src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
author paulson <lp15@cam.ac.uk>
Wed Apr 01 14:48:38 2015 +0100 (2015-04-01)
changeset 59870 68d6b6aa4450
parent 59862 44b3f4fa33ca
child 60017 b785d6d06430
permissions -rw-r--r--
HOL Light Libraries for complex Arctan, Arcsin, Arccos
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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_at_exp: "exp complex_differentiable at z"
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  using DERIV_exp complex_differentiable_def by blast
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lemma complex_differentiable_within_exp: "exp complex_differentiable (at z within s)"
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  by (simp add: complex_differentiable_at_exp complex_differentiable_at_within)
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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)
lp15@59746
   302
  also have "... = ?rhs"
lp15@59746
   303
    apply (auto simp: of_real_numeral)
lp15@59746
   304
    apply (rule_tac [2] x="-(x+1)" in exI)
lp15@59746
   305
    apply (rule_tac x="-(x+1)" in exI)
lp15@59746
   306
    apply (simp_all add: algebra_simps)
lp15@59746
   307
    done
lp15@59746
   308
  finally show ?thesis .
lp15@59862
   309
qed
lp15@59746
   310
lp15@59862
   311
lemma ccos_eq_minus1:
lp15@59746
   312
  fixes z::complex
lp15@59746
   313
  shows "cos z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + pi)"
lp15@59746
   314
  using csin_eq_1 [of "z - of_real pi/2"]
lp15@59746
   315
  apply (simp add: sin_diff)
lp15@59746
   316
  apply (simp add: algebra_simps of_real_numeral equation_minus_iff)
lp15@59862
   317
  done
lp15@59746
   318
lp15@59746
   319
lemma sin_eq_1: "sin x = 1 \<longleftrightarrow> (\<exists>n::int. x = (2 * n + 1 / 2) * pi)"
lp15@59746
   320
                (is "_ = ?rhs")
lp15@59746
   321
proof -
lp15@59746
   322
  have "sin x = 1 \<longleftrightarrow> sin (complex_of_real x) = 1"
lp15@59746
   323
    by (metis of_real_1 one_complex.simps(1) real_sin_eq sin_of_real)
lp15@59746
   324
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + of_real pi/2)"
lp15@59746
   325
    by (simp only: csin_eq_1)
lp15@59746
   326
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + of_real pi/2)"
lp15@59746
   327
    apply (rule iff_exI)
lp15@59746
   328
    apply (auto simp: algebra_simps of_real_numeral)
lp15@59746
   329
    apply (rule injD [OF inj_of_real [where 'a = complex]])
lp15@59746
   330
    apply (auto simp: of_real_numeral)
lp15@59746
   331
    done
lp15@59746
   332
  also have "... = ?rhs"
lp15@59746
   333
    by (auto simp: algebra_simps)
lp15@59746
   334
  finally show ?thesis .
lp15@59862
   335
qed
lp15@59746
   336
lp15@59746
   337
lemma sin_eq_minus1: "sin x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 3/2) * pi)"  (is "_ = ?rhs")
lp15@59746
   338
proof -
lp15@59746
   339
  have "sin x = -1 \<longleftrightarrow> sin (complex_of_real x) = -1"
lp15@59746
   340
    by (metis Re_complex_of_real of_real_def scaleR_minus1_left sin_of_real)
lp15@59746
   341
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + 3/2*pi)"
lp15@59746
   342
    by (simp only: csin_eq_minus1)
lp15@59746
   343
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + 3/2*pi)"
lp15@59746
   344
    apply (rule iff_exI)
lp15@59746
   345
    apply (auto simp: algebra_simps)
lp15@59746
   346
    apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
lp15@59746
   347
    done
lp15@59746
   348
  also have "... = ?rhs"
lp15@59746
   349
    by (auto simp: algebra_simps)
lp15@59746
   350
  finally show ?thesis .
lp15@59862
   351
qed
lp15@59746
   352
lp15@59746
   353
lemma cos_eq_minus1: "cos x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 1) * pi)"
lp15@59746
   354
                      (is "_ = ?rhs")
lp15@59746
   355
proof -
lp15@59746
   356
  have "cos x = -1 \<longleftrightarrow> cos (complex_of_real x) = -1"
lp15@59746
   357
    by (metis Re_complex_of_real of_real_def scaleR_minus1_left cos_of_real)
lp15@59746
   358
  also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + pi)"
lp15@59746
   359
    by (simp only: ccos_eq_minus1)
lp15@59746
   360
  also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + pi)"
lp15@59746
   361
    apply (rule iff_exI)
lp15@59746
   362
    apply (auto simp: algebra_simps)
lp15@59746
   363
    apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
lp15@59746
   364
    done
lp15@59746
   365
  also have "... = ?rhs"
lp15@59746
   366
    by (auto simp: algebra_simps)
lp15@59746
   367
  finally show ?thesis .
lp15@59862
   368
qed
lp15@59746
   369
lp15@59746
   370
lemma dist_exp_ii_1: "norm(exp(ii * of_real t) - 1) = 2 * abs(sin(t / 2))"
lp15@59862
   371
  apply (simp add: exp_Euler cmod_def power2_diff sin_of_real cos_of_real algebra_simps)
lp15@59746
   372
  using cos_double_sin [of "t/2"]
lp15@59746
   373
  apply (simp add: real_sqrt_mult)
lp15@59746
   374
  done
lp15@59746
   375
lp15@59746
   376
lemma sinh_complex:
lp15@59746
   377
  fixes z :: complex
lp15@59746
   378
  shows "(exp z - inverse (exp z)) / 2 = -ii * sin(ii * z)"
lp15@59746
   379
  by (simp add: sin_exp_eq divide_simps exp_minus of_real_numeral)
lp15@59746
   380
lp15@59746
   381
lemma sin_ii_times:
lp15@59746
   382
  fixes z :: complex
lp15@59746
   383
  shows "sin(ii * z) = ii * ((exp z - inverse (exp z)) / 2)"
lp15@59746
   384
  using sinh_complex by auto
lp15@59746
   385
lp15@59746
   386
lemma sinh_real:
lp15@59746
   387
  fixes x :: real
lp15@59746
   388
  shows "of_real((exp x - inverse (exp x)) / 2) = -ii * sin(ii * of_real x)"
lp15@59746
   389
  by (simp add: exp_of_real sin_ii_times of_real_numeral)
lp15@59746
   390
lp15@59746
   391
lemma cosh_complex:
lp15@59746
   392
  fixes z :: complex
lp15@59746
   393
  shows "(exp z + inverse (exp z)) / 2 = cos(ii * z)"
lp15@59746
   394
  by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lp15@59746
   395
lp15@59746
   396
lemma cosh_real:
lp15@59746
   397
  fixes x :: real
lp15@59746
   398
  shows "of_real((exp x + inverse (exp x)) / 2) = cos(ii * of_real x)"
lp15@59746
   399
  by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lp15@59746
   400
lp15@59746
   401
lemmas cos_ii_times = cosh_complex [symmetric]
lp15@59746
   402
lp15@59862
   403
lemma norm_cos_squared:
lp15@59746
   404
    "norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
lp15@59746
   405
  apply (cases z)
lp15@59746
   406
  apply (simp add: cos_add cmod_power2 cos_of_real sin_of_real)
lp15@59746
   407
  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
lp15@59746
   408
  apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
lp15@59746
   409
  apply (simp add: sin_squared_eq)
lp15@59746
   410
  apply (simp add: power2_eq_square algebra_simps divide_simps)
lp15@59746
   411
  done
lp15@59746
   412
lp15@59746
   413
lemma norm_sin_squared:
lp15@59746
   414
    "norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
lp15@59746
   415
  apply (cases z)
lp15@59746
   416
  apply (simp add: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double)
lp15@59746
   417
  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
lp15@59746
   418
  apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
lp15@59746
   419
  apply (simp add: cos_squared_eq)
lp15@59746
   420
  apply (simp add: power2_eq_square algebra_simps divide_simps)
lp15@59862
   421
  done
lp15@59746
   422
lp15@59746
   423
lemma exp_uminus_Im: "exp (- Im z) \<le> exp (cmod z)"
lp15@59746
   424
  using abs_Im_le_cmod linear order_trans by fastforce
lp15@59746
   425
lp15@59862
   426
lemma norm_cos_le:
lp15@59746
   427
  fixes z::complex
lp15@59746
   428
  shows "norm(cos z) \<le> exp(norm z)"
lp15@59746
   429
proof -
lp15@59746
   430
  have "Im z \<le> cmod z"
lp15@59746
   431
    using abs_Im_le_cmod abs_le_D1 by auto
lp15@59746
   432
  with exp_uminus_Im show ?thesis
lp15@59746
   433
