746
   499
    apply (auto simp: scaleR_conv_of_real norm_mult norm_power sin_exp_eq norm_divide)
lp15@59746
   500
    apply (rule order_trans [OF norm_triangle_ineq4])
lp15@59746
   501
    apply (rule mono)
lp15@59746
   502
    apply (auto simp: abs_if mult_left_le_one_le)
lp15@59746
   503
    apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
lp15@59746
   504
    apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
lp15@59746
   505
    done
lp15@59746
   506
  show "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
lp15@59746
   507
    apply (auto simp: scaleR_conv_of_real norm_mult norm_power cos_exp_eq norm_divide)
lp15@59746
   508
    apply (rule order_trans [OF norm_triangle_ineq])
lp15@59746
   509
    apply (rule mono)
lp15@59746
   510
    apply (auto simp: abs_if mult_left_le_one_le)
lp15@59746
   511
    apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
lp15@59746
   512
    apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
lp15@59746
   513
    done
lp15@59746
   514
qed
lp15@59862
   515
lp15@59862
   516
lemma Taylor_sin:
lp15@59862
   517
  "norm(sin z - (\<Sum>k\<le>n. complex_of_real (sin_coeff k) * z ^ k))
lp15@59746
   518
   \<le> exp\<bar>Im z\<bar> * (norm z) ^ (Suc n) / (fact n)"
lp15@59746
   519
proof -
lp15@59746
   520
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   521
      by arith
lp15@59746
   522
  have *: "cmod (sin z -
lp15@59746
   523
                 (\<Sum>i\<le>n. (-1) ^ (i div 2) * (if even i then sin 0 else cos 0) * z ^ i / (fact i)))
lp15@59862
   524
           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
lp15@59746
   525
  proof (rule complex_taylor [of "closed_segment 0 z" n "\<lambda>k x. (-1)^(k div 2) * (if even k then sin x else cos x)" "exp\<bar>Im z\<bar>" 0 z,
lp15@59746
   526
simplified])
lp15@59746
   527
  show "convex (closed_segment 0 z)"
lp15@59746
   528
    by (rule convex_segment [of 0 z])
lp15@59746
   529
  next
lp15@59746
   530
    fix k x
lp15@59746
   531
    show "((\<lambda>x. (- 1) ^ (k div 2) * (if even k then sin x else cos x)) has_field_derivative
lp15@59746
   532
            (- 1) ^ (Suc k div 2) * (if odd k then sin x else cos x))
lp15@59746
   533
            (at x within closed_segment 0 z)"
lp15@59746
   534
      apply (auto simp: power_Suc)
lp15@59746
   535
      apply (intro derivative_eq_intros | simp)+
lp15@59746
   536
      done
lp15@59746
   537
  next
lp15@59746
   538
    fix x
lp15@59746
   539
    assume "x \<in> closed_segment 0 z"
lp15@59746
   540
    then show "cmod ((- 1) ^ (Suc n div 2) * (if odd n then sin x else cos x)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   541
      by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
lp15@59746
   542
  next
lp15@59746
   543
    show "0 \<in> closed_segment 0 z"
lp15@59746
   544
      by (auto simp: closed_segment_def)
lp15@59746
   545
  next
lp15@59746
   546
    show "z \<in> closed_segment 0 z"
lp15@59746
   547
      apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   548
      using of_real_1 zero_le_one by blast
lp15@59862
   549
  qed
lp15@59746
   550
  have **: "\<And>k. complex_of_real (sin_coeff k) * z ^ k
lp15@59746
   551
            = (-1)^(k div 2) * (if even k then sin 0 else cos 0) * z^k / of_nat (fact k)"
lp15@59746
   552
    by (auto simp: sin_coeff_def elim!: oddE)
lp15@59746
   553
  show ?thesis
lp15@59746
   554
    apply (rule order_trans [OF _ *])
lp15@59746
   555
    apply (simp add: **)
lp15@59746
   556
    done
lp15@59746
   557
qed
lp15@59746
   558
lp15@59862
   559
lemma Taylor_cos:
lp15@59862
   560
  "norm(cos z - (\<Sum>k\<le>n. complex_of_real (cos_coeff k) * z ^ k))
lp15@59746
   561
   \<le> exp\<bar>Im z\<bar> * (norm z) ^ Suc n / (fact n)"
lp15@59746
   562
proof -
lp15@59746
   563
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   564
      by arith
lp15@59746
   565
  have *: "cmod (cos z -
lp15@59746
   566
                 (\<Sum>i\<le>n. (-1) ^ (Suc i div 2) * (if even i then cos 0 else sin 0) * z ^ i / (fact i)))
lp15@59862
   567
           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
lp15@59746
   568
  proof (rule complex_taylor [of "closed_segment 0 z" n "\<lambda>k x. (-1)^(Suc k div 2) * (if even k then cos x else sin x)" "exp\<bar>Im z\<bar>" 0 z,
lp15@59746
   569
simplified])
lp15@59746
   570
  show "convex (closed_segment 0 z)"
lp15@59746
   571
    by (rule convex_segment [of 0 z])
lp15@59746
   572
  next
lp15@59746
   573
    fix k x
lp15@59746
   574
    assume "x \<in> closed_segment 0 z" "k \<le> n"
lp15@59746
   575
    show "((\<lambda>x. (- 1) ^ (Suc k div 2) * (if even k then cos x else sin x)) has_field_derivative
lp15@59746
   576
            (- 1) ^ Suc (k div 2) * (if odd k then cos x else sin x))
lp15@59746
   577
             (at x within closed_segment 0 z)"
lp15@59746
   578
      apply (auto simp: power_Suc)
lp15@59746
   579
      apply (intro derivative_eq_intros | simp)+
lp15@59746
   580
      done
lp15@59746
   581
  next
lp15@59746
   582
    fix x
lp15@59746
   583
    assume "x \<in> closed_segment 0 z"
lp15@59746
   584
    then show "cmod ((- 1) ^ Suc (n div 2) * (if odd n then cos x else sin x)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   585
      by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
lp15@59746
   586
  next
lp15@59746
   587
    show "0 \<in> closed_segment 0 z"
lp15@59746
   588
      by (auto simp: closed_segment_def)
lp15@59746
   589
  next
lp15@59746
   590
    show "z \<in> closed_segment 0 z"
lp15@59746
   591
      apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   592
      using of_real_1 zero_le_one by blast
lp15@59862
   593
  qed
lp15@59746
   594
  have **: "\<And>k. complex_of_real (cos_coeff k) * z ^ k
lp15@59746
   595
            = (-1)^(Suc k div 2) * (if even k then cos 0 else sin 0) * z^k / of_nat (fact k)"
lp15@59746
   596
    by (auto simp: cos_coeff_def elim!: evenE)
lp15@59746
   597
  show ?thesis
lp15@59746
   598
    apply (rule order_trans [OF _ *])
lp15@59746
   599
    apply (simp add: **)
lp15@59746
   600
    done
lp15@59746
   601
qed
lp15@59746
   602
lp15@59746
   603
end (* of context *)
lp15@59746
   604
lp15@59751
   605
text{*32-bit Approximation to e*}
lp15@59751
   606
lemma e_approx_32: "abs(exp(1) - 5837465777 / 2147483648) \<le> (inverse(2 ^ 32)::real)"
lp15@59751
   607
  using Taylor_exp [of 1 14] exp_le
lp15@59751
   608
  apply (simp add: setsum_left_distrib in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
lp15@59751
   609
  apply (simp only: pos_le_divide_eq [symmetric], linarith)
lp15@59751
   610
  done
lp15@59751
   611
lp15@60017
   612
lemma e_less_3: "exp 1 < (3::real)"
lp15@60017
   613
  using e_approx_32
lp15@60017
   614
  by (simp add: abs_if split: split_if_asm)
lp15@60017
   615
lp15@60017
   616
lemma ln3_gt_1: "ln 3 > (1::real)"
lp15@60017
   617
  by (metis e_less_3 exp_less_cancel_iff exp_ln_iff less_trans ln_exp)
lp15@60017
   618
lp15@60017
   619
lp15@59746
   620
subsection{*The argument of a complex number*}
lp15@59746
   621
lp15@59746
   622
definition Arg :: "complex \<Rightarrow> real" where
lp15@59746
   623
 "Arg z \<equiv> if z = 0 then 0
lp15@59746
   624
           else THE t. 0 \<le> t \<and> t < 2*pi \<and>
lp15@59746
   625
                    z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   626
lp15@59746
   627
lemma Arg_0 [simp]: "Arg(0) = 0"
lp15@59746
   628
  by (simp add: Arg_def)
lp15@59746
   629
lp15@59746
   630
lemma Arg_unique_lemma:
lp15@59746
   631
  assumes z:  "z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   632
      and z': "z = of_real(norm z) * exp(ii * of_real t')"
lp15@59746
   633
      and t:  "0 \<le> t"  "t < 2*pi"
lp15@59746
   634
      and t': "0 \<le> t'" "t' < 2*pi"
lp15@59746
   635
      and nz: "z \<noteq> 0"
lp15@59746
   636
  shows "t' = t"
lp15@59746
   637
proof -
lp15@59746
   638
  have [dest]: "\<And>x y z::real. x\<ge>0 \<Longrightarrow> x+y < z \<Longrightarrow> y<z"
lp15@59746
   639
    by arith
lp15@59746
   640
  have "of_real (cmod z) * exp (\<i> * of_real t') = of_real (cmod z) * exp (\<i> * of_real t)"
lp15@59746
   641
    by (metis z z')
lp15@59746
   642
  then have "exp (\<i> * of_real t') = exp (\<i> * of_real t)"
lp15@59746
   643
    by (metis nz mult_left_cancel mult_zero_left z)
lp15@59746
   644
  then have "sin t' = sin t \<and> cos t' = cos t"
lp15@59746
   645
    apply (simp add: exp_Euler sin_of_real cos_of_real)
lp15@59746
   646
    by (metis Complex_eq complex.sel)
lp15@59746
   647
  then obtain n::int where n: "t' = t + 2 * real n * pi"
lp15@59746
   648
    by (auto simp: sin_cos_eq_iff)
lp15@59746
   649
  then have "n=0"
lp15@59746
   650
    apply (rule_tac z=n in int_cases)
lp15@59746
   651
    using t t'
lp15@59746
   652
    apply (auto simp: mult_less_0_iff algebra_simps)
lp15@59746
   653
    done
lp15@59746
   654
  then show "t' = t"
lp15@59746
   655
      by (simp add: n)
lp15@59746
   656
qed
lp15@59746
   657
lp15@59746
   658
lemma Arg: "0 \<le> Arg z & Arg z < 2*pi & z = of_real(norm z) * exp(ii * of_real(Arg z))"
lp15@59746
   659
proof (cases "z=0")
lp15@59746
   660
  case True then show ?thesis
lp15@59746
   661
    by (simp add: Arg_def)
lp15@59746
   662
next
lp15@59746
   663
  case False
lp15@59746
   664
  obtain t where t: "0 \<le> t" "t < 2*pi"
lp15@59746
   665
             and ReIm: "Re z / cmod z = cos t" "Im z / cmod z = sin t"
lp15@59746
   666
    using sincos_total_2pi [OF complex_unit_circle [OF False]]
lp15@59746
   667
    by blast
lp15@59746
   668
  have z: "z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   669
    apply (rule complex_eqI)
lp15@59746
   670
    using t False ReIm
lp15@59746
   671
    apply (auto simp: exp_Euler sin_of_real cos_of_real divide_simps)
lp15@59746
   672
    done
lp15@59746
   673
  show ?thesis
lp15@59746
   674
    apply (simp add: Arg_def False)
lp15@59746
   675
    apply (rule theI [where a=t])
lp15@59746
   676
    using t z False
lp15@59746
   677
    apply (auto intro: Arg_unique_lemma)
lp15@59746
   678
    done
lp15@59746
   679
qed
lp15@59746
   680
lp15@59746
   681
lp15@59746
   682
corollary
lp15@59746
   683
  shows Arg_ge_0: "0 \<le> Arg z"
lp15@59746
   684
    and Arg_lt_2pi: "Arg z < 2*pi"
lp15@59746
   685
    and Arg_eq: "z = of_real(norm z) * exp(ii * of_real(Arg z))"
lp15@59746
   686
  using Arg by auto
lp15@59746
   687
lp15@59746
   688
lemma complex_norm_eq_1_exp: "norm z = 1 \<longleftrightarrow> (\<exists>t. z = exp(ii * of_real t))"
lp15@59746
   689
  using Arg [of z] by auto
lp15@59746
   690
lp15@59746
   691
lemma Arg_unique: "\<lbrakk>of_real r * exp(ii * of_real a) = z; 0 < r; 0 \<le> a; a < 2*pi\<rbrakk> \<Longrightarrow> Arg z = a"
lp15@59746
   692
  apply (rule Arg_unique_lemma [OF _ Arg_eq])
lp15@59746
   693
  using Arg [of z]
lp15@59746
   694
  apply (auto simp: norm_mult)
lp15@59746
   695
  done
lp15@59746
   696
lp15@59746
   697
lemma Arg_minus: "z \<noteq> 0 \<Longrightarrow> Arg (-z) = (if Arg z < pi then Arg z + pi else Arg z - pi)"
lp15@59746
   698
  apply (rule Arg_unique [of "norm z"])
lp15@59746
   699
  apply (rule complex_eqI)
lp15@59746
   700
  using Arg_ge_0 [of z] Arg_eq [of z] Arg_lt_2pi [of z] Arg_eq [of z]
lp15@59746
   701
  apply auto
lp15@59746
   702
  apply (auto simp: Re_exp Im_exp cos_diff sin_diff cis_conv_exp [symmetric])
lp15@59746
   703
  apply (metis Re_rcis Im_rcis rcis_def)+
lp15@59746
   704
  done
lp15@59746
   705
lp15@59746
   706
lemma Arg_times_of_real [simp]: "0 < r \<Longrightarrow> Arg (of_real r * z) = Arg z"
lp15@59746
   707
  apply (cases "z=0", simp)
lp15@59746
   708
  apply (rule Arg_unique [of "r * norm z"])
lp15@59746
   709
  using Arg
lp15@59746
   710
  apply auto
lp15@59746
   711
  done
lp15@59746
   712
lp15@59746
   713
lemma Arg_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg (z * of_real r) = Arg z"
lp15@59746
   714
  by (metis Arg_times_of_real mult.commute)
lp15@59746
   715
lp15@59746
   716
lemma Arg_divide_of_real [simp]: "0 < r \<Longrightarrow> Arg (z / of_real r) = Arg z"
lp15@59746
   717
  by (metis Arg_times_of_real2 less_numeral_extra(3) nonzero_eq_divide_eq of_real_eq_0_iff)
lp15@59746
   718
lp15@59746
   719
lemma Arg_le_pi: "Arg z \<le> pi \<longleftrightarrow> 0 \<le> Im z"
lp15@59746
   720
proof (cases "z=0")
lp15@59746
   721
  case True then show ?thesis
lp15@59746
   722
    by simp
lp15@59746
   723
next
lp15@59746
   724
  case False
lp15@59746
   725
  have "0 \<le> Im z \<longleftrightarrow> 0 \<le> Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   726
    by (metis Arg_eq)
lp15@59746
   727
  also have "... = (0 \<le> Im (exp (\<i> * complex_of_real (Arg z))))"
lp15@59746
   728
    using False
lp15@59746
   729
    by (simp add: zero_le_mult_iff)
lp15@59746
   730
  also have "... \<longleftrightarrow> Arg z \<le> pi"
lp15@59746
   731
    by (simp add: Im_exp) (metis Arg_ge_0 Arg_lt_2pi sin_lt_zero sin_ge_zero not_le)
lp15@59746
   732
  finally show ?thesis