    apply (simp add: cos_exp_eq norm_divide)
lp15@59746
   434
    apply (rule order_trans [OF norm_triangle_ineq], simp)
lp15@59746
   435
    apply (metis add_mono exp_le_cancel_iff mult_2_right)
lp15@59746
   436
    done
lp15@59746
   437
qed
lp15@59746
   438
lp15@59862
   439
lemma norm_cos_plus1_le:
lp15@59746
   440
  fixes z::complex
lp15@59746
   441
  shows "norm(1 + cos z) \<le> 2 * exp(norm z)"
lp15@59746
   442
proof -
lp15@59746
   443
  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
   444
      by arith
lp15@59746
   445
  have *: "Im z \<le> cmod z"
lp15@59746
   446
    using abs_Im_le_cmod abs_le_D1 by auto
lp15@59746
   447
  have triangle3: "\<And>x y z. norm(x + y + z) \<le> norm(x) + norm(y) + norm(z)"
lp15@59746
   448
    by (simp add: norm_add_rule_thm)
lp15@59746
   449
  have "norm(1 + cos z) = cmod (1 + (exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
lp15@59746
   450
    by (simp add: cos_exp_eq)
lp15@59746
   451
  also have "... = cmod ((2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
lp15@59746
   452
    by (simp add: field_simps)
lp15@59746
   453
  also have "... = cmod (2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2"
lp15@59746
   454
    by (simp add: norm_divide)
lp15@59746
   455
  finally show ?thesis
lp15@59746
   456
    apply (rule ssubst, simp)
lp15@59746
   457
    apply (rule order_trans [OF triangle3], simp)
lp15@59746
   458
    using exp_uminus_Im *
lp15@59746
   459
    apply (auto intro: mono)
lp15@59746
   460
    done
lp15@59746
   461
qed
lp15@59746
   462
lp15@59746
   463
subsection{* Taylor series for complex exponential, sine and cosine.*}
lp15@59746
   464
lp15@59862
   465
context
lp15@59746
   466
begin
lp15@59746
   467
lp15@59746
   468
declare power_Suc [simp del]
lp15@59746
   469
lp15@59862
   470
lemma Taylor_exp:
lp15@59746
   471
  "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
   472
proof (rule complex_taylor [of _ n "\<lambda>k. exp" "exp\<bar>Re z\<bar>" 0 z, simplified])
lp15@59746
   473
  show "convex (closed_segment 0 z)"
lp15@59746
   474
    by (rule convex_segment [of 0 z])
lp15@59746
   475
next
lp15@59746
   476
  fix k x
lp15@59746
   477
  assume "x \<in> closed_segment 0 z" "k \<le> n"
lp15@59746
   478
  show "(exp has_field_derivative exp x) (at x within closed_segment 0 z)"
lp15@59746
   479
    using DERIV_exp DERIV_subset by blast
lp15@59746
   480
next
lp15@59746
   481
  fix x
lp15@59746
   482
  assume "x \<in> closed_segment 0 z"
lp15@59746
   483
  then show "Re x \<le> \<bar>Re z\<bar>"
lp15@59746
   484
    apply (auto simp: closed_segment_def scaleR_conv_of_real)
lp15@59746
   485
    by (meson abs_ge_self abs_ge_zero linear mult_left_le_one_le mult_nonneg_nonpos order_trans)
lp15@59746
   486
next
lp15@59746
   487
  show "0 \<in> closed_segment 0 z"
lp15@59746
   488
    by (auto simp: closed_segment_def)
lp15@59746
   489
next
lp15@59746
   490
  show "z \<in> closed_segment 0 z"
lp15@59746
   491
    apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   492
    using of_real_1 zero_le_one by blast
lp15@59862
   493
qed
lp15@59746
   494
lp15@59862
   495
lemma
lp15@59746
   496
  assumes "0 \<le> u" "u \<le> 1"
lp15@59862
   497
  shows cmod_sin_le_exp: "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   498
    and cmod_cos_le_exp: "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
lp15@59746
   499
proof -
lp15@59746
   500
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   501
    by arith
lp15@59746
   502
  show "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
lp15@59746
   503
    apply (auto simp: scaleR_conv_of_real norm_mult norm_power sin_exp_eq norm_divide)
lp15@59746
   504
    apply (rule order_trans [OF norm_triangle_ineq4])
lp15@59746
   505
    apply (rule mono)
lp15@59746
   506
    apply (auto simp: abs_if mult_left_le_one_le)
lp15@59746
   507
    apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
lp15@59746
   508
    apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
lp15@59746
   509
    done
lp15@59746
   510
  show "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
lp15@59746
   511
    apply (auto simp: scaleR_conv_of_real norm_mult norm_power cos_exp_eq norm_divide)
lp15@59746
   512
    apply (rule order_trans [OF norm_triangle_ineq])
lp15@59746
   513
    apply (rule mono)
lp15@59746
   514
    apply (auto simp: abs_if mult_left_le_one_le)
lp15@59746
   515
    apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
lp15@59746
   516
    apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
lp15@59746
   517
    done
lp15@59746
   518
qed
lp15@59862
   519
lp15@59862
   520
lemma Taylor_sin:
lp15@59862
   521
  "norm(sin z - (\<Sum>k\<le>n. complex_of_real (sin_coeff k) * z ^ k))
lp15@59746
   522
   \<le> exp\<bar>Im z\<bar> * (norm z) ^ (Suc n) / (fact n)"
lp15@59746
   523
proof -
lp15@59746
   524
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   525
      by arith
lp15@59746
   526
  have *: "cmod (sin z -
lp15@59746
   527
                 (\<Sum>i\<le>n. (-1) ^ (i div 2) * (if even i then sin 0 else cos 0) * z ^ i / (fact i)))
lp15@59862
   528
           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
lp15@59746
   529
  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
   530
simplified])
lp15@59746
   531
  show "convex (closed_segment 0 z)"
lp15@59746
   532
    by (rule convex_segment [of 0 z])
lp15@59746
   533
  next
lp15@59746
   534
    fix k x
lp15@59746
   535
    show "((\<lambda>x. (- 1) ^ (k div 2) * (if even k then sin x else cos x)) has_field_derivative
lp15@59746
   536
            (- 1) ^ (Suc k div 2) * (if odd k then sin x else cos x))
lp15@59746
   537
            (at x within closed_segment 0 z)"
lp15@59746
   538
      apply (auto simp: power_Suc)
lp15@59746
   539
      apply (intro derivative_eq_intros | simp)+
lp15@59746
   540
      done
lp15@59746
   541
  next
lp15@59746
   542
    fix x
lp15@59746
   543
    assume "x \<in> closed_segment 0 z"
lp15@59746
   544
    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
   545
      by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
lp15@59746
   546
  next
lp15@59746
   547
    show "0 \<in> closed_segment 0 z"
lp15@59746
   548
      by (auto simp: closed_segment_def)
lp15@59746
   549
  next
lp15@59746
   550
    show "z \<in> closed_segment 0 z"
lp15@59746
   551
      apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   552
      using of_real_1 zero_le_one by blast
lp15@59862
   553
  qed
lp15@59746
   554
  have **: "\<And>k. complex_of_real (sin_coeff k) * z ^ k
lp15@59746
   555
            = (-1)^(k div 2) * (if even k then sin 0 else cos 0) * z^k / of_nat (fact k)"
lp15@59746
   556
    by (auto simp: sin_coeff_def elim!: oddE)
lp15@59746
   557
  show ?thesis
lp15@59746
   558
    apply (rule order_trans [OF _ *])
lp15@59746
   559
    apply (simp add: **)
lp15@59746
   560
    done
lp15@59746
   561
qed
lp15@59746
   562
lp15@59862
   563
lemma Taylor_cos:
lp15@59862
   564
  "norm(cos z - (\<Sum>k\<le>n. complex_of_real (cos_coeff k) * z ^ k))
lp15@59746
   565
   \<le> exp\<bar>Im z\<bar> * (norm z) ^ Suc n / (fact n)"
lp15@59746
   566
proof -
lp15@59746
   567
  have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
lp15@59746
   568
      by arith
lp15@59746
   569
  have *: "cmod (cos z -
lp15@59746
   570
                 (\<Sum>i\<le>n. (-1) ^ (Suc i div 2) * (if even i then cos 0 else sin 0) * z ^ i / (fact i)))
lp15@59862
   571
           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
lp15@59746
   572
  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
   573
simplified])
lp15@59746
   574
  show "convex (closed_segment 0 z)"
lp15@59746
   575
    by (rule convex_segment [of 0 z])
lp15@59746
   576
  next
lp15@59746
   577
    fix k x
lp15@59746
   578
    assume "x \<in> closed_segment 0 z" "k \<le> n"
lp15@59746
   579
    show "((\<lambda>x. (- 1) ^ (Suc k div 2) * (if even k then cos x else sin x)) has_field_derivative
lp15@59746
   580
            (- 1) ^ Suc (k div 2) * (if odd k then cos x else sin x))
lp15@59746
   581
             (at x within closed_segment 0 z)"
lp15@59746
   582
      apply (auto simp: power_Suc)
lp15@59746
   583
      apply (intro derivative_eq_intros | simp)+
lp15@59746
   584
      done
lp15@59746
   585
  next
lp15@59746
   586
    fix x
lp15@59746
   587
    assume "x \<in> closed_segment 0 z"
lp15@59746
   588
    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
   589
      by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
lp15@59746
   590
  next
lp15@59746
   591
    show "0 \<in> closed_segment 0 z"
lp15@59746
   592
      by (auto simp: closed_segment_def)
lp15@59746
   593
  next
lp15@59746
   594
    show "z \<in> closed_segment 0 z"
lp15@59746
   595
      apply (simp add: closed_segment_def scaleR_conv_of_real)
lp15@59746
   596
      using of_real_1 zero_le_one by blast
lp15@59862
   597
  qed
lp15@59746
   598
  have **: "\<And>k. complex_of_real (cos_coeff k) * z ^ k
lp15@59746
   599
            = (-1)^(Suc k div 2) * (if even k then cos 0 else sin 0) * z^k / of_nat (fact k)"
lp15@59746
   600
    by (auto simp: cos_coeff_def elim!: evenE)
lp15@59746
   601
  show ?thesis
lp15@59746
   602
    apply (rule order_trans [OF _ *])
lp15@59746
   603
    apply (simp add: **)
lp15@59746
   604
    done
lp15@59746
   605
qed
lp15@59746
   606
lp15@59746
   607
end (* of context *)
lp15@59746
   608
lp15@59751
   609
text{*32-bit Approximation to e*}
lp15@59751
   610
lemma e_approx_32: "abs(exp(1) - 5837465777 / 2147483648) \<le> (inverse(2 ^ 32)::real)"
lp15@59751
   611
  using Taylor_exp [of 1 14] exp_le
lp15@59751
   612
  apply (simp add: setsum_left_distrib in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
lp15@59751
   613
  apply (simp only: pos_le_divide_eq [symmetric], linarith)
lp15@59751
   614
  done
lp15@59751
   615
lp15@59746
   616
subsection{*The argument of a complex number*}
lp15@59746
   617
lp15@59746
   618
definition Arg :: "complex \<Rightarrow> real" where
lp15@59746
   619
 "Arg z \<equiv> if z = 0 then 0
lp15@59746
   620
           else THE t. 0 \<le> t \<and> t < 2*pi \<and>
lp15@59746
   621
                    z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   622
lp15@59746
   623
lemma Arg_0 [simp]: "Arg(0) = 0"
lp15@59746
   624
  by (simp add: Arg_def)
lp15@59746
   625
lp15@59746
   626
lemma Arg_unique_lemma:
lp15@59746
   627
  assumes z:  "z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   628
      and z': "z = of_real(norm z) * exp(ii * of_real t')"
lp15@59746
   629
      and t:  "0 \<le> t"  "t < 2*pi"
lp15@59746
   630
      and t': "0 \<le> t'" "t' < 2*pi"
lp15@59746
   631
      and nz: "z \<noteq> 0"
lp15@59746
   632
  shows "t' = t"
lp15@59746
   633
proof -
lp15@59746
   634
  have [dest]: "\<And>x y z::real. x\<ge>0 \<Longrightarrow> x+y < z \<Longrightarrow> y<z"
lp15@59746
   635
    by arith
lp15@59746
   636
  have "of_real (cmod z) * exp (\<i> * of_real t') = of_real (cmod z) * exp (\<i> * of_real t)"
lp15@59746
   637
    by (metis z z')
lp15@59746
   638
  then have "exp (\<i> * of_real t') = exp (\<i> * of_real t)"
lp15@59746
   639
    by (metis nz mult_left_cancel mult_zero_left z)
lp15@59746
   640
  then have "sin t' = sin t \<and> cos t' = cos t"
lp15@59746
   641
    apply (simp add: exp_Euler sin_of_real cos_of_real)
lp15@59746
   642
    by (metis Complex_eq complex.sel)
lp15@59746
   643
  then obtain n::int where n: "t' = t + 2 * real n * pi"
lp15@59746
   644
    by (auto simp: sin_cos_eq_iff)
lp15@59746
   645
  then have "n=0"
lp15@59746
   646
    apply (rule_tac z=n in int_cases)
lp15@59746
   647
    using t t'
lp15@59746
   648
    apply (auto simp: mult_less_0_iff algebra_simps)
lp15@59746
   649
    done
lp15@59746
   650
  then show "t' = t"
lp15@59746
   651
      by (simp add: n)
lp15@59746
   652
qed
lp15@59746
   653
lp15@59746
   654
lemma Arg: "0 \<le> Arg z & Arg z < 2*pi & z = of_real(norm z) * exp(ii * of_real(Arg z))"
lp15@59746
   655
proof (cases "z=0")
lp15@59746
   656
  case True then show ?thesis
lp15@59746
   657
    by (simp add: Arg_def)
lp15@59746
   658
next
lp15@59746
   659
  case False
lp15@59746
   660
  obtain t where t: "0 \<le> t" "t < 2*pi"
lp15@59746
   661
             and ReIm: "Re z / cmod z = cos t" "Im z / cmod z = sin t"
lp15@59746
   662
    using sincos_total_2pi [OF complex_unit_circle [OF False]]
lp15@59746
   663
    by blast
lp15@59746
   664
  have z: "z = of_real(norm z) * exp(ii * of_real t)"
lp15@59746
   665
    apply (rule complex_eqI)
lp15@59746
   666
    using t False ReIm
lp15@59746
   667
    apply (auto simp: exp_Euler sin_of_real cos_of_real divide_simps)
lp15@59746
   668
    done
lp15@59746
   669
  show ?thesis
lp15@59746
   670
    apply (simp add: Arg_def False)
lp15@59746
   671
    apply (rule theI [where a=t])
lp15@59746
   672
    using t z False
lp15@59746
   673
    apply (auto intro: Arg_unique_lemma)
lp15@59746
   674
    done
lp15@59746
   675
qed
lp15@59746
   676
lp15@59746
   677
lp15@59746
   678
corollary
lp15@59746
   679
  shows Arg_ge_0: "0 \<le> Arg z"
lp15@59746
   680
    and Arg_lt_2pi: "Arg z < 2*pi"
lp15@59746
   681
    and Arg_eq: "z = of_real(norm z) * exp(ii * of_real(Arg z))"
lp15@59746
   682
  using Arg by auto
lp15@59746
   683
lp15@59746
   684
lemma complex_norm_eq_1_exp: "norm z = 1 \<longleftrightarrow> (\<exists>t. z = exp(ii * of_real t))"
lp15@59746
   685
  using Arg [of z] by auto
lp15@59746
   686
lp15@59746
   687
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
   688
  apply (rule Arg_unique_lemma [OF _ Arg_eq])
lp15@59746
   689
  using Arg [of z]
lp15@59746
   690
  apply (auto simp: norm_mult)
lp15@59746
   691
  done
lp15@59746
   692
lp15@59746
   693
lemma Arg_minus: "z \<noteq> 0 \<Longrightarrow> Arg (-z) = (if Arg z < pi then Arg z + pi else Arg z - pi)"
lp15@59746
   694
  apply (rule Arg_unique [of "norm z"])
lp15@59746
   695
  apply (rule complex_eqI)
lp15@59746
   696
  using Arg_ge_0 [of z] Arg_eq [of z] Arg_lt_2pi [of z] Arg_eq [of z]
lp15@59746
   697
  apply auto
lp15@59746
   698
  apply (auto simp: Re_exp Im_exp cos_diff sin_diff cis_conv_exp [symmetric])
lp15@59746
   699
  apply (metis Re_rcis Im_rcis rcis_def)+
lp15@59746
   700
  done
lp15@59746
   701
lp15@59746
   702
lemma Arg_times_of_real [simp]: "0 < r \<Longrightarrow> Arg (of_real r * z) = Arg z"
lp15@59746
   703
  apply (cases "z=0", simp)
lp15@59746
   704
  apply (rule Arg_unique [of "r * norm z"])
lp15@59746
   705
  using Arg
lp15@59746
   706
  apply auto
lp15@59746
   707
  done
lp15@59746
   708
lp15@59746
   709
lemma Arg_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg (z * of_real r) = Arg z"
lp15@59746
   710
  by (metis Arg_times_of_real mult.commute)
lp15@59746
   711
lp15@59746
   712
lemma Arg_divide_of_real [simp]: "0 < r \<Longrightarrow> Arg (z / of_real r) = Arg z"
lp15@59746
   713
  by (metis Arg_times_of_real2 less_numeral_extra(3) nonzero_eq_divide_eq of_real_eq_0_iff)
lp15@59746
   714
lp15@59746
   715
lemma Arg_le_pi: "Arg z \<le> pi \<longleftrightarrow> 0 \<le> Im z"
lp15@59746
   716
proof (cases "z=0")
lp15@59746
   717
  case True then show ?thesis
lp15@59746
   718
    by simp
lp15@59746
   719
next
lp15@59746
   720
  case False
lp15@59746
   721
  have "0 \<le> Im z \<longleftrightarrow> 0 \<le> Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   722
    by (metis Arg_eq)
lp15@59746
   723
  also have "... = (0 \<le> Im (exp (\<i> * complex_of_real (Arg z))))"
lp15@59746
   724
    using False
lp15@59746
   725
    by (simp add: zero_le_mult_iff)
lp15@59746
   726
  also have "... \<longleftrightarrow> Arg z \<le> pi"
lp15@59746
   727
    by (simp add: Im_exp) (metis Arg_ge_0 Arg_lt_2pi sin_lt_zero sin_ge_zero not_le)
lp15@59746
   728
  finally show ?thesis
lp15@59746
   729
    by blast
lp15@59746
   730
qed
lp15@59746
   731
lp15@59746
   732