lp15@59746
   733
    by blast
lp15@59746
   734
qed
lp15@59746
   735
lp15@59746
   736
lemma Arg_lt_pi: "0 < Arg z \<and> Arg z < pi \<longleftrightarrow> 0 < Im z"
lp15@59746
   737
proof (cases "z=0")
lp15@59746
   738
  case True then show ?thesis
lp15@59746
   739
    by simp
lp15@59746
   740
next
lp15@59746
   741
  case False
lp15@59746
   742
  have "0 < Im z \<longleftrightarrow> 0 < Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   743
    by (metis Arg_eq)
lp15@59746
   744
  also have "... = (0 < Im (exp (\<i> * complex_of_real (Arg z))))"
lp15@59746
   745
    using False
lp15@59746
   746
    by (simp add: zero_less_mult_iff)
lp15@59746
   747
  also have "... \<longleftrightarrow> 0 < Arg z \<and> Arg z < pi"
lp15@59746
   748
    using Arg_ge_0  Arg_lt_2pi sin_le_zero sin_gt_zero
lp15@59746
   749
    apply (auto simp: Im_exp)
lp15@59746
   750
    using le_less apply fastforce
lp15@59746
   751
    using not_le by blast
lp15@59746
   752
  finally show ?thesis
lp15@59746
   753
    by blast
lp15@59746
   754
qed
lp15@59746
   755
lp15@59746
   756
lemma Arg_eq_0: "Arg z = 0 \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re z"
lp15@59746
   757
proof (cases "z=0")
lp15@59746
   758
  case True then show ?thesis
lp15@59746
   759
    by simp
lp15@59746
   760
next
lp15@59746
   761
  case False
lp15@59746
   762
  have "z \<in> Reals \<and> 0 \<le> Re z \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   763
    by (metis Arg_eq)
lp15@59746
   764
  also have "... \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re (exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   765
    using False
lp15@59746
   766
    by (simp add: zero_le_mult_iff)
lp15@59746
   767
  also have "... \<longleftrightarrow> Arg z = 0"
lp15@59746
   768
    apply (auto simp: Re_exp)
lp15@59746
   769
    apply (metis Arg_lt_pi Arg_ge_0 Arg_le_pi cos_pi complex_is_Real_iff leD less_linear less_minus_one_simps(2) minus_minus neg_less_eq_nonneg order_refl)
lp15@59746
   770
    using Arg_eq [of z]
lp15@59746
   771
    apply (auto simp: Reals_def)
lp15@59746
   772
    done
lp15@59746
   773
  finally show ?thesis
lp15@59746
   774
    by blast
lp15@59746
   775
qed
lp15@59746
   776
lp15@60150
   777
corollary Arg_gt_0: 
lp15@60150
   778
  assumes "z \<in> \<real> \<Longrightarrow> Re z < 0"
lp15@60150
   779
    shows "Arg z > 0"
lp15@60150
   780
  using Arg_eq_0 Arg_ge_0 assms dual_order.strict_iff_order by fastforce
lp15@60150
   781
lp15@59746
   782
lemma Arg_of_real: "Arg(of_real x) = 0 \<longleftrightarrow> 0 \<le> x"
lp15@59746
   783
  by (simp add: Arg_eq_0)
lp15@59746
   784
lp15@59746
   785
lemma Arg_eq_pi: "Arg z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
lp15@59746
   786
  apply  (cases "z=0", simp)
lp15@59746
   787
  using Arg_eq_0 [of "-z"]
lp15@59746
   788
  apply (auto simp: complex_is_Real_iff Arg_minus)
lp15@59746
   789
  apply (simp add: complex_Re_Im_cancel_iff)
lp15@59746
   790
  apply (metis Arg_minus pi_gt_zero add.left_neutral minus_minus minus_zero)
lp15@59746
   791
  done
lp15@59746
   792
lp15@59746
   793
lemma Arg_eq_0_pi: "Arg z = 0 \<or> Arg z = pi \<longleftrightarrow> z \<in> \<real>"
lp15@59746
   794
  using Arg_eq_0 Arg_eq_pi not_le by auto
lp15@59746
   795
lp15@59746
   796
lemma Arg_inverse: "Arg(inverse z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
lp15@59746
   797
  apply (cases "z=0", simp)
lp15@59746
   798
  apply (rule Arg_unique [of "inverse (norm z)"])
lp15@59746
   799
  using Arg_ge_0 [of z] Arg_lt_2pi [of z] Arg_eq [of z] Arg_eq_0 [of z] Exp_two_pi_i
lp15@59746
   800
  apply (auto simp: of_real_numeral algebra_simps exp_diff divide_simps)
lp15@59746
   801
  done
lp15@59746
   802
lp15@59746
   803
lemma Arg_eq_iff:
lp15@59746
   804
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   805
     shows "Arg w = Arg z \<longleftrightarrow> (\<exists>x. 0 < x & w = of_real x * z)"
lp15@59746
   806
  using assms Arg_eq [of z] Arg_eq [of w]
lp15@59746
   807
  apply auto
lp15@59746
   808
  apply (rule_tac x="norm w / norm z" in exI)
lp15@59746
   809
  apply (simp add: divide_simps)
lp15@59746
   810
  by (metis mult.commute mult.left_commute)
lp15@59746
   811
lp15@59746
   812
lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
lp15@59746
   813
  using complex_is_Real_iff
lp15@59746
   814
  apply (simp add: Arg_eq_0)
lp15@59746
   815
  apply (auto simp: divide_simps not_sum_power2_lt_zero)
lp15@59746
   816
  done
lp15@59746
   817
lp15@59746
   818
lemma Arg_divide:
lp15@59746
   819
  assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
lp15@59746
   820
    shows "Arg(z / w) = Arg z - Arg w"
lp15@59746
   821
  apply (rule Arg_unique [of "norm(z / w)"])
lp15@59746
   822
  using assms Arg_eq [of z] Arg_eq [of w] Arg_ge_0 [of w] Arg_lt_2pi [of z]
lp15@59746
   823
  apply (auto simp: exp_diff norm_divide algebra_simps divide_simps)
lp15@59746
   824
  done
lp15@59746
   825
lp15@59746
   826
lemma Arg_le_div_sum:
lp15@59746
   827
  assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
lp15@59746
   828
    shows "Arg z = Arg w + Arg(z / w)"
lp15@59746
   829
  by (simp add: Arg_divide assms)
lp15@59746
   830
lp15@59746
   831
lemma Arg_le_div_sum_eq:
lp15@59746
   832
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   833
    shows "Arg w \<le> Arg z \<longleftrightarrow> Arg z = Arg w + Arg(z / w)"
lp15@59746
   834
  using assms
lp15@59746
   835
  by (auto simp: Arg_ge_0 intro: Arg_le_div_sum)
lp15@59746
   836
lp15@59746
   837
lemma Arg_diff:
lp15@59746
   838
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   839
    shows "Arg w - Arg z = (if Arg z \<le> Arg w then Arg(w / z) else Arg(w/z) - 2*pi)"
lp15@59746
   840
  using assms
lp15@59746
   841
  apply (auto simp: Arg_ge_0 Arg_divide not_le)
lp15@59746
   842
  using Arg_divide [of w z] Arg_inverse [of "w/z"]
lp15@59746
   843
  apply auto
lp15@59746
   844
  by (metis Arg_eq_0 less_irrefl minus_diff_eq right_minus_eq)
lp15@59746
   845
lp15@59746
   846
lemma Arg_add:
lp15@59746
   847
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   848
    shows "Arg w + Arg z = (if Arg w + Arg z < 2*pi then Arg(w * z) else Arg(w * z) + 2*pi)"
lp15@59746
   849
  using assms
lp15@59746
   850
  using Arg_diff [of "w*z" z] Arg_le_div_sum_eq [of z "w*z"]
lp15@59746
   851
  apply (auto simp: Arg_ge_0 Arg_divide not_le)
lp15@59746
   852
  apply (metis Arg_lt_2pi add.commute)
lp15@59746
   853
  apply (metis (no_types) Arg add.commute diff_0 diff_add_cancel diff_less_eq diff_minus_eq_add not_less)
lp15@59746
   854
  done
lp15@59746
   855
lp15@59746
   856
lemma Arg_times:
lp15@59746
   857
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   858
    shows "Arg (w * z) = (if Arg w + Arg z < 2*pi then Arg w + Arg z
lp15@59746
   859
                            else (Arg w + Arg z) - 2*pi)"
lp15@59746
   860
  using Arg_add [OF assms]
lp15@59746
   861
  by auto
lp15@59746
   862
lp15@59746
   863
lemma Arg_cnj: "Arg(cnj z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
lp15@59746
   864
  apply (cases "z=0", simp)
lp15@59746
   865
  apply (rule trans [of _ "Arg(inverse z)"])
lp15@59746
   866
  apply (simp add: Arg_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
lp15@59746
   867
  apply (metis norm_eq_zero of_real_power zero_less_power2)
lp15@59746
   868
  apply (auto simp: of_real_numeral Arg_inverse)
lp15@59746
   869
  done
lp15@59746
   870
lp15@59746
   871
lemma Arg_real: "z \<in> \<real> \<Longrightarrow> Arg z = (if 0 \<le> Re z then 0 else pi)"
lp15@59746
   872
  using Arg_eq_0 Arg_eq_0_pi
lp15@59746
   873
  by auto
lp15@59746
   874
lp15@59746
   875
lemma Arg_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg(exp z) = Im z"
lp15@59746
   876
  by (rule Arg_unique [of  "exp(Re z)"]) (auto simp: Exp_eq_polar)
lp15@59746
   877
lp15@59751
   878
lp15@59751
   879
subsection{*Analytic properties of tangent function*}
lp15@59751
   880
lp15@59751
   881
lemma cnj_tan: "cnj(tan z) = tan(cnj z)"
lp15@59751
   882
  by (simp add: cnj_cos cnj_sin tan_def)
lp15@59751
   883
lp15@59751
   884
lemma complex_differentiable_at_tan: "~(cos z = 0) \<Longrightarrow> tan complex_differentiable at z"
lp15@59751
   885
  unfolding complex_differentiable_def
lp15@59751
   886
  using DERIV_tan by blast
lp15@59751
   887
lp15@59751
   888
lemma complex_differentiable_within_tan: "~(cos z = 0)
lp15@59751
   889
         \<Longrightarrow> tan complex_differentiable (at z within s)"
lp15@59751
   890
  using complex_differentiable_at_tan complex_differentiable_at_within by blast
lp15@59751
   891
lp15@59751
   892
lemma continuous_within_tan: "~(cos z = 0) \<Longrightarrow> continuous (at z within s) tan"
lp15@59751
   893
  using continuous_at_imp_continuous_within isCont_tan by blast
lp15@59751
   894
lp15@59751
   895
lemma continuous_on_tan [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> continuous_on s tan"
lp15@59751
   896
  by (simp add: continuous_at_imp_continuous_on)
lp15@59751
   897
lp15@59751
   898
lemma holomorphic_on_tan: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> tan holomorphic_on s"
lp15@59751
   899
  by (simp add: complex_differentiable_within_tan holomorphic_on_def)
lp15@59751
   900
lp15@59751
   901
lp15@59751
   902
subsection{*Complex logarithms (the conventional principal value)*}
lp15@59751
   903
lp15@60020
   904
instantiation complex :: ln
lp15@60020
   905
begin
lp15@60017
   906
lp15@60020
   907
definition ln_complex :: "complex \<Rightarrow> complex"
lp15@60020
   908
  where "ln_complex \<equiv> \<lambda>z. THE w. exp w = z & -pi < Im(w) & Im(w) \<le> pi"
lp15@59751
   909
lp15@59751
   910
lemma
lp15@59751
   911
  assumes "z \<noteq> 0"
lp15@60020
   912
    shows exp_Ln [simp]:  "exp(ln z) = z"
lp15@60020
   913
      and mpi_less_Im_Ln: "-pi < Im(ln z)"
lp15@60020
   914
      and Im_Ln_le_pi:    "Im(ln z) \<le> pi"
lp15@59751
   915
proof -
lp15@59751
   916
  obtain \<psi> where z: "z / (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
lp15@59751
   917
    using complex_unimodular_polar [of "z / (norm z)"] assms
lp15@59751
   918
    by (auto simp: norm_divide divide_simps)
lp15@59751
   919
  obtain \<phi> where \<phi>: "- pi < \<phi>" "\<phi> \<le> pi" "sin \<phi> = sin \<psi>" "cos \<phi> = cos \<psi>"
lp15@59751
   920
    using sincos_principal_value [of "\<psi>"] assms
lp15@59751
   921
    by (auto simp: norm_divide divide_simps)
lp15@60020
   922
  have "exp(ln z) = z & -pi < Im(ln z) & Im(ln z) \<le> pi" unfolding ln_complex_def
lp15@59751
   923
    apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
lp15@59751
   924
    using z assms \<phi>
lp15@59751
   925
    apply (auto simp: field_simps exp_complex_eqI Exp_eq_polar cis.code)
lp15@59751
   926
    done
lp15@60020
   927
  then show "exp(ln z) = z" "-pi < Im(ln z)" "Im(ln z) \<le> pi"
lp15@59751
   928
    by auto
lp15@59751
   929
qed
lp15@59751
   930
lp15@59751
   931
lemma Ln_exp [simp]:
lp15@59751
   932
  assumes "-pi < Im(z)" "Im(z) \<le> pi"
lp15@60020
   933
    shows "ln(exp z) = z"
lp15@59751
   934
  apply (rule exp_complex_eqI)
lp15@59751
   935
  using assms mpi_less_Im_Ln  [of "exp z"] Im_Ln_le_pi [of "exp z"]
lp15@59751
   936
  apply auto
lp15@59751
   937
  done
lp15@59751
   938
lp15@60020
   939
subsection{*Relation to Real Logarithm*}
lp15@60020
   940
lp15@60020
   941
lemma Ln_of_real:
lp15@60020
   942
  assumes "0 < z"
lp15@60020
   943
    shows "ln(of_real z::complex) = of_real(ln z)"
lp15@60020
   944
proof -
lp15@60020
   945
  have "ln(of_real (exp (ln z))::complex) = ln (exp (of_real (ln z)))"
lp15@60020
   946
    by (simp add: exp_of_real)
lp15@60020
   947
  also have "... = of_real(ln z)"
lp15@60020
   948
    using assms
lp15@60020
   949
    by (subst Ln_exp) auto
lp15@60020
   950
  finally show ?thesis
lp15@60020
   951
    using assms by simp
lp15@60020
   952
qed
lp15@60020
   953
lp15@60020
   954
corollary Ln_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Re z > 0 \<Longrightarrow> ln z \<in> \<real>"
lp15@60020
   955
  by (auto simp: Ln_of_real elim: Reals_cases)
lp15@60020
   956
lp15@60150
   957
corollary Im_Ln_of_real [simp]: "r > 0 \<Longrightarrow> Im (ln (of_real r)) = 0"
lp15@60150
   958
  by (simp add: Ln_of_real)
lp15@60150
   959
lp15@60150
   960
lemma cmod_Ln_Reals [simp]: "z \<in> Reals \<Longrightarrow> 0 < Re z \<Longrightarrow> cmod (ln z) = norm (ln (Re z))"
lp15@60150
   961
  using Ln_of_real by force
lp15@60150
   962
lp15@60020
   963
lemma Ln_1: "ln 1 = (0::complex)"
lp15@60020
   964
proof -
lp15@60020
   965
  have "ln (exp 0) = (0::complex)"
lp15@60020
   966
    by (metis (mono_tags, hide_lams) Ln_of_real exp_zero ln_one of_real_0 of_real_1 zero_less_one)
lp15@60020
   967
  then show ?thesis
lp15@60020
   968
    by simp
lp15@60020
   969
qed
lp15@60020
   970
lp15@60020
   971
instance
lp15@60020
   972
  by intro_classes (rule ln_complex_def Ln_1)
lp15@60020
   973
lp15@60020
   974
end
lp15@60020
   975