lemma Arg_lt_pi: "0 < Arg z \<and> Arg z < pi \<longleftrightarrow> 0 < Im z"
lp15@59746
   733
proof (cases "z=0")
lp15@59746
   734
  case True then show ?thesis
lp15@59746
   735
    by simp
lp15@59746
   736
next
lp15@59746
   737
  case False
lp15@59746
   738
  have "0 < Im z \<longleftrightarrow> 0 < Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   739
    by (metis Arg_eq)
lp15@59746
   740
  also have "... = (0 < Im (exp (\<i> * complex_of_real (Arg z))))"
lp15@59746
   741
    using False
lp15@59746
   742
    by (simp add: zero_less_mult_iff)
lp15@59746
   743
  also have "... \<longleftrightarrow> 0 < Arg z \<and> Arg z < pi"
lp15@59746
   744
    using Arg_ge_0  Arg_lt_2pi sin_le_zero sin_gt_zero
lp15@59746
   745
    apply (auto simp: Im_exp)
lp15@59746
   746
    using le_less apply fastforce
lp15@59746
   747
    using not_le by blast
lp15@59746
   748
  finally show ?thesis
lp15@59746
   749
    by blast
lp15@59746
   750
qed
lp15@59746
   751
lp15@59746
   752
lemma Arg_eq_0: "Arg z = 0 \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re z"
lp15@59746
   753
proof (cases "z=0")
lp15@59746
   754
  case True then show ?thesis
lp15@59746
   755
    by simp
lp15@59746
   756
next
lp15@59746
   757
  case False
lp15@59746
   758
  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
   759
    by (metis Arg_eq)
lp15@59746
   760
  also have "... \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re (exp (\<i> * complex_of_real (Arg z)))"
lp15@59746
   761
    using False
lp15@59746
   762
    by (simp add: zero_le_mult_iff)
lp15@59746
   763
  also have "... \<longleftrightarrow> Arg z = 0"
lp15@59746
   764
    apply (auto simp: Re_exp)
lp15@59746
   765
    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
   766
    using Arg_eq [of z]
lp15@59746
   767
    apply (auto simp: Reals_def)
lp15@59746
   768
    done
lp15@59746
   769
  finally show ?thesis
lp15@59746
   770
    by blast
lp15@59746
   771
qed
lp15@59746
   772
lp15@59746
   773
lemma Arg_of_real: "Arg(of_real x) = 0 \<longleftrightarrow> 0 \<le> x"
lp15@59746
   774
  by (simp add: Arg_eq_0)
lp15@59746
   775
lp15@59746
   776
lemma Arg_eq_pi: "Arg z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
lp15@59746
   777
  apply  (cases "z=0", simp)
lp15@59746
   778
  using Arg_eq_0 [of "-z"]
lp15@59746
   779
  apply (auto simp: complex_is_Real_iff Arg_minus)
lp15@59746
   780
  apply (simp add: complex_Re_Im_cancel_iff)
lp15@59746
   781
  apply (metis Arg_minus pi_gt_zero add.left_neutral minus_minus minus_zero)
lp15@59746
   782
  done
lp15@59746
   783
lp15@59746
   784
lemma Arg_eq_0_pi: "Arg z = 0 \<or> Arg z = pi \<longleftrightarrow> z \<in> \<real>"
lp15@59746
   785
  using Arg_eq_0 Arg_eq_pi not_le by auto
lp15@59746
   786
lp15@59746
   787
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
   788
  apply (cases "z=0", simp)
lp15@59746
   789
  apply (rule Arg_unique [of "inverse (norm z)"])
lp15@59746
   790
  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
   791
  apply (auto simp: of_real_numeral algebra_simps exp_diff divide_simps)
lp15@59746
   792
  done
lp15@59746
   793
lp15@59746
   794
lemma Arg_eq_iff:
lp15@59746
   795
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   796
     shows "Arg w = Arg z \<longleftrightarrow> (\<exists>x. 0 < x & w = of_real x * z)"
lp15@59746
   797
  using assms Arg_eq [of z] Arg_eq [of w]
lp15@59746
   798
  apply auto
lp15@59746
   799
  apply (rule_tac x="norm w / norm z" in exI)
lp15@59746
   800
  apply (simp add: divide_simps)
lp15@59746
   801
  by (metis mult.commute mult.left_commute)
lp15@59746
   802
lp15@59746
   803
lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
lp15@59746
   804
  using complex_is_Real_iff
lp15@59746
   805
  apply (simp add: Arg_eq_0)
lp15@59746
   806
  apply (auto simp: divide_simps not_sum_power2_lt_zero)
lp15@59746
   807
  done
lp15@59746
   808
lp15@59746
   809
lemma Arg_divide:
lp15@59746
   810
  assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
lp15@59746
   811
    shows "Arg(z / w) = Arg z - Arg w"
lp15@59746
   812
  apply (rule Arg_unique [of "norm(z / w)"])
lp15@59746
   813
  using assms Arg_eq [of z] Arg_eq [of w] Arg_ge_0 [of w] Arg_lt_2pi [of z]
lp15@59746
   814
  apply (auto simp: exp_diff norm_divide algebra_simps divide_simps)
lp15@59746
   815
  done
lp15@59746
   816
lp15@59746
   817
lemma Arg_le_div_sum:
lp15@59746
   818
  assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
lp15@59746
   819
    shows "Arg z = Arg w + Arg(z / w)"
lp15@59746
   820
  by (simp add: Arg_divide assms)
lp15@59746
   821
lp15@59746
   822
lemma Arg_le_div_sum_eq:
lp15@59746
   823
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   824
    shows "Arg w \<le> Arg z \<longleftrightarrow> Arg z = Arg w + Arg(z / w)"
lp15@59746
   825
  using assms
lp15@59746
   826
  by (auto simp: Arg_ge_0 intro: Arg_le_div_sum)
lp15@59746
   827
lp15@59746
   828
lemma Arg_diff:
lp15@59746
   829
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   830
    shows "Arg w - Arg z = (if Arg z \<le> Arg w then Arg(w / z) else Arg(w/z) - 2*pi)"
lp15@59746
   831
  using assms
lp15@59746
   832
  apply (auto simp: Arg_ge_0 Arg_divide not_le)
lp15@59746
   833
  using Arg_divide [of w z] Arg_inverse [of "w/z"]
lp15@59746
   834
  apply auto
lp15@59746
   835
  by (metis Arg_eq_0 less_irrefl minus_diff_eq right_minus_eq)
lp15@59746
   836
lp15@59746
   837
lemma Arg_add:
lp15@59746
   838
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   839
    shows "Arg w + Arg z = (if Arg w + Arg z < 2*pi then Arg(w * z) else Arg(w * z) + 2*pi)"
lp15@59746
   840
  using assms
lp15@59746
   841
  using Arg_diff [of "w*z" z] Arg_le_div_sum_eq [of z "w*z"]
lp15@59746
   842
  apply (auto simp: Arg_ge_0 Arg_divide not_le)
lp15@59746
   843
  apply (metis Arg_lt_2pi add.commute)
lp15@59746
   844
  apply (metis (no_types) Arg add.commute diff_0 diff_add_cancel diff_less_eq diff_minus_eq_add not_less)
lp15@59746
   845
  done
lp15@59746
   846
lp15@59746
   847
lemma Arg_times:
lp15@59746
   848
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59746
   849
    shows "Arg (w * z) = (if Arg w + Arg z < 2*pi then Arg w + Arg z
lp15@59746
   850
                            else (Arg w + Arg z) - 2*pi)"
lp15@59746
   851
  using Arg_add [OF assms]
lp15@59746
   852
  by auto
lp15@59746
   853
lp15@59746
   854
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
   855
  apply (cases "z=0", simp)
lp15@59746
   856
  apply (rule trans [of _ "Arg(inverse z)"])
lp15@59746
   857
  apply (simp add: Arg_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
lp15@59746
   858
  apply (metis norm_eq_zero of_real_power zero_less_power2)
lp15@59746
   859
  apply (auto simp: of_real_numeral Arg_inverse)
lp15@59746
   860
  done
lp15@59746
   861
lp15@59746
   862
lemma Arg_real: "z \<in> \<real> \<Longrightarrow> Arg z = (if 0 \<le> Re z then 0 else pi)"
lp15@59746
   863
  using Arg_eq_0 Arg_eq_0_pi
lp15@59746
   864
  by auto
lp15@59746
   865
lp15@59746
   866
lemma Arg_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg(exp z) = Im z"
lp15@59746
   867
  by (rule Arg_unique [of  "exp(Re z)"]) (auto simp: Exp_eq_polar)
lp15@59746
   868
lp15@59751
   869
lp15@59751
   870
subsection{*Analytic properties of tangent function*}
lp15@59751
   871
lp15@59751
   872
lemma cnj_tan: "cnj(tan z) = tan(cnj z)"
lp15@59751
   873
  by (simp add: cnj_cos cnj_sin tan_def)
lp15@59751
   874
lp15@59751
   875
lemma complex_differentiable_at_tan: "~(cos z = 0) \<Longrightarrow> tan complex_differentiable at z"
lp15@59751
   876
  unfolding complex_differentiable_def
lp15@59751
   877
  using DERIV_tan by blast
lp15@59751
   878
lp15@59751
   879
lemma complex_differentiable_within_tan: "~(cos z = 0)
lp15@59751
   880
         \<Longrightarrow> tan complex_differentiable (at z within s)"
lp15@59751
   881
  using complex_differentiable_at_tan complex_differentiable_at_within by blast
lp15@59751
   882
lp15@59751
   883
lemma continuous_within_tan: "~(cos z = 0) \<Longrightarrow> continuous (at z within s) tan"
lp15@59751
   884
  using continuous_at_imp_continuous_within isCont_tan by blast
lp15@59751
   885
lp15@59751
   886
lemma continuous_on_tan [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> continuous_on s tan"
lp15@59751
   887
  by (simp add: continuous_at_imp_continuous_on)
lp15@59751
   888
lp15@59751
   889
lemma holomorphic_on_tan: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> tan holomorphic_on s"
lp15@59751
   890
  by (simp add: complex_differentiable_within_tan holomorphic_on_def)
lp15@59751
   891
lp15@59751
   892
lp15@59751
   893
subsection{*Complex logarithms (the conventional principal value)*}
lp15@59751
   894
lp15@59751
   895
definition Ln where
lp15@59751
   896
   "Ln \<equiv> \<lambda>z. THE w. exp w = z & -pi < Im(w) & Im(w) \<le> pi"
lp15@59751
   897
lp15@59751
   898
lemma
lp15@59751
   899
  assumes "z \<noteq> 0"
lp15@59751
   900
    shows exp_Ln [simp]: "exp(Ln z) = z"
lp15@59751
   901
      and mpi_less_Im_Ln: "-pi < Im(Ln z)"
lp15@59751
   902
      and Im_Ln_le_pi:    "Im(Ln z) \<le> pi"
lp15@59751
   903
proof -
lp15@59751
   904
  obtain \<psi> where z: "z / (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
lp15@59751
   905
    using complex_unimodular_polar [of "z / (norm z)"] assms
lp15@59751
   906
    by (auto simp: norm_divide divide_simps)
lp15@59751
   907
  obtain \<phi> where \<phi>: "- pi < \<phi>" "\<phi> \<le> pi" "sin \<phi> = sin \<psi>" "cos \<phi> = cos \<psi>"
lp15@59751
   908
    using sincos_principal_value [of "\<psi>"] assms
lp15@59751
   909
    by (auto simp: norm_divide divide_simps)
lp15@59751
   910
  have "exp(Ln z) = z & -pi < Im(Ln z) & Im(Ln z) \<le> pi" unfolding Ln_def
lp15@59751
   911
    apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
lp15@59751
   912
    using z assms \<phi>
lp15@59751
   913
    apply (auto simp: field_simps exp_complex_eqI Exp_eq_polar cis.code)
lp15@59751
   914
    done
lp15@59751
   915
  then show "exp(Ln z) = z" "-pi < Im(Ln z)" "Im(Ln z) \<le> pi"
lp15@59751
   916
    by auto
lp15@59751
   917
qed
lp15@59751
   918
lp15@59751
   919
lemma Ln_exp [simp]:
lp15@59751
   920
  assumes "-pi < Im(z)" "Im(z) \<le> pi"
lp15@59751
   921
    shows "Ln(exp z) = z"
lp15@59751
   922
  apply (rule exp_complex_eqI)
lp15@59751
   923
  using assms mpi_less_Im_Ln  [of "exp z"] Im_Ln_le_pi [of "exp z"]
lp15@59751
   924
  apply auto
lp15@59751
   925
  done
lp15@59751
   926
lp15@59751
   927
lemma Ln_eq_iff: "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (Ln w = Ln z \<longleftrightarrow> w = z)"
lp15@59751
   928
  by (metis exp_Ln)
lp15@59751
   929
lp15@59751
   930
lemma Ln_unique: "exp(z) = w \<Longrightarrow> -pi < Im(z) \<Longrightarrow> Im(z) \<le> pi \<Longrightarrow> Ln w = z"
lp15@59751
   931
  using Ln_exp by blast
lp15@59751
   932
lp15@59751
   933
lemma Re_Ln [simp]: "z \<noteq> 0 \<Longrightarrow> Re(Ln z) = ln(norm z)"
lp15@59751
   934
by (metis exp_Ln assms ln_exp norm_exp_eq_Re)
lp15@59751
   935
lp15@59751
   936
lemma exists_complex_root:
lp15@59751
   937
  fixes a :: complex
lp15@59751
   938
  shows "n \<noteq> 0 \<Longrightarrow> \<exists>z. z ^ n = a"
lp15@59751
   939
  apply (cases "a=0", simp)
lp15@59751
   940
  apply (rule_tac x= "exp(Ln(a) / n)" in exI)
lp15@59751
   941
  apply (auto simp: exp_of_nat_mult [symmetric])
lp15@59751
   942
  done
lp15@59751
   943
lp15@59862
   944
subsection{*The Unwinding Number and the Ln-product Formula*}
lp15@59862
   945
lp15@59862
   946
text{*Note that in this special case the unwinding number is -1, 0 or 1.*}
lp15@59862
   947
lp15@59862
   948
definition unwinding :: "complex \<Rightarrow> complex" where
lp15@59862
   949
   "unwinding(z) = (z - Ln(exp z)) / (of_real(2*pi) * ii)"
lp15@59862
   950
lp15@59862
   951
lemma unwinding_2pi: "(2*pi) * ii * unwinding(z) = z - Ln(exp z)"
lp15@59862
   952
  by (simp add: unwinding_def)
lp15@59862
   953
lp15@59862
   954
lemma Ln_times_unwinding:
lp15@59862
   955
    "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
   956
  using unwinding_2pi by (simp add: exp_add)
lp15@59862
   957
lp15@59862
   958
lp15@59751
   959
subsection{*Derivative of Ln away from the branch cut*}
lp15@59751
   960
lp15@59751
   961
lemma
lp15@59751
   962
  assumes "Im(z) = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
   963
    shows has_field_derivative_Ln: "(Ln has_field_derivative inverse(z)) (at z)"
lp15@59751
   964
      and Im_Ln_less_pi:           "Im (Ln z) < pi"
lp15@59751
   965
proof -
lp15@59751
   966
  have znz: "z \<noteq> 0"
lp15@59751
   967
    using assms by auto
lp15@59751
   968
  then show *: "Im (Ln z) < pi" using assms
lp15@59751
   969
    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
   970
  show "(Ln has_field_derivative inverse(z)) (at z)"
lp15@59751
   971
    apply (rule has_complex_derivative_inverse_strong_x
lp15@59751
   972
              [where f = exp and s = "{w. -pi < Im(w) & Im(w) < pi}"])
lp15@59751
   973
    using znz *
lp15@59751
   974
    apply (auto simp: continuous_on_exp open_Collect_conj open_halfspace_Im_gt open_halfspace_Im_lt)
lp15@59751
   975
    apply (metis DERIV_exp exp_Ln)
lp15@59751
   976
    apply (metis mpi_less_Im_Ln)
lp15@59751
   977
    done
lp15@59751
   978
qed
lp15@59751
   979
lp15@59751
   980
declare has_field_derivative_Ln [derivative_intros]
lp15@59751
   981
declare has_field_derivative_Ln [THEN DERIV_chain2, derivative_intros]
lp15@59751
   982
lp15@59751
   983
lemma complex_differentiable_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln complex_differentiable at z"
lp15@59751
   984
  using complex_differentiable_def has_field_derivative_Ln by blast
lp15@59751
   985
lp15@59751
   986
lemma complex_differentiable_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z))
lp15@59751
   987
         \<Longrightarrow> Ln complex_differentiable (at z within s)"
lp15@59751
   988
  using complex_differentiable_at_Ln complex_differentiable_within_subset by blast
lp15@59751
   989
lp15@59751
   990
lemma continuous_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) Ln"
lp15@59751
   991
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Ln)
lp15@59751
   992
lp15@59862
   993
lemma isCont_Ln' [simp]:
lp15@59862
   994
   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. Ln (f x)) z"
lp15@59862
   995
  by (blast intro: isCont_o2 [OF _ continuous_at_Ln])
lp15@59862
   996
lp15@59751
   997
lemma continuous_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) Ln"
lp15@59751
   998
  using continuous_at_Ln continuous_at_imp_continuous_within by blast
lp15@59751
   999
lp15@59751
  1000
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
  1001
  by (simp add: continuous_at_imp_continuous_on continuous_within_Ln)
lp15@59751
  1002
lp15@59751
  1003
lemma holomorphic_on_Ln: "(\<And>z. z \<in> s \<Longrightarrow> Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln holomorphic_on s"
lp15@59751
  1004
  by (simp add: complex_differentiable_within_Ln holomorphic_on_def)
lp15@59751
  1005
lp15@59751
  1006
lp15@59751
  1007
subsection{*Relation to Real Logarithm*}
lp15@59751
  1008
lp15@59862
  1009
lemma Ln_of_real:
lp15@59751
  1010
  assumes "0 < z"
lp15@59751
  1011
    shows "Ln(of_real z) = of_real(ln z)"
lp15@59751
  1012
proof -
lp15@59751
  1013
  have "Ln(of_real (exp (ln z))) = Ln (exp (of_real (ln z)))"
lp15@59751
  1014
    by (simp add: exp_of_real)
lp15@59751
  1015
  also have "... = of_real(ln z)"
lp15@59751
  1016
    using assms
lp15@59751
  1017
    by (subst Ln_exp) auto
lp15@59751
  1018
  finally show ?thesis
lp15@59751
  1019
    using assms by simp
lp15@59751
  1020
qed
lp15@59751
  1021
lp15@59862
  1022
corollary Ln_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Re z > 0 \<Longrightarrow> Ln z \<in> \<real>"
lp15@59862
  1023
  by (auto simp: Ln_of_real elim: Reals_cases)