lp15@60020
   976
abbreviation Ln :: "complex \<Rightarrow> complex"
lp15@60020
   977
  where "Ln \<equiv> ln"
lp15@60020
   978
lp15@59751
   979
lemma Ln_eq_iff: "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (Ln w = Ln z \<longleftrightarrow> w = z)"
lp15@59751
   980
  by (metis exp_Ln)
lp15@59751
   981
lp15@59751
   982
lemma Ln_unique: "exp(z) = w \<Longrightarrow> -pi < Im(z) \<Longrightarrow> Im(z) \<le> pi \<Longrightarrow> Ln w = z"
lp15@59751
   983
  using Ln_exp by blast
lp15@59751
   984
lp15@59751
   985
lemma Re_Ln [simp]: "z \<noteq> 0 \<Longrightarrow> Re(Ln z) = ln(norm z)"
lp15@60150
   986
  by (metis exp_Ln assms ln_exp norm_exp_eq_Re)
lp15@60150
   987
lp15@60150
   988
corollary ln_cmod_le: 
lp15@60150
   989
  assumes z: "z \<noteq> 0"
lp15@60150
   990
    shows "ln (cmod z) \<le> cmod (Ln z)"
lp15@60150
   991
  using norm_exp [of "Ln z", simplified exp_Ln [OF z]]
lp15@60150
   992
  by (metis Re_Ln complex_Re_le_cmod z)
lp15@59751
   993
lp15@59751
   994
lemma exists_complex_root:
lp15@59751
   995
  fixes a :: complex
lp15@59751
   996
  shows "n \<noteq> 0 \<Longrightarrow> \<exists>z. z ^ n = a"
lp15@59751
   997
  apply (cases "a=0", simp)
lp15@59751
   998
  apply (rule_tac x= "exp(Ln(a) / n)" in exI)
lp15@59751
   999
  apply (auto simp: exp_of_nat_mult [symmetric])
lp15@59751
  1000
  done
lp15@59751
  1001
lp15@60017
  1002
lp15@59862
  1003
subsection{*The Unwinding Number and the Ln-product Formula*}
lp15@59862
  1004
lp15@59862
  1005
text{*Note that in this special case the unwinding number is -1, 0 or 1.*}
lp15@59862
  1006
lp15@59862
  1007
definition unwinding :: "complex \<Rightarrow> complex" where
lp15@59862
  1008
   "unwinding(z) = (z - Ln(exp z)) / (of_real(2*pi) * ii)"
lp15@59862
  1009
lp15@59862
  1010
lemma unwinding_2pi: "(2*pi) * ii * unwinding(z) = z - Ln(exp z)"
lp15@59862
  1011
  by (simp add: unwinding_def)
lp15@59862
  1012
lp15@59862
  1013
lemma Ln_times_unwinding:
lp15@59862
  1014
    "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z) - (2*pi) * ii * unwinding(Ln w + Ln z)"
lp15@59862
  1015
  using unwinding_2pi by (simp add: exp_add)
lp15@59862
  1016
lp15@59862
  1017
lp15@59751
  1018
subsection{*Derivative of Ln away from the branch cut*}
lp15@59751
  1019
lp15@59751
  1020
lemma
lp15@59751
  1021
  assumes "Im(z) = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
  1022
    shows has_field_derivative_Ln: "(Ln has_field_derivative inverse(z)) (at z)"
lp15@59751
  1023
      and Im_Ln_less_pi:           "Im (Ln z) < pi"
lp15@59751
  1024
proof -
lp15@59751
  1025
  have znz: "z \<noteq> 0"
lp15@59751
  1026
    using assms by auto
lp15@59751
  1027
  then show *: "Im (Ln z) < pi" using assms
lp15@59751
  1028
    by (metis exp_Ln Im_Ln_le_pi Im_exp Re_exp abs_of_nonneg cmod_eq_Re cos_pi mult.right_neutral mult_minus_right mult_zero_right neg_less_0_iff_less norm_exp_eq_Re not_less not_less_iff_gr_or_eq sin_pi)
lp15@59751
  1029
  show "(Ln has_field_derivative inverse(z)) (at z)"
lp15@59751
  1030
    apply (rule has_complex_derivative_inverse_strong_x
lp15@59751
  1031
              [where f = exp and s = "{w. -pi < Im(w) & Im(w) < pi}"])
lp15@59751
  1032
    using znz *
lp15@59751
  1033
    apply (auto simp: continuous_on_exp open_Collect_conj open_halfspace_Im_gt open_halfspace_Im_lt)
lp15@59751
  1034
    apply (metis DERIV_exp exp_Ln)
lp15@59751
  1035
    apply (metis mpi_less_Im_Ln)
lp15@59751
  1036
    done
lp15@59751
  1037
qed
lp15@59751
  1038
lp15@59751
  1039
declare has_field_derivative_Ln [derivative_intros]
lp15@59751
  1040
declare has_field_derivative_Ln [THEN DERIV_chain2, derivative_intros]
lp15@59751
  1041
lp15@59751
  1042
lemma complex_differentiable_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln complex_differentiable at z"
lp15@59751
  1043
  using complex_differentiable_def has_field_derivative_Ln by blast
lp15@59751
  1044
lp15@59751
  1045
lemma complex_differentiable_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z))
lp15@59751
  1046
         \<Longrightarrow> Ln complex_differentiable (at z within s)"
lp15@59751
  1047
  using complex_differentiable_at_Ln complex_differentiable_within_subset by blast
lp15@59751
  1048
lp15@59751
  1049
lemma continuous_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) Ln"
lp15@59751
  1050
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Ln)
lp15@59751
  1051
lp15@59862
  1052
lemma isCont_Ln' [simp]:
lp15@59862
  1053
   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. Ln (f x)) z"
lp15@59862
  1054
  by (blast intro: isCont_o2 [OF _ continuous_at_Ln])
lp15@59862
  1055
lp15@59751
  1056
lemma continuous_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) Ln"
lp15@59751
  1057
  using continuous_at_Ln continuous_at_imp_continuous_within by blast
lp15@59751
  1058
lp15@59751
  1059
lemma continuous_on_Ln [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous_on s Ln"
lp15@59751
  1060
  by (simp add: continuous_at_imp_continuous_on continuous_within_Ln)
lp15@59751
  1061
lp15@59751
  1062
lemma holomorphic_on_Ln: "(\<And>z. z \<in> s \<Longrightarrow> Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln holomorphic_on s"
lp15@59751
  1063
  by (simp add: complex_differentiable_within_Ln holomorphic_on_def)
lp15@59751
  1064
lp15@59751
  1065
lp15@59751
  1066
subsection{*Quadrant-type results for Ln*}
lp15@59751
  1067
lp15@59751
  1068
lemma cos_lt_zero_pi: "pi/2 < x \<Longrightarrow> x < 3*pi/2 \<Longrightarrow> cos x < 0"
lp15@59751
  1069
  using cos_minus_pi cos_gt_zero_pi [of "x-pi"]
lp15@59751
  1070
  by simp
lp15@59751
  1071
lp15@59751
  1072
lemma Re_Ln_pos_lt:
lp15@59751
  1073
  assumes "z \<noteq> 0"
lp15@59751
  1074
    shows "abs(Im(Ln z)) < pi/2 \<longleftrightarrow> 0 < Re(z)"
lp15@59751
  1075
proof -
lp15@59751
  1076
  { fix w
lp15@59751
  1077
    assume "w = Ln z"
lp15@59751
  1078
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1079
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1080
      by auto
lp15@59751
  1081
    then have "abs(Im w) < pi/2 \<longleftrightarrow> 0 < Re(exp w)"
lp15@59751
  1082
      apply (auto simp: Re_exp zero_less_mult_iff cos_gt_zero_pi)
lp15@59751
  1083
      using cos_lt_zero_pi [of "-(Im w)"] cos_lt_zero_pi [of "(Im w)"]
lp15@59751
  1084
      apply (simp add: abs_if split: split_if_asm)
lp15@59751
  1085
      apply (metis (no_types) cos_minus cos_pi_half eq_divide_eq_numeral1(1) eq_numeral_simps(4)
lp15@59751
  1086
               less_numeral_extra(3) linorder_neqE_linordered_idom minus_mult_minus minus_mult_right
lp15@59751
  1087
               mult_numeral_1_right)
lp15@59751
  1088
      done
lp15@59751
  1089
  }
lp15@59751
  1090
  then show ?thesis using assms
lp15@59751
  1091
    by auto
lp15@59751
  1092
qed
lp15@59751
  1093
lp15@59751
  1094
lemma Re_Ln_pos_le:
lp15@59751
  1095
  assumes "z \<noteq> 0"
lp15@59751
  1096
    shows "abs(Im(Ln z)) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(z)"
lp15@59751
  1097
proof -
lp15@59751
  1098
  { fix w
lp15@59751
  1099
    assume "w = Ln z"
lp15@59751
  1100
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1101
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1102
      by auto
lp15@59751
  1103
    then have "abs(Im w) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(exp w)"
lp15@59751
  1104
      apply (auto simp: Re_exp zero_le_mult_iff cos_ge_zero)
lp15@59751
  1105
      using cos_lt_zero_pi [of "- (Im w)"] cos_lt_zero_pi [of "(Im w)"] not_le
lp15@59751
  1106
      apply (auto simp: abs_if split: split_if_asm)
lp15@59751
  1107
      done
lp15@59751
  1108
  }
lp15@59751
  1109
  then show ?thesis using assms
lp15@59751
  1110
    by auto
lp15@59751
  1111
qed
lp15@59751
  1112
lp15@59751
  1113
lemma Im_Ln_pos_lt:
lp15@59751
  1114
  assumes "z \<noteq> 0"
lp15@59751
  1115
    shows "0 < Im(Ln z) \<and> Im(Ln z) < pi \<longleftrightarrow> 0 < Im(z)"
lp15@59751
  1116
proof -
lp15@59751
  1117
  { fix w
lp15@59751
  1118
    assume "w = Ln z"
lp15@59751
  1119
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1120
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1121
      by auto
lp15@59751
  1122
    then have "0 < Im w \<and> Im w < pi \<longleftrightarrow> 0 < Im(exp w)"
lp15@59751
  1123
      using sin_gt_zero [of "- (Im w)"] sin_gt_zero [of "(Im w)"]
lp15@59751
  1124
      apply (auto simp: Im_exp zero_less_mult_iff)
lp15@59751
  1125
      using less_linear apply fastforce
lp15@59751
  1126
      using less_linear apply fastforce
lp15@59751
  1127
      done
lp15@59751
  1128
  }
lp15@59751
  1129
  then show ?thesis using assms
lp15@59751
  1130
    by auto
lp15@59751
  1131
qed
lp15@59751
  1132
lp15@59751
  1133
lemma Im_Ln_pos_le:
lp15@59751
  1134
  assumes "z \<noteq> 0"
lp15@59751
  1135
    shows "0 \<le> Im(Ln z) \<and> Im(Ln z) \<le> pi \<longleftrightarrow> 0 \<le> Im(z)"
lp15@59751
  1136
proof -
lp15@59751
  1137
  { fix w
lp15@59751
  1138
    assume "w = Ln z"
lp15@59751
  1139
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1140
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1141
      by auto
lp15@59751
  1142
    then have "0 \<le> Im w \<and> Im w \<le> pi \<longleftrightarrow> 0 \<le> Im(exp w)"
lp15@59751
  1143
      using sin_ge_zero [of "- (Im w)"] sin_ge_zero [of "(Im w)"]
lp15@59751
  1144
      apply (auto simp: Im_exp zero_le_mult_iff sin_ge_zero)
lp15@59751
  1145
      apply (metis not_le not_less_iff_gr_or_eq pi_not_less_zero sin_eq_0_pi)
lp15@59751
  1146
      done }
lp15@59751
  1147
  then show ?thesis using assms
lp15@59751
  1148
    by auto
lp15@59751
  1149
qed
lp15@59751
  1150
lp15@59751
  1151
lemma Re_Ln_pos_lt_imp: "0 < Re(z) \<Longrightarrow> abs(Im(Ln z)) < pi/2"
lp15@59751
  1152
  by (metis Re_Ln_pos_lt less_irrefl zero_complex.simps(1))
lp15@59751
  1153
lp15@59751
  1154
lemma Im_Ln_pos_lt_imp: "0 < Im(z) \<Longrightarrow> 0 < Im(Ln z) \<and> Im(Ln z) < pi"
lp15@59751
  1155
  by (metis Im_Ln_pos_lt not_le order_refl zero_complex.simps(2))
lp15@59751
  1156
lp15@59751
  1157
lemma Im_Ln_eq_0: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = 0 \<longleftrightarrow> 0 < Re(z) \<and> Im(z) = 0)"
lp15@59751
  1158
  by (metis exp_Ln Im_Ln_less_pi Im_Ln_pos_le Im_Ln_pos_lt Re_complex_of_real add.commute add.left_neutral
lp15@59751
  1159
       complex_eq exp_of_real le_less mult_zero_right norm_exp_eq_Re norm_le_zero_iff not_le of_real_0)
lp15@59751
  1160
lp15@59751
  1161
lemma Im_Ln_eq_pi: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = pi \<longleftrightarrow> Re(z) < 0 \<and> Im(z) = 0)"
lp15@59751
  1162
  by (metis Im_Ln_eq_0 Im_Ln_less_pi Im_Ln_pos_le Im_Ln_pos_lt add.right_neutral complex_eq mult_zero_right not_less not_less_iff_gr_or_eq of_real_0)
lp15@59751
  1163
lp15@59751
  1164
lp15@59751
  1165
subsection{*More Properties of Ln*}
lp15@59751
  1166
lp15@59751
  1167
lemma cnj_Ln: "(Im z = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> cnj(Ln z) = Ln(cnj z)"
lp15@59751
  1168
  apply (cases "z=0", auto)
lp15@59751
  1169
  apply (rule exp_complex_eqI)
lp15@59751
  1170
  apply (auto simp: abs_if split: split_if_asm)
lp15@60017
  1171
  apply (metis Im_Ln_less_pi add_mono_thms_linordered_field(5) cnj.simps mult_2 neg_equal_0_iff_equal)
lp15@60017
  1172
  apply (metis complex_cnj_zero_iff diff_minus_eq_add diff_strict_mono minus_less_iff mpi_less_Im_Ln mult.commute mult_2_right)
lp15@59751
  1173
  by (metis exp_Ln exp_cnj)
lp15@59751
  1174
lp15@59751
  1175
lemma Ln_inverse: "(Im(z) = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> Ln(inverse z) = -(Ln z)"
lp15@59751
  1176
  apply (cases "z=0", auto)
lp15@59751
  1177
  apply (rule exp_complex_eqI)
lp15@59751
  1178
  using mpi_less_Im_Ln [of z] mpi_less_Im_Ln [of "inverse z"]
lp15@59751
  1179
  apply (auto simp: abs_if exp_minus split: split_if_asm)
lp15@59751
  1180
  apply (metis Im_Ln_less_pi Im_Ln_pos_le add_less_cancel_left add_strict_mono
lp15@59751
  1181
               inverse_inverse_eq inverse_zero le_less mult.commute mult_2_right)
lp15@59751
  1182
  done
lp15@59751
  1183
lp15@59751
  1184
lemma Ln_minus1 [simp]: "Ln(-1) = ii * pi"
lp15@59751
  1185
  apply (rule exp_complex_eqI)
lp15@59751
  1186
  using Im_Ln_le_pi [of "-1"] mpi_less_Im_Ln [of "-1"] cis_conv_exp cis_pi
lp15@59751
  1187
  apply (auto simp: abs_if)
lp15@59751
  1188
  done
lp15@59751
  1189
lp15@59751
  1190
lemma Ln_ii [simp]: "Ln ii = ii * of_real pi/2"
lp15@59751
  1191
  using Ln_exp [of "ii * (of_real pi/2)"]
lp15@59751
  1192
  unfolding exp_Euler
lp15@59751
  1193
  by simp
lp15@59751
  1194
lp15@59751
  1195
lemma Ln_minus_ii [simp]: "Ln(-ii) = - (ii * pi/2)"
lp15@59751
  1196
proof -
lp15@59751
  1197
  have  "Ln(-ii) = Ln(1/ii)"
lp15@59751
  1198
    by simp
lp15@59751
  1199
  also have "... = - (Ln ii)"
lp15@59751
  1200
    by (metis Ln_inverse ii.sel(2) inverse_eq_divide zero_neq_one)
lp15@59751
  1201
  also have "... = - (ii * pi/2)"
lp15@60150
  1202
    by simp
lp15@59751
  1203
  finally show ?thesis .