lp15@59862
  1024
lp15@59751
  1025
lp15@59751
  1026
subsection{*Quadrant-type results for Ln*}
lp15@59751
  1027
lp15@59751
  1028
lemma cos_lt_zero_pi: "pi/2 < x \<Longrightarrow> x < 3*pi/2 \<Longrightarrow> cos x < 0"
lp15@59751
  1029
  using cos_minus_pi cos_gt_zero_pi [of "x-pi"]
lp15@59751
  1030
  by simp
lp15@59751
  1031
lp15@59751
  1032
lemma Re_Ln_pos_lt:
lp15@59751
  1033
  assumes "z \<noteq> 0"
lp15@59751
  1034
    shows "abs(Im(Ln z)) < pi/2 \<longleftrightarrow> 0 < Re(z)"
lp15@59751
  1035
proof -
lp15@59751
  1036
  { fix w
lp15@59751
  1037
    assume "w = Ln z"
lp15@59751
  1038
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1039
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1040
      by auto
lp15@59751
  1041
    then have "abs(Im w) < pi/2 \<longleftrightarrow> 0 < Re(exp w)"
lp15@59751
  1042
      apply (auto simp: Re_exp zero_less_mult_iff cos_gt_zero_pi)
lp15@59751
  1043
      using cos_lt_zero_pi [of "-(Im w)"] cos_lt_zero_pi [of "(Im w)"]
lp15@59751
  1044
      apply (simp add: abs_if split: split_if_asm)
lp15@59751
  1045
      apply (metis (no_types) cos_minus cos_pi_half eq_divide_eq_numeral1(1) eq_numeral_simps(4)
lp15@59751
  1046
               less_numeral_extra(3) linorder_neqE_linordered_idom minus_mult_minus minus_mult_right
lp15@59751
  1047
               mult_numeral_1_right)
lp15@59751
  1048
      done
lp15@59751
  1049
  }
lp15@59751
  1050
  then show ?thesis using assms
lp15@59751
  1051
    by auto
lp15@59751
  1052
qed
lp15@59751
  1053
lp15@59751
  1054
lemma Re_Ln_pos_le:
lp15@59751
  1055
  assumes "z \<noteq> 0"
lp15@59751
  1056
    shows "abs(Im(Ln z)) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(z)"
lp15@59751
  1057
proof -
lp15@59751
  1058
  { fix w
lp15@59751
  1059
    assume "w = Ln z"
lp15@59751
  1060
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1061
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1062
      by auto
lp15@59751
  1063
    then have "abs(Im w) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(exp w)"
lp15@59751
  1064
      apply (auto simp: Re_exp zero_le_mult_iff cos_ge_zero)
lp15@59751
  1065
      using cos_lt_zero_pi [of "- (Im w)"] cos_lt_zero_pi [of "(Im w)"] not_le
lp15@59751
  1066
      apply (auto simp: abs_if split: split_if_asm)
lp15@59751
  1067
      done
lp15@59751
  1068
  }
lp15@59751
  1069
  then show ?thesis using assms
lp15@59751
  1070
    by auto
lp15@59751
  1071
qed
lp15@59751
  1072
lp15@59751
  1073
lemma Im_Ln_pos_lt:
lp15@59751
  1074
  assumes "z \<noteq> 0"
lp15@59751
  1075
    shows "0 < Im(Ln z) \<and> Im(Ln z) < pi \<longleftrightarrow> 0 < Im(z)"
lp15@59751
  1076
proof -
lp15@59751
  1077
  { fix w
lp15@59751
  1078
    assume "w = Ln z"
lp15@59751
  1079
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1080
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1081
      by auto
lp15@59751
  1082
    then have "0 < Im w \<and> Im w < pi \<longleftrightarrow> 0 < Im(exp w)"
lp15@59751
  1083
      using sin_gt_zero [of "- (Im w)"] sin_gt_zero [of "(Im w)"]
lp15@59751
  1084
      apply (auto simp: Im_exp zero_less_mult_iff)
lp15@59751
  1085
      using less_linear apply fastforce
lp15@59751
  1086
      using less_linear apply fastforce
lp15@59751
  1087
      done
lp15@59751
  1088
  }
lp15@59751
  1089
  then show ?thesis using assms
lp15@59751
  1090
    by auto
lp15@59751
  1091
qed
lp15@59751
  1092
lp15@59751
  1093
lemma Im_Ln_pos_le:
lp15@59751
  1094
  assumes "z \<noteq> 0"
lp15@59751
  1095
    shows "0 \<le> Im(Ln z) \<and> Im(Ln z) \<le> pi \<longleftrightarrow> 0 \<le> Im(z)"
lp15@59751
  1096
proof -
lp15@59751
  1097
  { fix w
lp15@59751
  1098
    assume "w = Ln z"
lp15@59751
  1099
    then have w: "Im w \<le> pi" "- pi < Im w"
lp15@59751
  1100
      using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
lp15@59751
  1101
      by auto
lp15@59751
  1102
    then have "0 \<le> Im w \<and> Im w \<le> pi \<longleftrightarrow> 0 \<le> Im(exp w)"
lp15@59751
  1103
      using sin_ge_zero [of "- (Im w)"] sin_ge_zero [of "(Im w)"]
lp15@59751
  1104
      apply (auto simp: Im_exp zero_le_mult_iff sin_ge_zero)
lp15@59751
  1105
      apply (metis not_le not_less_iff_gr_or_eq pi_not_less_zero sin_eq_0_pi)
lp15@59751
  1106
      done }
lp15@59751
  1107
  then show ?thesis using assms
lp15@59751
  1108
    by auto
lp15@59751
  1109
qed
lp15@59751
  1110
lp15@59751
  1111
lemma Re_Ln_pos_lt_imp: "0 < Re(z) \<Longrightarrow> abs(Im(Ln z)) < pi/2"
lp15@59751
  1112
  by (metis Re_Ln_pos_lt less_irrefl zero_complex.simps(1))
lp15@59751
  1113
lp15@59751
  1114
lemma Im_Ln_pos_lt_imp: "0 < Im(z) \<Longrightarrow> 0 < Im(Ln z) \<and> Im(Ln z) < pi"
lp15@59751
  1115
  by (metis Im_Ln_pos_lt not_le order_refl zero_complex.simps(2))
lp15@59751
  1116
lp15@59751
  1117
lemma Im_Ln_eq_0: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = 0 \<longleftrightarrow> 0 < Re(z) \<and> Im(z) = 0)"
lp15@59751
  1118
  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
  1119
       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
  1120
lp15@59751
  1121
lemma Im_Ln_eq_pi: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = pi \<longleftrightarrow> Re(z) < 0 \<and> Im(z) = 0)"
lp15@59751
  1122
  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
  1123
lp15@59751
  1124
lp15@59751
  1125
subsection{*More Properties of Ln*}
lp15@59751
  1126
lp15@59751
  1127
lemma cnj_Ln: "(Im z = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> cnj(Ln z) = Ln(cnj z)"
lp15@59751
  1128
  apply (cases "z=0", auto)
lp15@59751
  1129
  apply (rule exp_complex_eqI)
lp15@59751
  1130
  apply (auto simp: abs_if split: split_if_asm)
lp15@59751
  1131
  apply (metis Im_Ln_less_pi add_mono_thms_linordered_field(5) cnj.simps(1) cnj.simps(2) mult_2 neg_equal_0_iff_equal)
lp15@59751
  1132
  apply (metis add_mono_thms_linordered_field(5) complex_cnj_zero_iff diff_0_right diff_minus_eq_add minus_diff_eq mpi_less_Im_Ln mult.commute mult_2_right neg_less_iff_less)
lp15@59751
  1133
  by (metis exp_Ln exp_cnj)
lp15@59751
  1134
lp15@59751
  1135
lemma Ln_inverse: "(Im(z) = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> Ln(inverse z) = -(Ln z)"
lp15@59751
  1136
  apply (cases "z=0", auto)
lp15@59751
  1137
  apply (rule exp_complex_eqI)
lp15@59751
  1138
  using mpi_less_Im_Ln [of z] mpi_less_Im_Ln [of "inverse z"]
lp15@59751
  1139
  apply (auto simp: abs_if exp_minus split: split_if_asm)
lp15@59751
  1140
  apply (metis Im_Ln_less_pi Im_Ln_pos_le add_less_cancel_left add_strict_mono
lp15@59751
  1141
               inverse_inverse_eq inverse_zero le_less mult.commute mult_2_right)
lp15@59751
  1142
  done
lp15@59751
  1143
lp15@59751
  1144
lemma Ln_1 [simp]: "Ln(1) = 0"
lp15@59751
  1145
proof -
lp15@59751
  1146
  have "Ln (exp 0) = 0"
lp15@59862
  1147
    by (metis exp_zero ln_exp Ln_of_real of_real_0 of_real_1 zero_less_one)
lp15@59751
  1148
  then show ?thesis
lp15@59751
  1149
    by simp
lp15@59751
  1150
qed
lp15@59751
  1151
lp15@59751
  1152
lemma Ln_minus1 [simp]: "Ln(-1) = ii * pi"
lp15@59751
  1153
  apply (rule exp_complex_eqI)
lp15@59751
  1154
  using Im_Ln_le_pi [of "-1"] mpi_less_Im_Ln [of "-1"] cis_conv_exp cis_pi
lp15@59751
  1155
  apply (auto simp: abs_if)
lp15@59751
  1156
  done
lp15@59751
  1157
lp15@59751
  1158
lemma Ln_ii [simp]: "Ln ii = ii * of_real pi/2"
lp15@59751
  1159
  using Ln_exp [of "ii * (of_real pi/2)"]
lp15@59751
  1160
  unfolding exp_Euler
lp15@59751
  1161
  by simp
lp15@59751
  1162
lp15@59751
  1163
lemma Ln_minus_ii [simp]: "Ln(-ii) = - (ii * pi/2)"
lp15@59751
  1164
proof -
lp15@59751
  1165
  have  "Ln(-ii) = Ln(1/ii)"
lp15@59751
  1166
    by simp
lp15@59751
  1167
  also have "... = - (Ln ii)"
lp15@59751
  1168
    by (metis Ln_inverse ii.sel(2) inverse_eq_divide zero_neq_one)
lp15@59751
  1169
  also have "... = - (ii * pi/2)"
lp15@59751
  1170
    by (simp add: Ln_ii)
lp15@59751
  1171
  finally show ?thesis .