lp15@59751
  1204
qed
lp15@59751
  1205
lp15@59751
  1206
lemma Ln_times:
lp15@59751
  1207
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59751
  1208
    shows "Ln(w * z) =
lp15@59751
  1209
                (if Im(Ln w + Ln z) \<le> -pi then
lp15@59751
  1210
                  (Ln(w) + Ln(z)) + ii * of_real(2*pi)
lp15@59751
  1211
                else if Im(Ln w + Ln z) > pi then
lp15@59751
  1212
                  (Ln(w) + Ln(z)) - ii * of_real(2*pi)
lp15@59751
  1213
                else Ln(w) + Ln(z))"
lp15@59751
  1214
  using pi_ge_zero Im_Ln_le_pi [of w] Im_Ln_le_pi [of z]
lp15@59751
  1215
  using assms mpi_less_Im_Ln [of w] mpi_less_Im_Ln [of z]
lp15@59751
  1216
  by (auto simp: of_real_numeral exp_add exp_diff sin_double cos_double exp_Euler intro!: Ln_unique)
lp15@59751
  1217
lp15@60150
  1218
corollary Ln_times_simple:
lp15@59751
  1219
    "\<lbrakk>w \<noteq> 0; z \<noteq> 0; -pi < Im(Ln w) + Im(Ln z); Im(Ln w) + Im(Ln z) \<le> pi\<rbrakk>
lp15@59751
  1220
         \<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z)"
lp15@59751
  1221
  by (simp add: Ln_times)
lp15@59751
  1222
lp15@60150
  1223
corollary Ln_times_of_real:
lp15@60150
  1224
    "\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(of_real r * z) = ln r + Ln(z)"
lp15@60150
  1225
  using mpi_less_Im_Ln Im_Ln_le_pi
lp15@60150
  1226
  by (force simp: Ln_times)
lp15@60150
  1227
lp15@60150
  1228
corollary Ln_divide_of_real:
lp15@60150
  1229
    "\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(z / of_real r) = Ln(z) - ln r"
lp15@60150
  1230
using Ln_times_of_real [of "inverse r" z]
lp15@60150
  1231
by (simp add: ln_inverse Ln_of_real mult.commute divide_inverse of_real_inverse [symmetric] 
lp15@60150
  1232
         del: of_real_inverse)
lp15@60150
  1233
lp15@59751
  1234
lemma Ln_minus:
lp15@59751
  1235
  assumes "z \<noteq> 0"
lp15@59751
  1236
    shows "Ln(-z) = (if Im(z) \<le> 0 \<and> ~(Re(z) < 0 \<and> Im(z) = 0)
lp15@59751
  1237
                     then Ln(z) + ii * pi
lp15@59751
  1238
                     else Ln(z) - ii * pi)" (is "_ = ?rhs")
lp15@59751
  1239
  using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
lp15@59751
  1240
        Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z]
lp15@59751
  1241
    by (auto simp: of_real_numeral exp_add exp_diff exp_Euler intro!: Ln_unique)
lp15@59751
  1242
lp15@59751
  1243
lemma Ln_inverse_if:
lp15@59751
  1244
  assumes "z \<noteq> 0"
lp15@59751
  1245
    shows "Ln (inverse z) =
lp15@59751
  1246
            (if (Im(z) = 0 \<longrightarrow> 0 < Re z)
lp15@59751
  1247
             then -(Ln z)
lp15@59751
  1248
             else -(Ln z) + \<i> * 2 * complex_of_real pi)"
lp15@59751
  1249
proof (cases "(Im(z) = 0 \<longrightarrow> 0 < Re z)")
lp15@59751
  1250
  case True then show ?thesis
lp15@59751
  1251
    by (simp add: Ln_inverse)
lp15@59751
  1252
next
lp15@59751
  1253
  case False
lp15@59751
  1254
  then have z: "Im z = 0" "Re z < 0"
lp15@59751
  1255
    using assms
lp15@59751
  1256
    apply auto
lp15@59751
  1257
    by (metis cnj.code complex_cnj_cnj not_less_iff_gr_or_eq zero_complex.simps(1) zero_complex.simps(2))
lp15@59751
  1258
  have "Ln(inverse z) = Ln(- (inverse (-z)))"
lp15@59751
  1259
    by simp
lp15@59751
  1260
  also have "... = Ln (inverse (-z)) + \<i> * complex_of_real pi"
lp15@59751
  1261
    using assms z
lp15@59751
  1262
    apply (simp add: Ln_minus)
lp15@59751
  1263
    apply (simp add: field_simps)
lp15@59751
  1264
    done
lp15@59751
  1265
  also have "... = - Ln (- z) + \<i> * complex_of_real pi"
lp15@59751
  1266
    apply (subst Ln_inverse)
lp15@59751
  1267
    using z assms by auto
lp15@59751
  1268
  also have "... = - (Ln z) + \<i> * 2 * complex_of_real pi"
lp15@59751
  1269
    apply (subst Ln_minus [OF assms])
lp15@59751
  1270
    using assms z
lp15@59751
  1271
    apply simp
lp15@59751
  1272
    done
lp15@59751
  1273
  finally show ?thesis
lp15@59751
  1274
    using assms z
lp15@59751
  1275
    by simp
lp15@59751
  1276
qed
lp15@59751
  1277
lp15@59751
  1278
lemma Ln_times_ii:
lp15@59751
  1279
  assumes "z \<noteq> 0"
lp15@59751
  1280
    shows  "Ln(ii * z) = (if 0 \<le> Re(z) | Im(z) < 0
lp15@59751
  1281
                          then Ln(z) + ii * of_real pi/2
lp15@59751
  1282
                          else Ln(z) - ii * of_real(3 * pi/2))"
lp15@59751
  1283
  using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
lp15@59751
  1284
        Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z] Re_Ln_pos_le [of z]
lp15@59751
  1285
  by (auto simp: of_real_numeral Ln_times)
lp15@59751
  1286
lp15@59751
  1287
lp15@60150
  1288
subsection{*Relation between Ln and Arg, and hence continuity of Arg*}
lp15@60150
  1289
lp15@60150
  1290
lemma Arg_Ln: 
lp15@60150
  1291
  assumes "0 < Arg z" shows "Arg z = Im(Ln(-z)) + pi"
lp15@60150
  1292
proof (cases "z = 0")
lp15@60150
  1293
  case True
lp15@60150
  1294
  with assms show ?thesis
lp15@60150
  1295
    by simp
lp15@60150
  1296
next
lp15@60150
  1297
  case False
lp15@60150
  1298
  then have "z / of_real(norm z) = exp(ii * of_real(Arg z))"
lp15@60150
  1299
    using Arg [of z]
lp15@60150
  1300
    by (metis abs_norm_cancel nonzero_mult_divide_cancel_left norm_of_real zero_less_norm_iff)
lp15@60150
  1301
  then have "- z / of_real(norm z) = exp (\<i> * (of_real (Arg z) - pi))"
lp15@60150
  1302
    using cis_conv_exp cis_pi
lp15@60150
  1303
    by (auto simp: exp_diff algebra_simps)
lp15@60150
  1304
  then have "ln (- z / of_real(norm z)) = ln (exp (\<i> * (of_real (Arg z) - pi)))"
lp15@60150
  1305
    by simp
lp15@60150
  1306
  also have "... = \<i> * (of_real(Arg z) - pi)"
lp15@60150
  1307
    using Arg [of z] assms pi_not_less_zero
lp15@60150
  1308
    by auto
lp15@60150
  1309
  finally have "Arg z =  Im (Ln (- z / of_real (cmod z))) + pi"
lp15@60150
  1310
    by simp
lp15@60150
  1311
  also have "... = Im (Ln (-z) - ln (cmod z)) + pi"
lp15@60150
  1312
    by (metis diff_0_right minus_diff_eq zero_less_norm_iff Ln_divide_of_real False)
lp15@60150
  1313
  also have "... = Im (Ln (-z)) + pi"
lp15@60150
  1314
    by simp
lp15@60150
  1315
  finally show ?thesis .
lp15@60150
  1316
qed
lp15@60150
  1317
lp15@60150
  1318
lemma continuous_at_Arg: 
lp15@60150
  1319
  assumes "z \<in> \<real> \<Longrightarrow> Re z < 0"
lp15@60150
  1320
    shows "continuous (at z) Arg"
lp15@60150
  1321
proof -
lp15@60150
  1322
  have *: "isCont (\<lambda>z. Im (Ln (- z)) + pi) z"
lp15@60150
  1323
    by (rule Complex.isCont_Im isCont_Ln' continuous_intros | simp add: assms complex_is_Real_iff)+
lp15@60150
  1324
  then show ?thesis
lp15@60150
  1325
    apply (simp add: continuous_at)
lp15@60150
  1326
    apply (rule Lim_transform_within_open [of "-{z. z \<in> \<real> & 0 \<le> Re z}" _ "\<lambda>z. Im(Ln(-z)) + pi"])
lp15@60150
  1327
    apply (simp add: closed_def [symmetric] closed_Collect_conj closed_complex_Reals closed_halfspace_Re_ge)
lp15@60150
  1328
    apply (simp_all add: assms not_le Arg_Ln [OF Arg_gt_0])
lp15@60150
  1329
    done
lp15@60150
  1330
qed
lp15@60150
  1331
lp15@60150
  1332
text{*Relation between Arg and arctangent in upper halfplane*}
lp15@60150
  1333
lemma Arg_arctan_upperhalf: 
lp15@60150
  1334
  assumes "0 < Im z"
lp15@60150
  1335
    shows "Arg z = pi/2 - arctan(Re z / Im z)"
lp15@60150
  1336
proof (cases "z = 0")
lp15@60150
  1337
  case True with assms show ?thesis
lp15@60150
  1338
    by simp
lp15@60150
  1339
next
lp15@60150
  1340
  case False
lp15@60150
  1341
  show ?thesis
lp15@60150
  1342
    apply (rule Arg_unique [of "norm z"])
lp15@60150
  1343
    using False assms arctan [of "Re z / Im z"] pi_ge_two pi_half_less_two
lp15@60150
  1344
    apply (auto simp: exp_Euler cos_diff sin_diff)
lp15@60150
  1345
    using norm_complex_def [of z, symmetric]
lp15@60150
  1346
    apply (simp add: of_real_numeral sin_of_real cos_of_real sin_arctan cos_arctan field_simps real_sqrt_divide)
lp15@60150
  1347
    apply (metis complex_eq mult.assoc ring_class.ring_distribs(2))
lp15@60150
  1348
    done
lp15@60150
  1349
qed
lp15@60150
  1350
lp15@60150
  1351
lemma Arg_eq_Im_Ln: 
lp15@60150
  1352
  assumes "0 \<le> Im z" "0 < Re z" 
lp15@60150
  1353
    shows "Arg z = Im (Ln z)"
lp15@60150
  1354
proof (cases "z = 0 \<or> Im z = 0")
lp15@60150
  1355
  case True then show ?thesis
lp15@60150
  1356
    using assms Arg_eq_0 complex_is_Real_iff  
lp15@60150
  1357
    apply auto
lp15@60150
  1358
    by (metis Arg_eq_0_pi Arg_eq_pi Im_Ln_eq_0 Im_Ln_eq_pi less_numeral_extra(3) zero_complex.simps(1))
lp15@60150
  1359
next
lp15@60150
  1360
  case False 
lp15@60150
  1361
  then have "Arg z > 0"
lp15@60150
  1362
    using Arg_gt_0 complex_is_Real_iff by blast
lp15@60150
  1363
  then show ?thesis
lp15@60150
  1364
    using assms False 
lp15@60150
  1365
    by (subst Arg_Ln) (auto simp: Ln_minus)
lp15@60150
  1366
qed
lp15@60150
  1367
lp15@60150
  1368
lemma continuous_within_upperhalf_Arg: 
lp15@60150
  1369
  assumes "z \<noteq> 0"
lp15@60150
  1370
    shows "continuous (at z within {z. 0 \<le> Im z}) Arg"
lp15@60150
  1371
proof (cases "z \<in> \<real> & 0 \<le> Re z")
lp15@60150
  1372
  case False then show ?thesis
lp15@60150
  1373
    using continuous_at_Arg continuous_at_imp_continuous_within by auto
lp15@60150
  1374
next
lp15@60150
  1375
  case True
lp15@60150
  1376
  then have z: "z \<in> \<real>" "0 < Re z"
lp15@60150
  1377
    using assms  by (auto simp: complex_is_Real_iff complex_neq_0)
lp15@60150
  1378
  then have [simp]: "Arg z = 0" "Im (Ln z) = 0"
lp15@60150
  1379
    by (auto simp: Arg_eq_0 Im_Ln_eq_0 assms complex_is_Real_iff)
lp15@60150
  1380
  show ?thesis  
lp15@60150
  1381
  proof (clarsimp simp add: continuous_within Lim_within dist_norm)
lp15@60150
  1382
    fix e::real
lp15@60150
  1383
    assume "0 < e"
lp15@60150
  1384
    moreover have "continuous (at z) (\<lambda>x. Im (Ln x))"
lp15@60150
  1385
      using z  by (rule continuous_intros | simp)
lp15@60150
  1386
    ultimately
lp15@60150
  1387
    obtain d where d: "d>0" "\<And>x. x \<noteq> z \<Longrightarrow> cmod (x - z) < d \<Longrightarrow> \<bar>Im (Ln x)\<bar> < e"
lp15@60150
  1388
      by (auto simp: continuous_within Lim_within dist_norm)
lp15@60150
  1389
    { fix x
lp15@60150
  1390
      assume "cmod (x - z) < Re z / 2"
lp15@60150
  1391
      then have "\<bar>Re x - Re z\<bar> < Re z / 2"
lp15@60150
  1392
        by (metis le_less_trans abs_Re_le_cmod minus_complex.simps(1))
lp15@60150
  1393
      then have "0 < Re x"
lp15@60150
  1394
        using z by linarith
lp15@60150
  1395
    }
lp15@60150
  1396
    then show "\<exists>d>0. \<forall>x. 0 \<le> Im x \<longrightarrow> x \<noteq> z \<and> cmod (x - z) < d \<longrightarrow> \<bar>Arg x\<bar> < e"
lp15@60150
  1397
      apply (rule_tac x="min d (Re z / 2)" in exI)
lp15@60150
  1398
      using z d
lp15@60150
  1399
      apply (auto simp: Arg_eq_Im_Ln)
lp15@60150
  1400
      done
lp15@60150
  1401
  qed
lp15@60150
  1402
qed
lp15@60150
  1403
lp15@60150
  1404
lemma continuous_on_upperhalf_Arg: "continuous_on ({z. 0 \<le> Im z} - {0}) Arg"
lp15@60150
  1405
  apply (auto simp: continuous_on_eq_continuous_within)
lp15@60150
  1406
  by (metis Diff_subset continuous_within_subset continuous_within_upperhalf_Arg)
lp15@60150
  1407
lp15@60150
  1408
lemma open_Arg_less_Int: 
lp15@60150
  1409
  assumes "0 \<le> s" "t \<le> 2*pi"
lp15@60150
  1410
    shows "open ({y. s < Arg y} \<inter> {y. Arg y < t})"
lp15@60150
  1411
proof -
lp15@60150
  1412
  have 1: "continuous_on (UNIV - {z \<in> \<real>. 0 \<le> Re z}) Arg"
lp15@60150
  1413
    using continuous_at_Arg continuous_at_imp_continuous_within 
lp15@60150
  1414
    by (auto simp: continuous_on_eq_continuous_within set_diff_eq)
lp15@60150
  1415
  have 2: "open (UNIV - {z \<in> \<real>. 0 \<le> Re z})"
lp15@60150
  1416
    by (simp add: closed_Collect_conj closed_complex_Reals closed_halfspace_Re_ge open_Diff)
lp15@60150
  1417
  have "open ({z. s < z} \<inter> {z. z < t})"
lp15@60150
  1418
    using open_lessThan [of t] open_greaterThan [of s]
lp15@60150
  1419
    by (metis greaterThan_def lessThan_def open_Int)
lp15@60150
  1420
  moreover have "{y. s < Arg y} \<inter> {y. Arg y < t} \<subseteq> - {z \<in> \<real>. 0 \<le> Re z}"
lp15@60150
  1421
    using assms
lp15@60150
  1422
    by (auto simp: Arg_real)
lp15@60150
  1423
  ultimately show ?thesis
lp15@60150
  1424
    using continuous_imp_open_vimage [OF 1 2, of  "{z. Re z > s} \<inter> {z. Re z < t}"] 
lp15@60150
  1425
    by auto
lp15@60150
  1426
qed
lp15@60150
  1427
lp15@60150
  1428
lemma open_Arg_gt: "open {z. t < Arg z}"
lp15@60150
  1429
proof (cases "t < 0")
lp15@60150
  1430
  case True then have "{z. t < Arg z} = UNIV"
lp15@60150
  1431
    using Arg_ge_0 less_le_trans by auto
lp15@60150
  1432
  then show ?thesis
lp15@60150
  1433
    by simp
lp15@60150
  1434
next
lp15@60150
  1435
  case False then show ?thesis
lp15@60150
  1436
    using open_Arg_less_Int [of t "2*pi"] Arg_lt_2pi
lp15@60150
  1437
    by auto
lp15@60150
  1438
qed
lp15@60150
  1439
lp15@60150
  1440
lemma closed_Arg_le: "closed {z. Arg z \<le> t}"
lp15@60150
  1441
  using open_Arg_gt [of t]
lp15@60150
  1442
  by (simp add: closed_def Set.Collect_neg_eq [symmetric] not_le)
lp15@60017
  1443
lp15@60017
  1444
subsection{*Complex Powers*}
lp15@60017
  1445
lp15@60017
  1446
lemma powr_to_1 [simp]: "z powr 1 = (z::complex)"
lp15@60020
  1447
  by (simp add: powr_def)
lp15@60017
  1448
lp15@60017
  1449
lemma powr_nat:
lp15@60017
  1450
  fixes n::nat and z::complex shows "z powr n = (if z = 0 then 0 else z^n)"
lp15@60020
  1451
  by (simp add: exp_of_nat_mult powr_def)
lp15@60017
  1452
lp15@60017
  1453
lemma powr_add:
lp15@60017
  1454
  fixes w::complex shows "w powr (z1 + z2) = w powr z1 * w powr z2"
lp15@60017
  1455
  by (simp add: powr_def algebra_simps exp_add)
lp15@60017
  1456
lp15@60017
  1457
lemma powr_minus:
lp15@60017
  1458
  fixes w::complex shows  "w powr (-z) = inverse(w powr z)"
lp15@60017
  1459
  by (simp add: powr_def exp_minus)
lp15@60017
  1460
lp15@60017
  1461
lemma powr_diff:
lp15@60017
  1462
  fixes w::complex shows  "w powr (z1 - z2) = w powr z1 / w powr z2"
lp15@60017
  1463
  by (simp add: powr_def algebra_simps exp_diff)
lp15@60017
  1464
lp15@60017
  1465
lemma norm_powr_real: "w \<in> \<real> \<Longrightarrow> 0 < Re w \<Longrightarrow> norm(w powr z) = exp(Re z * ln(Re w))"
lp15@60020
  1466
  apply (simp add: powr_def)
lp15@60017
  1467
  using Im_Ln_eq_0 complex_is_Real_iff norm_complex_def
lp15@60017
  1468
  by auto
lp15@60017
  1469
lp15@60017
  1470
lemma powr_real_real:
lp15@60017
  1471
    "\<lbrakk>w \<in> \<real>; z \<in> \<real>; 0 < Re w\<rbrakk> \<Longrightarrow> w powr z = exp(Re z * ln(Re w))"
lp15@60020
  1472
  apply (simp add: powr_def)
lp15@60017
  1473
  by (metis complex_eq complex_is_Real_iff diff_0 diff_0_right diff_minus_eq_add exp_ln exp_not_eq_zero
lp15@60017
  1474
       exp_of_real Ln_of_real mult_zero_right of_real_0 of_real_mult)
lp15@60017
  1475
lp15@60017
  1476
lemma powr_of_real:
lp15@60020
  1477
  fixes x::real and y::real
lp15@60020
  1478
  shows "0 < x \<Longrightarrow> of_real x powr (of_real y::complex) = of_real (x powr y)"
lp15@60020
  1479
  by (simp add: powr_def) (metis exp_of_real of_real_mult Ln_of_real)
lp15@60017
  1480
lp15@60017
  1481
lemma norm_powr_real_mono:
lp15@60020
  1482
    "\<lbrakk>w \<in> \<real>; 1 < Re w\<rbrakk>
lp15@60020
  1483
     \<Longrightarrow> cmod(w powr z1) \<le> cmod(w powr z2) \<longleftrightarrow> Re z1 \<le> Re z2"
lp15@60020
  1484
  by (auto simp: powr_def algebra_simps Reals_def Ln_of_real)
lp15@60017
  1485
lp15@60017
  1486
lemma powr_times_real:
lp15@60017
  1487
    "\<lbrakk>x \<in> \<real>; y \<in> \<real>; 0 \<le> Re x; 0 \<le> Re y\<rbrakk>
lp15@60017
  1488
           \<Longrightarrow> (x * y) powr z = x powr z * y powr z"
lp15@60020
  1489
  by (auto simp: Reals_def powr_def Ln_times exp_add algebra_simps less_eq_real_def Ln_of_real)
lp15@60017
  1490
lp15@60017
  1491
lemma has_field_derivative_powr:
lp15@60017
  1492
    "(Im z = 0 \<Longrightarrow> 0 < Re z)
lp15@60017
  1493
     \<Longrightarrow> ((\<lambda>z. z powr s) has_field_derivative (s * z powr (s - 1))) (at z)"
lp15@60017
  1494
  apply (cases "z=0", auto)
lp15@60020
  1495
  apply (simp add: powr_def)
lp15@60017
  1496
  apply (rule DERIV_transform_at [where d = "norm z" and f = "\<lambda>z. exp (s * Ln z)"])
lp15@60020
  1497
  apply (auto simp: dist_complex_def)
lp15@60017
  1498
  apply (intro derivative_eq_intros | simp add: assms)+
lp15@60017
  1499
  apply (simp add: field_simps exp_diff)
lp15@60017
  1500
  done
lp15@60017
  1501
lp15@60017
  1502
lemma has_field_derivative_powr_right:
lp15@60017
  1503
    "w \<noteq> 0 \<Longrightarrow> ((\<lambda>z. w powr z) has_field_derivative Ln w * w powr z) (at z)"
lp15@60020
  1504
  apply (simp add: powr_def)
lp15@60017
  1505
  apply (intro derivative_eq_intros | simp add: assms)+
lp15@60017
  1506
  done
lp15@60017
  1507
lp15@60017
  1508
lemma complex_differentiable_powr_right:
lp15@60017
  1509
    "w \<noteq> 0 \<Longrightarrow> (\<lambda>z. w powr z) complex_differentiable (at z)"
lp15@60017
  1510
using complex_differentiable_def has_field_derivative_powr_right by blast
lp15@60017
  1511
lp15@60017
  1512
lemma holomorphic_on_powr_right:
lp15@60017
  1513
    "f holomorphic_on s \<Longrightarrow> w \<noteq> 0 \<Longrightarrow> (\<lambda>z. w powr (f z)) holomorphic_on s"
lp15@60017
  1514
    unfolding holomorphic_on_def
lp15@60017
  1515
    using DERIV_chain' complex_differentiable_def has_field_derivative_powr_right by fastforce
lp15@60017
  1516
lp15@60017
  1517
lemma norm_powr_real_powr:
lp15@60017
  1518
  "w \<in> \<real> \<Longrightarrow> 0 < Re w \<Longrightarrow> norm(w powr z) = Re w powr Re z"
lp15@60020
  1519
  by (auto simp add: norm_powr_real powr_def Im_Ln_eq_0 complex_is_Real_iff in_Reals_norm)
lp15@60017
  1520
lp15@60150
  1521
lp15@60150
  1522
subsection{*Some Limits involving Logarithms*}
lp15@60150
  1523
        
lp15@60150
  1524
lemma lim_Ln_over_power:
lp15@60150
  1525
  fixes s::complex
lp15@60150
  1526
  assumes "0 < Re s"
lp15@60150
  1527
    shows "((\<lambda>n. Ln n / (n powr s)) ---> 0) sequentially"
lp15@60150
  1528
proof (simp add: lim_sequentially dist_norm, clarify)
lp15@60150
  1529
  fix e::real 
lp15@60150
  1530
  assume e: "0 < e"
lp15@60150
  1531
  have "\<exists>xo>0. \<forall>x\<ge>xo. 0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
lp15@60150
  1532
  proof (rule_tac x="2/(e * (Re s)\<^sup>2)" in exI, safe)
lp15@60150
  1533
    show "0 < 2 / (e * (Re s)\<^sup>2)"
lp15@60150
  1534
      using e assms by (simp add: field_simps)
lp15@60150
  1535
  next
lp15@60150
  1536
    fix x::real
lp15@60150
  1537
    assume x: "2 / (e * (Re s)\<^sup>2) \<le> x"
lp15@60150
  1538
    then have "x>0"
lp15@60150
  1539
    using e assms
lp15@60150
  1540
      by (metis less_le_trans mult_eq_0_iff mult_pos_pos pos_less_divide_eq power2_eq_square
lp15@60150
  1541
                zero_less_numeral)
lp15@60150
  1542
    then show "0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
lp15@60150
  1543
      using e assms x
lp15@60150
  1544
      apply (auto simp: field_simps)
lp15@60150
  1545
      apply (rule_tac y = "e * (x\<^sup>2 * (Re s)\<^sup>2)" in le_less_trans)
lp15@60150
  1546
      apply (auto simp: power2_eq_square field_simps add_pos_pos)
lp15@60150
  1547
      done
lp15@60150
  1548
  qed
lp15@60150
  1549
  then have "\<exists>xo>0. \<forall>x\<ge>xo. x / e < 1 + (Re s * x) + (1/2) * (Re s * x)^2"
lp15@60150
  1550
    using e  by (simp add: field_simps)
lp15@60150
  1551
  then have "\<exists>xo>0. \<forall>x\<ge>xo. x / e < exp (Re s * x)"
lp15@60150
  1552
    using assms
lp15@60150
  1553
    by (force intro: less_le_trans [OF _ exp_lower_taylor_quadratic])
lp15@60150
  1554
  then have "\<exists>xo>0. \<forall>x\<ge>xo. x < e * exp (Re s * x)"
lp15@60150
  1555
    using e   by (auto simp: field_simps)
lp15@60150
  1556
  with e show "\<exists>no. \<forall>n\<ge>no. norm (Ln (of_nat n) / of_nat n powr s) < e"
lp15@60150
  1557
    apply (auto simp: norm_divide norm_powr_real divide_simps)
lp15@60150
  1558
    apply (rule_tac x="nat (ceiling (exp xo))" in exI)
lp15@60150
  1559
    apply clarify
lp15@60150
  1560
    apply (drule_tac x="ln n" in spec)
lp15@60150
  1561
    apply (auto simp: real_of_nat_def exp_less_mono nat_ceiling_le_eq not_le)
lp15@60150
  1562
    apply (metis exp_less_mono exp_ln not_le of_nat_0_less_iff)
lp15@60150
  1563
    done
lp15@60150
  1564
qed
lp15@60150
  1565
lp15@60150
  1566
lemma lim_Ln_over_n: "((\<lambda>n. Ln(of_nat n) / of_nat n) ---> 0) sequentially"
lp15@60150
  1567
  using lim_Ln_over_power [of 1]
lp15@60150
  1568
  by simp
lp15@60150
  1569
lp15@60150
  1570
lemma Ln_Reals_eq: "x \<in> Reals \<Longrightarrow> Re x > 0 \<Longrightarrow> Ln x = of_real (ln (Re x))"
lp15@60017
  1571
  using Ln_of_real by force
lp15@60017
  1572
lp15@60150
  1573
lemma powr_Reals_eq: "x \<in> Reals \<Longrightarrow> Re x > 0 \<Longrightarrow> x powr complex_of_real y = of_real (x powr y)"
lp15@60150
  1574
  by (simp add: powr_of_real)
lp15@60150
  1575
lp15@60150
  1576
lemma lim_ln_over_power:
lp15@60150
  1577
  fixes s :: real
lp15@60150
  1578
  assumes "0 < s"
lp15@60150
  1579
    shows "((\<lambda>n. ln n / (n powr s)) ---> 0) sequentially"
lp15@60150
  1580
  using lim_Ln_over_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
lp15@60150
  1581
  apply (subst filterlim_sequentially_Suc [symmetric])
lp15@60150
  1582
  apply (simp add: lim_sequentially dist_norm
lp15@60150
  1583
          Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
lp15@60150
  1584
  done
lp15@60150
  1585
lp15@60150
  1586
lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) ---> 0) sequentially"
lp15@60150
  1587
  using lim_ln_over_power [of 1, THEN filterlim_sequentially_Suc [THEN iffD2]]
lp15@60150
  1588
  apply (subst filterlim_sequentially_Suc [symmetric])
lp15@60150
  1589
  apply (simp add: lim_sequentially dist_norm real_of_nat_def)
lp15@60150
  1590
  done
lp15@60150
  1591
lp15@60150
  1592
lemma lim_1_over_complex_power:
lp15@60150
  1593
  assumes "0 < Re s"
lp15@60150
  1594
    shows "((\<lambda>n. 1 / (of_nat n powr s)) ---> 0) sequentially"
lp15@60150
  1595
proof -
lp15@60150
  1596
  have "\<forall>n>0. 3 \<le> n \<longrightarrow> 1 \<le> ln (real_of_nat n)"
lp15@60150
  1597
    using ln3_gt_1