lp15@59751
  1172
qed
lp15@59751
  1173
lp15@59751
  1174
lemma Ln_times:
lp15@59751
  1175
  assumes "w \<noteq> 0" "z \<noteq> 0"
lp15@59751
  1176
    shows "Ln(w * z) =
lp15@59751
  1177
                (if Im(Ln w + Ln z) \<le> -pi then
lp15@59751
  1178
                  (Ln(w) + Ln(z)) + ii * of_real(2*pi)
lp15@59751
  1179
                else if Im(Ln w + Ln z) > pi then
lp15@59751
  1180
                  (Ln(w) + Ln(z)) - ii * of_real(2*pi)
lp15@59751
  1181
                else Ln(w) + Ln(z))"
lp15@59751
  1182
  using pi_ge_zero Im_Ln_le_pi [of w] Im_Ln_le_pi [of z]
lp15@59751
  1183
  using assms mpi_less_Im_Ln [of w] mpi_less_Im_Ln [of z]
lp15@59751
  1184
  by (auto simp: of_real_numeral exp_add exp_diff sin_double cos_double exp_Euler intro!: Ln_unique)
lp15@59751
  1185
lp15@59751
  1186
lemma Ln_times_simple:
lp15@59751
  1187
    "\<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
  1188
         \<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z)"
lp15@59751
  1189
  by (simp add: Ln_times)
lp15@59751
  1190
lp15@59751
  1191
lemma Ln_minus:
lp15@59751
  1192
  assumes "z \<noteq> 0"
lp15@59751
  1193
    shows "Ln(-z) = (if Im(z) \<le> 0 \<and> ~(Re(z) < 0 \<and> Im(z) = 0)
lp15@59751
  1194
                     then Ln(z) + ii * pi
lp15@59751
  1195
                     else Ln(z) - ii * pi)" (is "_ = ?rhs")
lp15@59751
  1196
  using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
lp15@59751
  1197
        Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z]
lp15@59751
  1198
    by (auto simp: of_real_numeral exp_add exp_diff exp_Euler intro!: Ln_unique)
lp15@59751
  1199
lp15@59751
  1200
lemma Ln_inverse_if:
lp15@59751
  1201
  assumes "z \<noteq> 0"
lp15@59751
  1202
    shows "Ln (inverse z) =
lp15@59751
  1203
            (if (Im(z) = 0 \<longrightarrow> 0 < Re z)
lp15@59751
  1204
             then -(Ln z)
lp15@59751
  1205
             else -(Ln z) + \<i> * 2 * complex_of_real pi)"
lp15@59751
  1206
proof (cases "(Im(z) = 0 \<longrightarrow> 0 < Re z)")
lp15@59751
  1207
  case True then show ?thesis
lp15@59751
  1208
    by (simp add: Ln_inverse)
lp15@59751
  1209
next
lp15@59751
  1210
  case False
lp15@59751
  1211
  then have z: "Im z = 0" "Re z < 0"
lp15@59751
  1212
    using assms
lp15@59751
  1213
    apply auto
lp15@59751
  1214
    by (metis cnj.code complex_cnj_cnj not_less_iff_gr_or_eq zero_complex.simps(1) zero_complex.simps(2))
lp15@59751
  1215
  have "Ln(inverse z) = Ln(- (inverse (-z)))"
lp15@59751
  1216
    by simp
lp15@59751
  1217
  also have "... = Ln (inverse (-z)) + \<i> * complex_of_real pi"
lp15@59751
  1218
    using assms z
lp15@59751
  1219
    apply (simp add: Ln_minus)
lp15@59751
  1220
    apply (simp add: field_simps)
lp15@59751
  1221
    done
lp15@59751
  1222
  also have "... = - Ln (- z) + \<i> * complex_of_real pi"
lp15@59751
  1223
    apply (subst Ln_inverse)
lp15@59751
  1224
    using z assms by auto
lp15@59751
  1225
  also have "... = - (Ln z) + \<i> * 2 * complex_of_real pi"
lp15@59751
  1226
    apply (subst Ln_minus [OF assms])
lp15@59751
  1227
    using assms z
lp15@59751
  1228
    apply simp
lp15@59751
  1229
    done
lp15@59751
  1230
  finally show ?thesis
lp15@59751
  1231
    using assms z
lp15@59751
  1232
    by simp
lp15@59751
  1233
qed
lp15@59751
  1234
lp15@59751
  1235
lemma Ln_times_ii:
lp15@59751
  1236
  assumes "z \<noteq> 0"
lp15@59751
  1237
    shows  "Ln(ii * z) = (if 0 \<le> Re(z) | Im(z) < 0
lp15@59751
  1238
                          then Ln(z) + ii * of_real pi/2
lp15@59751
  1239
                          else Ln(z) - ii * of_real(3 * pi/2))"
lp15@59751
  1240
  using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
lp15@59751
  1241
        Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z] Re_Ln_pos_le [of z]
lp15@59751
  1242
  by (auto simp: of_real_numeral Ln_times)
lp15@59751
  1243
lp15@59751
  1244
lp15@59751
  1245
subsection{*Relation between Square Root and exp/ln, hence its derivative*}
lp15@59751
  1246
lp15@59751
  1247
lemma csqrt_exp_Ln:
lp15@59751
  1248
  assumes "z \<noteq> 0"
lp15@59751
  1249
    shows "csqrt z = exp(Ln(z) / 2)"
lp15@59751
  1250
proof -
lp15@59751
  1251
  have "(exp (Ln z / 2))\<^sup>2 = (exp (Ln z))"
lp15@59751
  1252
    by (metis exp_double nonzero_mult_divide_cancel_left times_divide_eq_right zero_neq_numeral)
lp15@59751
  1253
  also have "... = z"
lp15@59751
  1254
    using assms exp_Ln by blast
lp15@59751
  1255
  finally have "csqrt z = csqrt ((exp (Ln z / 2))\<^sup>2)"
lp15@59751
  1256
    by simp
lp15@59751
  1257
  also have "... = exp (Ln z / 2)"
lp15@59751
  1258
    apply (subst csqrt_square)
lp15@59751
  1259
    using cos_gt_zero_pi [of "(Im (Ln z) / 2)"] Im_Ln_le_pi mpi_less_Im_Ln assms
lp15@59751
  1260
    apply (auto simp: Re_exp Im_exp zero_less_mult_iff zero_le_mult_iff, fastforce+)
lp15@59751
  1261
    done
lp15@59751
  1262
  finally show ?thesis using assms csqrt_square
lp15@59751
  1263
    by simp
lp15@59751
  1264
qed
lp15@59751
  1265
lp15@59751
  1266
lemma csqrt_inverse:
lp15@59751
  1267
  assumes "Im(z) = 0 \<Longrightarrow> 0 < Re z"
lp15@59751
  1268
    shows "csqrt (inverse z) = inverse (csqrt z)"
lp15@59751
  1269
proof (cases "z=0", simp)
lp15@59751
  1270
  assume "z \<noteq> 0 "
lp15@59751
  1271
  then show ?thesis
lp15@59751
  1272
    using assms
lp15@59751
  1273
    by (simp add: csqrt_exp_Ln Ln_inverse exp_minus)
lp15@59751
  1274
qed
lp15@59751
  1275
lp15@59751
  1276
lemma cnj_csqrt:
lp15@59751
  1277
  assumes "Im z = 0 \<Longrightarrow> 0 \<le> Re(z)"
lp15@59751
  1278
    shows "cnj(csqrt z) = csqrt(cnj z)"
lp15@59751
  1279
proof (cases "z=0", simp)
lp15@59751
  1280
  assume z: "z \<noteq> 0"
lp15@59751
  1281
  then have "Im z = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
  1282
    using assms cnj.code complex_cnj_zero_iff by fastforce
lp15@59751
  1283
  then show ?thesis
lp15@59751
  1284
   using z by (simp add: csqrt_exp_Ln cnj_Ln exp_cnj)
lp15@59751
  1285
qed
lp15@59751
  1286
lp15@59751
  1287
lemma has_field_derivative_csqrt:
lp15@59751
  1288
  assumes "Im z = 0 \<Longrightarrow> 0 < Re(z)"
lp15@59751
  1289
    shows "(csqrt has_field_derivative inverse(2 * csqrt z)) (at z)"
lp15@59751
  1290
proof -
lp15@59751
  1291
  have z: "z \<noteq> 0"
lp15@59751
  1292
    using assms by auto
lp15@59751
  1293
  then have *: "inverse z = inverse (2*z) * 2"
lp15@59751
  1294
    by (simp add: divide_simps)
lp15@59751
  1295
  show ?thesis
lp15@59751
  1296
    apply (rule DERIV_transform_at [where f = "\<lambda>z. exp(Ln(z) / 2)" and d = "norm z"])
lp15@59751
  1297
    apply (intro derivative_eq_intros | simp add: assms)+
lp15@59751
  1298
    apply (rule *)
lp15@59751
  1299
    using z
lp15@59751
  1300
    apply (auto simp: field_simps csqrt_exp_Ln [symmetric])
lp15@59751
  1301
    apply (metis power2_csqrt power2_eq_square)
lp15@59751
  1302
    apply (metis csqrt_exp_Ln dist_0_norm less_irrefl)
lp15@59751
  1303
    done
lp15@59751
  1304
qed
lp15@59751
  1305
lp15@59751
  1306
lemma complex_differentiable_at_csqrt:
lp15@59751
  1307
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable at z"
lp15@59751
  1308
  using complex_differentiable_def has_field_derivative_csqrt by blast