lp15@60150
  1598
    by (force intro: order_trans [of _ "ln 3"] ln3_gt_1)
lp15@60150
  1599
  moreover have "(\<lambda>n. cmod (Ln (of_nat n) / of_nat n powr s)) ----> 0"
lp15@60150
  1600
    using lim_Ln_over_power [OF assms]
lp15@60150
  1601
    by (metis tendsto_norm_zero_iff)
lp15@60150
  1602
  ultimately show ?thesis
lp15@60150
  1603
    apply (auto intro!: Lim_null_comparison [where g = "\<lambda>n. norm (Ln(of_nat n) / of_nat n powr s)"])
lp15@60150
  1604
    apply (auto simp: norm_divide divide_simps eventually_sequentially)
lp15@60150
  1605
    done
lp15@60150
  1606
qed
lp15@60150
  1607
lp15@60150
  1608
lemma lim_1_over_real_power:
lp15@60150
  1609
  fixes s :: real
lp15@60150
  1610
  assumes "0 < s"
lp15@60150
  1611
    shows "((\<lambda>n. 1 / (of_nat n powr s)) ---> 0) sequentially"
lp15@60150
  1612
  using lim_1_over_complex_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
lp15@60150
  1613
  apply (subst filterlim_sequentially_Suc [symmetric])
lp15@60150
  1614
  apply (simp add: lim_sequentially dist_norm)
lp15@60150
  1615
  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
lp15@60150
  1616
  done
lp15@60150
  1617
lp15@60150
  1618
lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) ---> 0) sequentially"
lp15@60150
  1619
proof (clarsimp simp add: lim_sequentially dist_norm norm_divide divide_simps)
lp15@60150
  1620
  fix r::real
lp15@60150
  1621
  assume "0 < r"
lp15@60150
  1622
  have ir: "inverse (exp (inverse r)) > 0"
lp15@60150
  1623
    by simp
lp15@60150
  1624
  obtain n where n: "1 < of_nat n * inverse (exp (inverse r))"
lp15@60150
  1625
    using ex_less_of_nat_mult [of _ 1, OF ir]
lp15@60150
  1626
    by auto
lp15@60150
  1627
  then have "exp (inverse r) < of_nat n"
lp15@60150
  1628
    by (simp add: divide_simps)
lp15@60150
  1629
  then have "ln (exp (inverse r)) < ln (of_nat n)"
lp15@60150
  1630
    by (metis exp_gt_zero less_trans ln_exp ln_less_cancel_iff)
lp15@60150
  1631
  with `0 < r` have "1 < r * ln (real_of_nat n)"
lp15@60150
  1632
    by (simp add: field_simps)
lp15@60150
  1633
  moreover have "n > 0" using n
lp15@60150
  1634
    using neq0_conv by fastforce
lp15@60150
  1635
  ultimately show "\<exists>no. \<forall>n. Ln (of_nat n) \<noteq> 0 \<longrightarrow> no \<le> n \<longrightarrow> 1 < r * cmod (Ln (of_nat n))"
lp15@60150
  1636
    using n `0 < r`
lp15@60150
  1637
    apply (rule_tac x=n in exI)
lp15@60150
  1638
    apply (auto simp: divide_simps)
lp15@60150
  1639
    apply (erule less_le_trans, auto)
lp15@60150
  1640
    done
lp15@60150
  1641
qed
lp15@60150
  1642
lp15@60150
  1643
lemma lim_1_over_ln: "((\<lambda>n. 1 / ln(real_of_nat n)) ---> 0) sequentially"
lp15@60150
  1644
  using lim_1_over_Ln [THEN filterlim_sequentially_Suc [THEN iffD2]] assms
lp15@60150
  1645
  apply (subst filterlim_sequentially_Suc [symmetric])
lp15@60150
  1646
  apply (simp add: lim_sequentially dist_norm)
lp15@60150
  1647
  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
lp15@60150
  1648
  done
lp15@60150
  1649
lp15@60017
  1650
lp15@59751
  1651
subsection{*Relation between Square Root and exp/ln, hence its derivative*}
lp15@59751
  1652
lp15@59751
  1653
lemma csqrt_exp_Ln:
lp15@59751
  1654
  assumes "z \<noteq> 0"
lp15@59751
  1655
    shows "csqrt z = exp(Ln(z) / 2)"
lp15@59751
  1656
proof -
lp15@59751
  1657
  have "(exp (Ln z / 2))\<^sup>2 = (exp (Ln z))"
lp15@59751
  1658
    by (metis exp_double nonzero_mult_divide_cancel_left times_divide_eq_right zero_neq_numeral)
lp15@59751
  1659
  also have "... = z"
lp15@59751
  1660
    using assms exp_Ln by blast
lp15@59751
  1661
  finally have "csqrt z = csqrt ((exp (Ln z / 2))\<^sup>2)"
lp15@59751
  1662
    by simp
lp15@59751
  1663
  also have "... = exp (Ln z / 2)"
lp15@59751
  1664
    apply (subst csqrt_square)
lp15@59751
  1665
    using cos_gt_zero_pi [of "(Im (Ln z) / 2)"] Im_Ln_le_pi mpi_less_Im_Ln assms
lp15@59751
  1666
    apply (auto simp: Re_exp Im_exp zero_less_mult_iff zero_le_mult_iff, fastforce+)
lp15@59751
  1667
    done
lp15@59751
  1668
  finally show ?thesis using assms csqrt_square
lp15@59751
  1669
    by simp
lp15@59751
  1670
qed
lp15@59751
  1671
lp15@59751
  1672
lemma csqrt_inverse:
lp15@59751
  1673
  assumes "Im(z) = 0 \<Longrightarrow> 0 < Re z"
lp15@59751
  1674
    shows "csqrt (inverse z) = inverse (csqrt z)"
lp15@59751
  1675
proof (cases "z=0", simp)
lp15@59751
  1676
  assume "z \<noteq> 0 "
lp15@59751
  1677
  then show ?thesis
lp15@59751
  1678
    using assms
lp15@59751
  1679
    by (simp add: csqrt_exp_Ln Ln_inverse exp_minus)
lp15@59751
  1680
qed
lp15@59751
  1681
lp15@59751
  1682
lemma cnj_csqrt:
lp15@59751
  1683
  assumes "Im z = 0 \<Longrightarrow> 0 \<le> Re(z)"
lp15@59751
  1684
    shows "cnj(csqrt z) = csqrt(cnj z)"
lp15@59751
  1685
proof (cases "z=0", simp)
lp15@59751
  1686
  assume z: "z \<noteq> 0"
lp15@59751
  1687
  then have "Im z = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
  1688
    using assms cnj.code complex_cnj_zero_iff by fastforce
lp15@59751
  1689
  then show ?thesis
lp15@59751
  1690
   using z by (simp add: csqrt_exp_Ln cnj_Ln exp_cnj)
lp15@59751
  1691
qed
lp15@59751
  1692
lp15@59751
  1693
lemma has_field_derivative_csqrt:
lp15@59751
  1694
  assumes "Im z = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
  1695
    shows "(csqrt has_field_derivative inverse(2 * csqrt z)) (at z)"
lp15@59751
  1696
proof -
lp15@59751
  1697
  have z: "z \<noteq> 0"
lp15@59751
  1698
    using assms by auto
lp15@59751
  1699
  then have *: "inverse z = inverse (2*z) * 2"
lp15@59751
  1700
    by (simp add: divide_simps)
lp15@59751
  1701
  show ?thesis
lp15@59751
  1702
    apply (rule DERIV_transform_at [where f = "\<lambda>z. exp(Ln(z) / 2)" and d = "norm z"])
lp15@59751
  1703
    apply (intro derivative_eq_intros | simp add: assms)+
lp15@59751
  1704
    apply (rule *)
lp15@59751
  1705
    using z
lp15@59751
  1706
    apply (auto simp: field_simps csqrt_exp_Ln [symmetric])
lp15@59751
  1707
    apply (metis power2_csqrt power2_eq_square)
lp15@59751
  1708
    apply (metis csqrt_exp_Ln dist_0_norm less_irrefl)
lp15@59751
  1709
    done
lp15@59751
  1710
qed
lp15@59751
  1711
lp15@59751
  1712
lemma complex_differentiable_at_csqrt:
lp15@59751
  1713
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable at z"
lp15@59751
  1714
  using complex_differentiable_def has_field_derivative_csqrt by blast
lp15@59751
  1715
lp15@59751
  1716
lemma complex_differentiable_within_csqrt:
lp15@59751
  1717
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable (at z within s)"
lp15@59751
  1718
  using complex_differentiable_at_csqrt complex_differentiable_within_subset by blast
lp15@59751
  1719
lp15@59751
  1720
lemma continuous_at_csqrt:
lp15@59751
  1721
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) csqrt"
lp15@59751
  1722
  by (simp add: complex_differentiable_within_csqrt complex_differentiable_imp_continuous_at)
lp15@59751
  1723
lp15@59862
  1724
corollary isCont_csqrt' [simp]:
lp15@59862
  1725
   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. csqrt (f x)) z"
lp15@59862
  1726
  by (blast intro: isCont_o2 [OF _ continuous_at_csqrt])
lp15@59862
  1727
lp15@59751
  1728
lemma continuous_within_csqrt:
lp15@59751
  1729
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) csqrt"
lp15@59751
  1730
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_csqrt)
lp15@59751
  1731
lp15@59751
  1732
lemma continuous_on_csqrt [continuous_intros]:
lp15@59751
  1733
    "(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous_on s csqrt"
lp15@59751
  1734
  by (simp add: continuous_at_imp_continuous_on continuous_within_csqrt)
lp15@59751
  1735
lp15@59751
  1736
lemma holomorphic_on_csqrt:
lp15@59751
  1737
    "(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt holomorphic_on s"
lp15@59751
  1738
  by (simp add: complex_differentiable_within_csqrt holomorphic_on_def)
lp15@59751
  1739
lp15@59751
  1740
lemma continuous_within_closed_nontrivial:
lp15@59751
  1741
    "closed s \<Longrightarrow> a \<notin> s ==> continuous (at a within s) f"
lp15@59751
  1742
  using open_Compl
lp15@59751
  1743
  by (force simp add: continuous_def eventually_at_topological filterlim_iff open_Collect_neg)
lp15@59751
  1744
lp15@59751
  1745
lemma continuous_within_csqrt_posreal:
lp15@59751
  1746
    "continuous (at z within (\<real> \<inter> {w. 0 \<le> Re(w)})) csqrt"
lp15@59751
  1747
proof (cases "Im z = 0 --> 0 < Re(z)")
lp15@59751
  1748
  case True then show ?thesis
lp15@59751
  1749
    by (blast intro: continuous_within_csqrt)
lp15@59751
  1750
next
lp15@59751
  1751
  case False
lp15@59751
  1752
  then have "Im z = 0" "Re z < 0 \<or> z = 0"
lp15@59751
  1753
    using False cnj.code complex_cnj_zero_iff by auto force
lp15@59751
  1754
  then show ?thesis
lp15@59751
  1755
    apply (auto simp: continuous_within_closed_nontrivial [OF closed_Real_halfspace_Re_ge])
lp15@59751
  1756
    apply (auto simp: continuous_within_eps_delta norm_conv_dist [symmetric])
lp15@59751
  1757
    apply (rule_tac x="e^2" in exI)
lp15@59751
  1758
    apply (auto simp: Reals_def)
lp15@59751
  1759
by (metis linear not_less real_sqrt_less_iff real_sqrt_pow2_iff real_sqrt_power)
lp15@59751
  1760
qed
lp15@59751
  1761
lp15@59870
  1762
subsection{*Complex arctangent*}
lp15@59870
  1763
lp15@59870
  1764
text{*branch cut gives standard bounds in real case.*}
lp15@59870
  1765
lp15@59870
  1766
definition Arctan :: "complex \<Rightarrow> complex" where
lp15@59870
  1767
    "Arctan \<equiv> \<lambda>z. (\<i>/2) * Ln((1 - \<i>*z) / (1 + \<i>*z))"
lp15@59870
  1768
lp15@59870
  1769
lemma Arctan_0 [simp]: "Arctan 0 = 0"
lp15@59870
  1770
  by (simp add: Arctan_def)
lp15@59870
  1771
lp15@59870
  1772
lemma Im_complex_div_lemma: "Im((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<longleftrightarrow> Re z = 0"
lp15@59870
  1773
  by (auto simp: Im_complex_div_eq_0 algebra_simps)
lp15@59870
  1774
lp15@59870
  1775
lemma Re_complex_div_lemma: "0 < Re((1 - \<i>*z) / (1 + \<i>*z)) \<longleftrightarrow> norm z < 1"
lp15@59870
  1776
  by (simp add: Re_complex_div_gt_0 algebra_simps cmod_def power2_eq_square)
lp15@59870
  1777
lp15@59870
  1778
lemma tan_Arctan:
lp15@59870
  1779
  assumes "z\<^sup>2 \<noteq> -1"
lp15@59870
  1780
    shows [simp]:"tan(Arctan z) = z"
lp15@59870
  1781
proof -
lp15@59870
  1782
  have "1 + \<i>*z \<noteq> 0"
lp15@59870
  1783
    by (metis assms complex_i_mult_minus i_squared minus_unique power2_eq_square power2_minus)
lp15@59870
  1784
  moreover
lp15@59870
  1785
  have "1 - \<i>*z \<noteq> 0"
lp15@59870
  1786
    by (metis assms complex_i_mult_minus i_squared power2_eq_square power2_minus right_minus_eq)
lp15@59870
  1787
  ultimately
lp15@59870
  1788
  show ?thesis
lp15@59870
  1789
    by (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus csqrt_exp_Ln [symmetric]
lp15@59870
  1790
                  divide_simps power2_eq_square [symmetric])
lp15@59870
  1791
qed
lp15@59870
  1792
lp15@59870
  1793
lemma Arctan_tan [simp]:
lp15@59870
  1794
  assumes "\<bar>Re z\<bar> < pi/2"
lp15@59870
  1795
    shows "Arctan(tan z) = z"
lp15@59870
  1796
proof -
lp15@59870
  1797
  have ge_pi2: "\<And>n::int. abs (of_int (2*n + 1) * pi/2) \<ge> pi/2"
lp15@59870
  1798
    by (case_tac n rule: int_cases) (auto simp: abs_mult)
lp15@59870
  1799
  have "exp (\<i>*z)*exp (\<i>*z) = -1 \<longleftrightarrow> exp (2*\<i>*z) = -1"
lp15@59870
  1800
    by (metis distrib_right exp_add mult_2)
lp15@59870
  1801
  also have "... \<longleftrightarrow> exp (2*\<i>*z) = exp (\<i>*pi)"
lp15@59870
  1802
    using cis_conv_exp cis_pi by auto
lp15@59870
  1803
  also have "... \<longleftrightarrow> exp (2*\<i>*z - \<i>*pi) = 1"
lp15@59870
  1804
    by (metis (no_types) diff_add_cancel diff_minus_eq_add exp_add exp_minus_inverse mult.commute)
lp15@59870
  1805
  also have "... \<longleftrightarrow> Re(\<i>*2*z - \<i>*pi) = 0 \<and> (\<exists>n::int. Im(\<i>*2*z - \<i>*pi) = of_int (2 * n) * pi)"
lp15@59870
  1806
    by (simp add: exp_eq_1)
lp15@59870
  1807
  also have "... \<longleftrightarrow> Im z = 0 \<and> (\<exists>n::int. 2 * Re z = of_int (2*n + 1) * pi)"
lp15@59870
  1808
    by (simp add: algebra_simps)
lp15@59870
  1809
  also have "... \<longleftrightarrow> False"
lp15@59870
  1810
    using assms ge_pi2
lp15@59870
  1811
    apply (auto simp: algebra_simps)
lp15@59870
  1812
    by (metis abs_mult_pos not_less not_real_of_nat_less_zero real_of_nat_numeral)
lp15@59870
  1813
  finally have *: "exp (\<i>*z)*exp (\<i>*z) + 1 \<noteq> 0"
lp15@59870
  1814
    by (auto simp: add.commute minus_unique)
lp15@59870
  1815
  show ?thesis
lp15@59870
  1816
    using assms *
lp15@59870
  1817
    apply (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus divide_simps
lp15@59870
  1818
                     ii_times_eq_iff power2_eq_square [symmetric])
lp15@59870
  1819
    apply (rule Ln_unique)
lp15@59870
  1820
    apply (auto simp: divide_simps exp_minus)
lp15@59870
  1821
    apply (simp add: algebra_simps exp_double [symmetric])
lp15@59870
  1822
    done
lp15@59870
  1823
qed
lp15@59870
  1824
lp15@59870
  1825
lemma
lp15@59870
  1826
  assumes "Re z = 0 \<Longrightarrow> abs(Im z) < 1"
lp15@59870
  1827
  shows Re_Arctan_bounds: "abs(Re(Arctan z)) < pi/2"
lp15@59870
  1828
    and has_field_derivative_Arctan: "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
lp15@59870
  1829
proof -
lp15@59870
  1830
  have nz0: "1 + \<i>*z \<noteq> 0"
lp15@59870
  1831
    using assms
lp15@60141
  1832
    by (metis abs_one complex_i_mult_minus diff_0_right diff_minus_eq_add ii.simps(1) ii.simps(2)
lp15@59870
  1833
              less_irrefl minus_diff_eq mult.right_neutral right_minus_eq)
lp15@59870
  1834
  have "z \<noteq> -\<i>" using assms
lp15@59870
  1835
    by auto
lp15@59870
  1836
  then have zz: "1 + z * z \<noteq> 0"
lp15@59870
  1837
    by (metis abs_one assms i_squared ii.simps less_irrefl minus_unique square_eq_iff)
lp15@59870
  1838
  have nz1: "1 - \<i>*z \<noteq> 0"
lp15@59870
  1839
    using assms by (force simp add: ii_times_eq_iff)
lp15@59870
  1840
  have nz2: "inverse (1 + \<i>*z) \<noteq> 0"
lp15@59870
  1841
    using assms
lp15@59870
  1842
    by (metis Im_complex_div_lemma Re_complex_div_lemma cmod_eq_Im divide_complex_def
lp15@59870
  1843
              less_irrefl mult_zero_right zero_complex.simps(1) zero_complex.simps(2))
lp15@59870
  1844
  have nzi: "((1 - \<i>*z) * inverse (1 + \<i>*z)) \<noteq> 0"
lp15@59870
  1845
    using nz1 nz2 by auto
lp15@59870
  1846
  have *: "Im ((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<Longrightarrow> 0 < Re ((1 - \<i>*z) / (1 + \<i>*z))"
lp15@59870
  1847
    apply (simp add: divide_complex_def)
lp15@59870
  1848
    apply (simp add: divide_simps split: split_if_asm)
lp15@59870
  1849
    using assms
lp15@59870
  1850
    apply (auto simp: algebra_simps abs_square_less_1 [unfolded power2_eq_square])
lp15@59870
  1851
    done
lp15@59870
  1852
  show "abs(Re(Arctan z)) < pi/2"
lp15@59870
  1853
    unfolding Arctan_def divide_complex_def
lp15@59870
  1854
    using mpi_less_Im_Ln [OF nzi]
lp15@59870
  1855
    by (auto simp: abs_if intro: Im_Ln_less_pi * [unfolded divide_complex_def])
lp15@59870
  1856
  show "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
lp15@59870
  1857
    unfolding Arctan_def scaleR_conv_of_real
lp15@59870
  1858
    apply (rule DERIV_cong)
lp15@59870
  1859
    apply (intro derivative_eq_intros | simp add: nz0 *)+
lp15@59870
  1860
    using nz0 nz1 zz
lp15@59870
  1861
    apply (simp add: divide_simps power2_eq_square)
lp15@59870
  1862
    apply (auto simp: algebra_simps)
lp15@59870
  1863
    done
lp15@59870
  1864
qed
lp15@59870
  1865
lp15@59870
  1866
lemma complex_differentiable_at_Arctan: "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable at z"
lp15@59870
  1867
  using has_field_derivative_Arctan
lp15@59870
  1868
  by (auto simp: complex_differentiable_def)
lp15@59870
  1869
lp15@59870
  1870
lemma complex_differentiable_within_Arctan:
lp15@59870
  1871
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable (at z within s)"
lp15@59870
  1872
  using complex_differentiable_at_Arctan complex_differentiable_at_within by blast
lp15@59870
  1873
lp15@59870
  1874
declare has_field_derivative_Arctan [derivative_intros]
lp15@59870
  1875
declare has_field_derivative_Arctan [THEN DERIV_chain2, derivative_intros]
lp15@59870
  1876
lp15@59870
  1877
lemma continuous_at_Arctan:
lp15@59870
  1878
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z) Arctan"
lp15@59870
  1879
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Arctan)
lp15@59870
  1880
lp15@59870
  1881
lemma continuous_within_Arctan:
lp15@59870
  1882
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z within s) Arctan"
lp15@59870
  1883
  using continuous_at_Arctan continuous_at_imp_continuous_within by blast
lp15@59870
  1884
lp15@59870
  1885
lemma continuous_on_Arctan [continuous_intros]:
lp15@59870
  1886
    "(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous_on s Arctan"
lp15@59870
  1887
  by (auto simp: continuous_at_imp_continuous_on continuous_within_Arctan)
lp15@59870
  1888
lp15@59870
  1889
lemma holomorphic_on_Arctan:
lp15@59870
  1890
    "(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan holomorphic_on s"
lp15@59870
  1891
  by (simp add: complex_differentiable_within_Arctan holomorphic_on_def)
lp15@59870
  1892
lp15@59870
  1893
lp15@59870
  1894
subsection {*Real arctangent*}
lp15@59870
  1895
lp15@59870
  1896
lemma norm_exp_ii_times [simp]: "norm (exp(\<i> * of_real y)) = 1"
lp15@59870
  1897
  by simp
lp15@59870
  1898
lp15@59870
  1899
lemma norm_exp_imaginary: "norm(exp z) = 1 \<Longrightarrow> Re z = 0"
lp15@59870
  1900
  by (simp add: complex_norm_eq_1_exp)
lp15@59870
  1901
lp15@59870
  1902
lemma Im_Arctan_of_real [simp]: "Im (Arctan (of_real x)) = 0"
lp15@59870
  1903
  unfolding Arctan_def divide_complex_def
lp15@59870
  1904
  apply (simp add: complex_eq_iff)
lp15@59870
  1905
  apply (rule norm_exp_imaginary)
lp15@59870
  1906
  apply (subst exp_Ln, auto)
lp15@59870
  1907
  apply (simp_all add: cmod_def complex_eq_iff)
lp15@59870
  1908
  apply (auto simp: divide_simps)
lp15@59870
  1909
  apply (metis power_one realpow_two_sum_zero_iff zero_neq_one, algebra)
lp15@59870
  1910
  done
lp15@59870
  1911
lp15@59870
  1912
lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
lp15@59870
  1913
proof (rule arctan_unique)
lp15@59870
  1914
  show "- (pi / 2) < Re (Arctan (complex_of_real x))"
lp15@59870
  1915
    apply (simp add: Arctan_def)
lp15@59870
  1916
    apply (rule Im_Ln_less_pi)
lp15@59870
  1917
    apply (auto simp: Im_complex_div_lemma)
lp15@59870
  1918
    done
lp15@59870
  1919
next
lp15@59870
  1920
  have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
lp15@59870
  1921
    by (simp add: divide_simps) ( simp add: complex_eq_iff)
lp15@59870
  1922
  show "Re (Arctan (complex_of_real x)) < pi / 2"
lp15@59870
  1923
    using mpi_less_Im_Ln [OF *]
lp15@59870
  1924
    by (simp add: Arctan_def)
lp15@59870
  1925
next
lp15@59870
  1926
  have "tan (Re (Arctan (of_real x))) = Re (tan (Arctan (of_real x)))"
lp15@59870
  1927
    apply (auto simp: tan_def Complex.Re_divide Re_sin Re_cos Im_sin Im_cos)
lp15@59870
  1928
    apply (simp add: field_simps)
lp15@59870
  1929
    by (simp add: power2_eq_square)
lp15@59870
  1930
  also have "... = x"
lp15@59870
  1931
    apply (subst tan_Arctan, auto)
lp15@59870
  1932
    by (metis diff_0_right minus_diff_eq mult_zero_left not_le of_real_1 of_real_eq_iff of_real_minus of_real_power power2_eq_square real_minus_mult_self_le zero_less_one)
lp15@59870
  1933
  finally show "tan (Re (Arctan (complex_of_real x))) = x" .
lp15@59870
  1934
qed
lp15@59870
  1935
lp15@59870
  1936
lemma Arctan_of_real: "Arctan (of_real x) = of_real (arctan x)"
lp15@59870
  1937
  unfolding arctan_eq_Re_Arctan divide_complex_def
lp15@59870
  1938
  by (simp add: complex_eq_iff)
lp15@59870
  1939
lp15@59870
  1940
lemma Arctan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Arctan z \<in> \<real>"
lp15@59870
  1941
  by (metis Reals_cases Reals_of_real Arctan_of_real)
lp15@59870
  1942
lp15@59870
  1943
declare arctan_one [simp]
lp15@59870
  1944
lp15@59870
  1945
lemma arctan_less_pi4_pos: "x < 1 \<Longrightarrow> arctan x < pi/4"
lp15@59870
  1946
  by (metis arctan_less_iff arctan_one)
lp15@59870
  1947
lp15@59870
  1948
lemma arctan_less_pi4_neg: "-1 < x \<Longrightarrow> -(pi/4) < arctan x"
lp15@59870
  1949
  by (metis arctan_less_iff arctan_minus arctan_one)
lp15@59870
  1950
lp15@59870
  1951
lemma arctan_less_pi4: "abs x < 1 \<Longrightarrow> abs(arctan x) < pi/4"
lp15@59870
  1952
  by (metis abs_less_iff arctan_less_pi4_pos arctan_minus)
lp15@59870
  1953
lp15@59870
  1954
lemma arctan_le_pi4: "abs x \<le> 1 \<Longrightarrow> abs(arctan x) \<le> pi/4"
lp15@59870
  1955
  by (metis abs_le_iff arctan_le_iff arctan_minus arctan_one)
lp15@59870
  1956
lp15@59870
  1957
lemma abs_arctan: "abs(arctan x) = arctan(abs x)"
lp15@59870
  1958
  by (simp add: abs_if arctan_minus)
lp15@59870
  1959
lp15@59870
  1960
lemma arctan_add_raw:
lp15@59870
  1961
  assumes "abs(arctan x + arctan y) < pi/2"
lp15@59870
  1962
    shows "arctan x + arctan y = arctan((x + y) / (1 - x * y))"
lp15@59870
  1963
proof (rule arctan_unique [symmetric])
lp15@59870
  1964
  show 12: "- (pi / 2) < arctan x + arctan y" "arctan x + arctan y < pi / 2"
lp15@59870
  1965
    using assms by linarith+
lp15@59870
  1966
  show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
lp15@59870
  1967
    using cos_gt_zero_pi [OF 12]
lp15@59870
  1968
    by (simp add: arctan tan_add)
lp15@59870
  1969
qed
lp15@59870
  1970
lp15@59870
  1971
lemma arctan_inverse:
lp15@59870
  1972
  assumes "0 < x"
lp15@59870
  1973
    shows "arctan(inverse x) = pi/2 - arctan x"
lp15@59870
  1974
proof -
lp15@59870
  1975
  have "arctan(inverse x) = arctan(inverse(tan(arctan x)))"
lp15@59870
  1976
    by (simp add: arctan)
lp15@59870
  1977
  also have "... = arctan (tan (pi / 2 - arctan x))"
lp15@59870
  1978
    by (simp add: tan_cot)
lp15@59870
  1979
  also have "... = pi/2 - arctan x"
lp15@59870
  1980
  proof -
lp15@59870
  1981
    have "0 < pi - arctan x"
lp15@59870
  1982
    using arctan_ubound [of x] pi_gt_zero by linarith
lp15@59870
  1983
    with assms show ?thesis
lp15@59870
  1984
      by (simp add: Transcendental.arctan_tan)
lp15@59870
  1985
  qed
lp15@59870
  1986
  finally show ?thesis .
lp15@59870
  1987
qed
lp15@59870
  1988
lp15@59870
  1989
lemma arctan_add_small:
lp15@59870
  1990
  assumes "abs(x * y) < 1"
lp15@59870
  1991
    shows "(arctan x + arctan y = arctan((x + y) / (1 - x * y)))"
lp15@59870
  1992
proof (cases "x = 0 \<or> y = 0")
lp15@59870
  1993
  case True then show ?thesis
lp15@59870
  1994
    by auto
lp15@59870
  1995
next
lp15@59870
  1996
  case False
lp15@59870
  1997
  then have *: "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
lp15@59870
  1998
    apply (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
lp15@59870
  1999
    apply (simp add: divide_simps abs_mult)
lp15@59870
  2000
    done
lp15@59870
  2001
  show ?thesis
lp15@59870
  2002
    apply (rule arctan_add_raw)
lp15@59870
  2003
    using * by linarith
lp15@59870
  2004
qed
lp15@59870
  2005
lp15@59870
  2006
lemma abs_arctan_le:
lp15@59870
  2007
  fixes x::real shows "abs(arctan x) \<le> abs x"
lp15@59870
  2008
proof -
lp15@59870
  2009
  { fix w::complex and z::complex
lp15@59870
  2010
    assume *: "w \<in> \<real>" "z \<in> \<real>"
lp15@59870
  2011
    have "cmod (Arctan w - Arctan z) \<le> 1 * cmod (w-z)"
lp15@59870
  2012
      apply (rule complex_differentiable_bound [OF convex_Reals, of Arctan _ 1])
lp15@59870
  2013
      apply (rule has_field_derivative_at_within [OF has_field_derivative_Arctan])
lp15@59870
  2014
      apply (force simp add: Reals_def)
lp15@59870
  2015
      apply (simp add: norm_divide divide_simps in_Reals_norm complex_is_Real_iff power2_eq_square)
lp15@59870
  2016
      using * by auto
lp15@59870
  2017
  }
lp15@59870
  2018
  then have "cmod (Arctan (of_real x) - Arctan 0) \<le> 1 * cmod (of_real x -0)"
lp15@59870
  2019
    using Reals_0 Reals_of_real by blast
lp15@59870
  2020
  then show ?thesis
lp15@59870
  2021
    by (simp add: Arctan_of_real)
lp15@59870
  2022
qed
lp15@59870
  2023
lp15@59870
  2024
lemma arctan_le_self: "0 \<le> x \<Longrightarrow> arctan x \<le> x"
lp15@59870
  2025
  by (metis abs_arctan_le abs_of_nonneg zero_le_arctan_iff)
lp15@59870
  2026
lp15@59870
  2027
lemma abs_tan_ge: "abs x < pi/2 \<Longrightarrow> abs x \<le> abs(tan x)"
lp15@59870
  2028
  by (metis abs_arctan_le abs_less_iff arctan_tan minus_less_iff)
lp15@59870
  2029
lp15@59870
  2030
lp15@59870
  2031
subsection{*Inverse Sine*}
lp15@59870
  2032
lp15@59870
  2033
definition Arcsin :: "complex \<Rightarrow> complex" where
lp15@59870
  2034
   "Arcsin \<equiv> \<lambda>z. -\<i> * Ln(\<i> * z + csqrt(1 - z\<^sup>2))"
lp15@59870
  2035
lp15@59870
  2036
lemma Arcsin_body_lemma: "\<i> * z + csqrt(1 - z\<^sup>2) \<noteq> 0"
lp15@59870
  2037
  using power2_csqrt [of "1 - z\<^sup>2"]
lp15@59870
  2038
  apply auto
lp15@59870
  2039
  by (metis complex_i_mult_minus diff_add_cancel diff_minus_eq_add diff_self mult.assoc mult.left_commute numeral_One power2_csqrt power2_eq_square zero_neq_numeral)
lp15@59870
  2040
lp15@59870
  2041
lemma Arcsin_range_lemma: "abs (Re z) < 1 \<Longrightarrow> 0 < Re(\<i> * z + csqrt(1 - z\<^sup>2))"
lp15@59870
  2042
  using Complex.cmod_power2 [of z, symmetric]
lp15@59870
  2043
  by (simp add: real_less_rsqrt algebra_simps Re_power2 cmod_square_less_1_plus)
lp15@59870
  2044
lp15@59870
  2045
lemma Re_Arcsin: "Re(Arcsin z) = Im (Ln (\<i> * z + csqrt(1 - z\<^sup>2)))"
lp15@59870
  2046
  by (simp add: Arcsin_def)
lp15@59870
  2047
lp15@59870
  2048
lemma Im_Arcsin: "Im(Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"
lp15@59870
  2049
  by (simp add: Arcsin_def Arcsin_body_lemma)
lp15@59870
  2050
lp15@59870
  2051
lemma isCont_Arcsin:
lp15@59870
  2052
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  2053
    shows "isCont Arcsin z"
lp15@59870
  2054
proof -
lp15@59870
  2055
  have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
lp15@59870
  2056
    using assms
lp15@59870
  2057
    by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
lp15@59870
  2058
  have cmz: "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
lp15@59870
  2059
    by (blast intro: assms cmod_square_less_1_plus)
lp15@59870
  2060
  show ?thesis
lp15@59870
  2061
    using assms
lp15@59870
  2062
    apply (simp add: Arcsin_def)
lp15@59870
  2063
    apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
lp15@59870
  2064
    apply (erule rez)
lp15@59870
  2065
    apply (auto simp: Re_power2 Im_power2 abs_square_less_1 [symmetric] real_less_rsqrt algebra_simps split: split_if_asm)
lp15@59870
  2066
    apply (simp add: norm_complex_def)
lp15@59870
  2067
    using cmod_power2 [of z, symmetric] cmz
lp15@59870
  2068
    apply (simp add: real_less_rsqrt)
lp15@59870
  2069
    done
lp15@59870
  2070
qed
lp15@59870
  2071
lp15@59870
  2072
lemma isCont_Arcsin' [simp]:
lp15@59870
  2073
  shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arcsin (f x)) z"
lp15@59870
  2074
  by (blast intro: isCont_o2 [OF _ isCont_Arcsin])
lp15@59870
  2075
lp15@59870
  2076
lemma sin_Arcsin [simp]: "sin(Arcsin z) = z"
lp15@60141
  2077
proof -
lp15@59870
  2078
  have "\<i>*z*2 + csqrt (1 - z\<^sup>2)*2 = 0 \<longleftrightarrow> (\<i>*z)*2 + csqrt (1 - z\<^sup>2)*2 = 0"
lp15@59870
  2079
    by (simp add: algebra_simps)  --{*Cancelling a factor of 2*}
lp15@59870
  2080
  moreover have "... \<longleftrightarrow> (\<i>*z) + csqrt (1 - z\<^sup>2) = 0"
lp15@59870
  2081
    by (metis Arcsin_body_lemma distrib_right no_zero_divisors zero_neq_numeral)
lp15@59870
  2082
  ultimately show ?thesis
lp15@59870
  2083
    apply (simp add: sin_exp_eq Arcsin_def Arcsin_body_lemma exp_minus divide_simps)
lp15@59870
  2084
    apply (simp add: algebra_simps)
lp15@59870
  2085
    apply (simp add: power2_eq_square [symmetric] algebra_simps)
lp15@59870
  2086
    done
lp15@59870
  2087
qed
lp15@59870
  2088
lp15@59870
  2089
lemma Re_eq_pihalf_lemma:
lp15@59870
  2090
    "\<bar>Re z\<bar> = pi/2 \<Longrightarrow> Im z = 0 \<Longrightarrow>
lp15@59870
  2091
      Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2) = 0 \<and> 0 \<le> Im ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
lp15@59870
  2092
  apply (simp add: cos_ii_times [symmetric] Re_cos Im_cos abs_if del: eq_divide_eq_numeral1)
lp15@59870
  2093
  by (metis cos_minus cos_pi_half)
lp15@59870
  2094
lp15@59870
  2095
lemma Re_less_pihalf_lemma:
lp15@59870
  2096
  assumes "\<bar>Re z\<bar> < pi / 2"
lp15@59870
  2097
    shows "0 < Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
lp15@59870
  2098
proof -
lp15@59870
  2099
  have "0 < cos (Re z)" using assms
lp15@59870
  2100
    using cos_gt_zero_pi by auto
lp15@59870
  2101
  then show ?thesis
lp15@59870
  2102
    by (simp add: cos_ii_times [symmetric] Re_cos Im_cos add_pos_pos)
lp15@59870
  2103
qed
lp15@59870
  2104
lp15@59870
  2105
lemma Arcsin_sin:
lp15@59870
  2106
    assumes "\<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)"
lp15@59870
  2107
      shows "Arcsin(sin z) = z"
lp15@59870
  2108
proof -
lp15@59870
  2109
  have "Arcsin(sin z) = - (\<i> * Ln (csqrt (1 - (\<i> * (exp (\<i>*z) - inverse (exp (\<i>*z))))\<^sup>2 / 4) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
lp15@59870
  2110
    by (simp add: sin_exp_eq Arcsin_def exp_minus)
lp15@59870
  2111
  also have "... = - (\<i> * Ln (csqrt (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2)\<^sup>2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
lp15@59870
  2112
    by (simp add: field_simps power2_eq_square)
lp15@59870
  2113
  also have "... = - (\<i> * Ln (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
lp15@59870
  2114
    apply (subst csqrt_square)
lp15@59870
  2115
    using assms Re_eq_pihalf_lemma Re_less_pihalf_lemma
lp15@59870
  2116
    apply auto
lp15@59870
  2117
    done
lp15@59870
  2118
  also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
lp15@59870
  2119
    by (simp add: field_simps power2_eq_square)
lp15@59870
  2120
  also have "... = z"
lp15@59870
  2121
    apply (subst Complex_Transcendental.Ln_exp)
lp15@59870
  2122
    using assms
lp15@59870
  2123
    apply (auto simp: abs_if simp del: eq_divide_eq_numeral1 split: split_if_asm)
lp15@59870
  2124
    done
lp15@59870
  2125
  finally show ?thesis .
lp15@59870
  2126
qed
lp15@59870
  2127
lp15@59870
  2128
lemma Arcsin_unique:
lp15@59870
  2129
    "\<lbrakk>sin z = w; \<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)\<rbrakk> \<Longrightarrow> Arcsin w = z"
lp15@59870
  2130
  by (metis Arcsin_sin)
lp15@59870
  2131
lp15@59870
  2132
lemma Arcsin_0 [simp]: "Arcsin 0 = 0"
lp15@59870
  2133
  by (metis Arcsin_sin norm_zero pi_half_gt_zero real_norm_def sin_zero zero_complex.simps(1))
lp15@59870
  2134
lp15@59870
  2135
lemma Arcsin_1 [simp]: "Arcsin 1 = pi/2"
lp15@59870
  2136
  by (metis Arcsin_sin Im_complex_of_real Re_complex_of_real numeral_One of_real_numeral pi_half_ge_zero real_sqrt_abs real_sqrt_pow2 real_sqrt_power sin_of_real sin_pi_half)
lp15@59870
  2137
lp15@59870
  2138
lemma Arcsin_minus_1 [simp]: "Arcsin(-1) = - (pi/2)"
lp15@59870
  2139
  by (metis Arcsin_1 Arcsin_sin Im_complex_of_real Re_complex_of_real abs_of_nonneg of_real_minus pi_half_ge_zero power2_minus real_sqrt_abs sin_Arcsin sin_minus)
lp15@59870
  2140
lp15@59870
  2141
lemma has_field_derivative_Arcsin:
lp15@59870
  2142
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  2143
    shows "(Arcsin has_field_derivative inverse(cos(Arcsin z))) (at z)"
lp15@59870
  2144
proof -
lp15@59870
  2145
  have "(sin (Arcsin z))\<^sup>2 \<noteq> 1"
lp15@59870
  2146
    using assms
lp15@59870
  2147
    apply atomize
lp15@59870
  2148
    apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
lp15@59870
  2149
    apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
lp15@59870
  2150
    by (metis abs_minus_cancel abs_one abs_power2 one_neq_neg_one)
lp15@59870
  2151
  then have "cos (Arcsin z) \<noteq> 0"
lp15@59870
  2152
    by (metis diff_0_right power_zero_numeral sin_squared_eq)
lp15@59870
  2153
  then show ?thesis
lp15@59870
  2154
    apply (rule has_complex_derivative_inverse_basic [OF DERIV_sin])
lp15@59870
  2155
    apply (auto intro: isCont_Arcsin open_ball [of z 1] assms)
lp15@59870
  2156
    done
lp15@59870
  2157
qed
lp15@59870
  2158
lp15@59870
  2159
declare has_field_derivative_Arcsin [derivative_intros]
lp15@59870
  2160
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]
lp15@59870
  2161
lp15@59870
  2162
lemma complex_differentiable_at_Arcsin:
lp15@59870
  2163
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable at z"
lp15@59870
  2164
  using complex_differentiable_def has_field_derivative_Arcsin by blast
lp15@59870
  2165
lp15@59870
  2166
lemma complex_differentiable_within_Arcsin:
lp15@59870
  2167
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable (at z within s)"
lp15@59870
  2168
  using complex_differentiable_at_Arcsin complex_differentiable_within_subset by blast
lp15@59870
  2169
lp15@59870
  2170
lemma continuous_within_Arcsin:
lp15@59870
  2171
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arcsin"
lp15@59870
  2172
  using continuous_at_imp_continuous_within isCont_Arcsin by blast
lp15@59870
  2173
lp15@59870
  2174
lemma continuous_on_Arcsin [continuous_intros]:
lp15@59870
  2175
    "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arcsin"
lp15@59870
  2176
  by (simp add: continuous_at_imp_continuous_on)
lp15@59870
  2177
lp15@59870
  2178
lemma holomorphic_on_Arcsin: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin holomorphic_on s"
lp15@59870
  2179
  by (simp add: complex_differentiable_within_Arcsin holomorphic_on_def)
lp15@59870
  2180
lp15@59870
  2181
lp15@59870
  2182
subsection{*Inverse Cosine*}
lp15@59870
  2183
lp15@59870
  2184
definition Arccos :: "complex \<Rightarrow> complex" where
lp15@59870
  2185
   "Arccos \<equiv> \<lambda>z. -\<i> * Ln(z + \<i> * csqrt(1 - z\<^sup>2))"
lp15@59870
  2186
lp15@59870
  2187
lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"
lp15@59870
  2188
  using Arcsin_range_lemma [of "-z"]
lp15@59870
  2189
  by simp
lp15@59870
  2190
lp15@59870
  2191
lemma Arccos_body_lemma: "z + \<i> * csqrt(1 - z\<^sup>2) \<noteq> 0"
lp15@59870
  2192
  using Arcsin_body_lemma [of z]
lp15@59870
  2193
  by (metis complex_i_mult_minus diff_add_cancel minus_diff_eq minus_unique mult.assoc mult.left_commute
lp15@59870
  2194
           power2_csqrt power2_eq_square zero_neq_one)
lp15@59870
  2195
lp15@59870
  2196
lemma Re_Arccos: "Re(Arccos z) = Im (Ln (z + \<i> * csqrt(1 - z\<^sup>2)))"
lp15@59870
  2197
  by (simp add: Arccos_def)
lp15@59870
  2198
lp15@59870
  2199
lemma Im_Arccos: "Im(Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"
lp15@59870
  2200
  by (simp add: Arccos_def Arccos_body_lemma)
lp15@59870
  2201
lp15@59870
  2202
text{*A very tricky argument to find!*}
lp15@59870
  2203
lemma abs_Re_less_1_preserve:
lp15@59870
  2204
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"  "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0"
lp15@59870
  2205
    shows "0 < Re (z + \<i> * csqrt (1 - z\<^sup>2))"
lp15@59870
  2206
proof (cases "Im z = 0")
lp15@59870
  2207
  case True
lp15@59870
  2208
  then show ?thesis
lp15@60141
  2209
    using assms
lp15@59870
  2210
    by (fastforce simp add: cmod_def Re_power2 Im_power2 algebra_simps abs_square_less_1 [symmetric])
lp15@59870
  2211
next
lp15@59870
  2212
  case False
lp15@59870
  2213
  have Imz: "Im z = - sqrt ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
lp15@59870
  2214
    using assms abs_Re_le_cmod [of "1-z\<^sup>2"]
lp15@59870
  2215
    by (simp add: Re_power2 algebra_simps)
lp15@59870
  2216
  have "(cmod z)\<^sup>2 - 1 \<noteq> cmod (1 - z\<^sup>2)"
lp15@59870
  2217
  proof (clarsimp simp add: cmod_def)
lp15@59870
  2218
    assume "(Re z)\<^sup>2 + (Im z)\<^sup>2 - 1 = sqrt ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
lp15@59870
  2219
    then have "((Re z)\<^sup>2 + (Im z)\<^sup>2 - 1)\<^sup>2 = ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
lp15@59870
  2220
      by simp
lp15@59870
  2221
    then show False using False
lp15@59870
  2222
      by (simp add: power2_eq_square algebra_simps)
lp15@59870
  2223
  qed
lp15@59870
  2224
  moreover have "(Im z)\<^sup>2 = ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
lp15@59870
  2225
    apply (subst Imz, simp)
lp15@59870
  2226
    apply (subst real_sqrt_pow2)
lp15@59870
  2227
    using abs_Re_le_cmod [of "1-z\<^sup>2"]
lp15@59870
  2228
    apply (auto simp: Re_power2 field_simps)
lp15@59870
  2229
    done
lp15@59870
  2230
  ultimately show ?thesis
lp15@59870
  2231
    by (simp add: Re_power2 Im_power2 cmod_power2)
lp15@59870
  2232
qed
lp15@59870
  2233
lp15@59870
  2234
lemma isCont_Arccos:
lp15@59870
  2235
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  2236
    shows "isCont Arccos z"
lp15@59870
  2237
proof -
lp15@59870
  2238
  have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
lp15@59870
  2239
    using assms
lp15@59870
  2240
    by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
lp15@59870
  2241
  show ?thesis
lp15@59870
  2242
    using assms
lp15@59870
  2243
    apply (simp add: Arccos_def)
lp15@59870
  2244
    apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
lp15@59870
  2245
    apply (erule rez)
lp15@59870
  2246
    apply (blast intro: abs_Re_less_1_preserve)
lp15@59870
  2247
    done
lp15@59870
  2248
qed
lp15@59870
  2249
lp15@59870
  2250
lemma isCont_Arccos' [simp]:
lp15@59870
  2251
  shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arccos (f x)) z"
lp15@59870
  2252
  by (blast intro: isCont_o2 [OF _ isCont_Arccos])
lp15@59870
  2253
lp15@59870
  2254
lemma cos_Arccos [simp]: "cos(Arccos z) = z"
lp15@59870
  2255
proof -
lp15@59870
  2256
  have "z*2 + \<i> * (2 * csqrt (1 - z\<^sup>2)) = 0 \<longleftrightarrow> z*2 + \<i> * csqrt (1 - z\<^sup>2)*2 = 0"
lp15@59870
  2257
    by (simp add: algebra_simps)  --{*Cancelling a factor of 2*}
lp15@59870
  2258
  moreover have "... \<longleftrightarrow> z + \<i> * csqrt (1 - z\<^sup>2) = 0"
lp15@59870
  2259
    by (metis distrib_right mult_eq_0_iff zero_neq_numeral)
lp15@59870
  2260
  ultimately show ?thesis
lp15@59870
  2261
    apply (simp add: cos_exp_eq Arccos_def Arccos_body_lemma exp_minus field_simps)
lp15@59870
  2262
    apply (simp add: power2_eq_square [symmetric])
lp15@59870
  2263
    done
lp15@59870
  2264
qed
lp15@59870
  2265
lp15@59870
  2266
lemma Arccos_cos:
lp15@59870
  2267
    assumes "0 < Re z & Re z < pi \<or>
lp15@59870
  2268
             Re z = 0 & 0 \<le> Im z \<or>
lp15@59870
  2269
             Re z = pi & Im z \<le> 0"
lp15@59870
  2270
      shows "Arccos(cos z) = z"
lp15@59870
  2271
proof -
lp15@60017
  2272
  have *: "((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z))) = sin z"
lp15@59870
  2273
    by (simp add: sin_exp_eq exp_minus field_simps power2_eq_square)