lp15@59751
  1309
lp15@59751
  1310
lemma complex_differentiable_within_csqrt:
lp15@59751
  1311
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable (at z within s)"
lp15@59751
  1312
  using complex_differentiable_at_csqrt complex_differentiable_within_subset by blast
lp15@59751
  1313
lp15@59751
  1314
lemma continuous_at_csqrt:
lp15@59751
  1315
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) csqrt"
lp15@59751
  1316
  by (simp add: complex_differentiable_within_csqrt complex_differentiable_imp_continuous_at)
lp15@59751
  1317
lp15@59862
  1318
corollary isCont_csqrt' [simp]:
lp15@59862
  1319
   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. csqrt (f x)) z"
lp15@59862
  1320
  by (blast intro: isCont_o2 [OF _ continuous_at_csqrt])
lp15@59862
  1321
lp15@59751
  1322
lemma continuous_within_csqrt:
lp15@59751
  1323
    "(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) csqrt"
lp15@59751
  1324
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_csqrt)
lp15@59751
  1325
lp15@59751
  1326
lemma continuous_on_csqrt [continuous_intros]:
lp15@59751
  1327
    "(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous_on s csqrt"
lp15@59751
  1328
  by (simp add: continuous_at_imp_continuous_on continuous_within_csqrt)
lp15@59751
  1329
lp15@59751
  1330
lemma holomorphic_on_csqrt:
lp15@59751
  1331
    "(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt holomorphic_on s"
lp15@59751
  1332
  by (simp add: complex_differentiable_within_csqrt holomorphic_on_def)
lp15@59751
  1333
lp15@59751
  1334
lemma continuous_within_closed_nontrivial:
lp15@59751
  1335
    "closed s \<Longrightarrow> a \<notin> s ==> continuous (at a within s) f"
lp15@59751
  1336
  using open_Compl
lp15@59751
  1337
  by (force simp add: continuous_def eventually_at_topological filterlim_iff open_Collect_neg)
lp15@59751
  1338
lp15@59751
  1339
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
lp15@59751
  1340
  using closed_halfspace_Re_ge
lp15@59751
  1341
  by (simp add: closed_Int closed_complex_Reals)
lp15@59751
  1342
lp15@59751
  1343
lemma continuous_within_csqrt_posreal:
lp15@59751
  1344
    "continuous (at z within (\<real> \<inter> {w. 0 \<le> Re(w)})) csqrt"
lp15@59751
  1345
proof (cases "Im z = 0 --> 0 < Re(z)")
lp15@59751
  1346
  case True then show ?thesis
lp15@59751
  1347
    by (blast intro: continuous_within_csqrt)
lp15@59751
  1348
next
lp15@59751
  1349
  case False
lp15@59751
  1350
  then have "Im z = 0" "Re z < 0 \<or> z = 0"
lp15@59751
  1351
    using False cnj.code complex_cnj_zero_iff by auto force
lp15@59751
  1352
  then show ?thesis
lp15@59751
  1353
    apply (auto simp: continuous_within_closed_nontrivial [OF closed_Real_halfspace_Re_ge])
lp15@59751
  1354
    apply (auto simp: continuous_within_eps_delta norm_conv_dist [symmetric])
lp15@59751
  1355
    apply (rule_tac x="e^2" in exI)
lp15@59751
  1356
    apply (auto simp: Reals_def)
lp15@59751
  1357
by (metis linear not_less real_sqrt_less_iff real_sqrt_pow2_iff real_sqrt_power)
lp15@59751
  1358
qed
lp15@59751
  1359
lp15@59870
  1360
subsection{*Complex arctangent*}
lp15@59870
  1361
lp15@59870
  1362
text{*branch cut gives standard bounds in real case.*}
lp15@59870
  1363
lp15@59870
  1364
definition Arctan :: "complex \<Rightarrow> complex" where
lp15@59870
  1365
    "Arctan \<equiv> \<lambda>z. (\<i>/2) * Ln((1 - \<i>*z) / (1 + \<i>*z))"
lp15@59870
  1366
lp15@59870
  1367
lemma Arctan_0 [simp]: "Arctan 0 = 0"
lp15@59870
  1368
  by (simp add: Arctan_def)
lp15@59870
  1369
lp15@59870
  1370
lemma Im_complex_div_lemma: "Im((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<longleftrightarrow> Re z = 0"
lp15@59870
  1371
  by (auto simp: Im_complex_div_eq_0 algebra_simps)
lp15@59870
  1372
lp15@59870
  1373
lemma Re_complex_div_lemma: "0 < Re((1 - \<i>*z) / (1 + \<i>*z)) \<longleftrightarrow> norm z < 1"
lp15@59870
  1374
  by (simp add: Re_complex_div_gt_0 algebra_simps cmod_def power2_eq_square)
lp15@59870
  1375
lp15@59870
  1376
lemma tan_Arctan:
lp15@59870
  1377
  assumes "z\<^sup>2 \<noteq> -1"
lp15@59870
  1378
    shows [simp]:"tan(Arctan z) = z"
lp15@59870
  1379
proof -
lp15@59870
  1380
  have "1 + \<i>*z \<noteq> 0"
lp15@59870
  1381
    by (metis assms complex_i_mult_minus i_squared minus_unique power2_eq_square power2_minus)
lp15@59870
  1382
  moreover
lp15@59870
  1383
  have "1 - \<i>*z \<noteq> 0"
lp15@59870
  1384
    by (metis assms complex_i_mult_minus i_squared power2_eq_square power2_minus right_minus_eq)
lp15@59870
  1385
  ultimately
lp15@59870
  1386
  show ?thesis
lp15@59870
  1387
    by (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus csqrt_exp_Ln [symmetric]
lp15@59870
  1388
                  divide_simps power2_eq_square [symmetric])
lp15@59870
  1389
qed
lp15@59870
  1390
lp15@59870
  1391
lemma Arctan_tan [simp]:
lp15@59870
  1392
  assumes "\<bar>Re z\<bar> < pi/2"
lp15@59870
  1393
    shows "Arctan(tan z) = z"
lp15@59870
  1394
proof -
lp15@59870
  1395
  have ge_pi2: "\<And>n::int. abs (of_int (2*n + 1) * pi/2) \<ge> pi/2"
lp15@59870
  1396
    by (case_tac n rule: int_cases) (auto simp: abs_mult)
lp15@59870
  1397
  have "exp (\<i>*z)*exp (\<i>*z) = -1 \<longleftrightarrow> exp (2*\<i>*z) = -1"
lp15@59870
  1398
    by (metis distrib_right exp_add mult_2)
lp15@59870
  1399
  also have "... \<longleftrightarrow> exp (2*\<i>*z) = exp (\<i>*pi)"
lp15@59870
  1400
    using cis_conv_exp cis_pi by auto
lp15@59870
  1401
  also have "... \<longleftrightarrow> exp (2*\<i>*z - \<i>*pi) = 1"
lp15@59870
  1402
    by (metis (no_types) diff_add_cancel diff_minus_eq_add exp_add exp_minus_inverse mult.commute)
lp15@59870
  1403
  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
  1404
    by (simp add: exp_eq_1)
lp15@59870
  1405
  also have "... \<longleftrightarrow> Im z = 0 \<and> (\<exists>n::int. 2 * Re z = of_int (2*n + 1) * pi)"
lp15@59870
  1406
    by (simp add: algebra_simps)
lp15@59870
  1407
  also have "... \<longleftrightarrow> False"
lp15@59870
  1408
    using assms ge_pi2
lp15@59870
  1409
    apply (auto simp: algebra_simps)
lp15@59870
  1410
    by (metis abs_mult_pos not_less not_real_of_nat_less_zero real_of_nat_numeral)
lp15@59870
  1411
  finally have *: "exp (\<i>*z)*exp (\<i>*z) + 1 \<noteq> 0"
lp15@59870
  1412
    by (auto simp: add.commute minus_unique)
lp15@59870
  1413
  show ?thesis
lp15@59870
  1414
    using assms *
lp15@59870
  1415
    apply (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus divide_simps
lp15@59870
  1416
                     ii_times_eq_iff power2_eq_square [symmetric])
lp15@59870
  1417
    apply (rule Ln_unique)
lp15@59870
  1418
    apply (auto simp: divide_simps exp_minus)
lp15@59870
  1419
    apply (simp add: algebra_simps exp_double [symmetric])
lp15@59870
  1420
    done
lp15@59870
  1421
qed
lp15@59870
  1422
lp15@59870
  1423
lemma
lp15@59870
  1424
  assumes "Re z = 0 \<Longrightarrow> abs(Im z) < 1"
lp15@59870
  1425
  shows Re_Arctan_bounds: "abs(Re(Arctan z)) < pi/2"
lp15@59870
  1426
    and has_field_derivative_Arctan: "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
lp15@59870
  1427
proof -
lp15@59870
  1428
  have nz0: "1 + \<i>*z \<noteq> 0"
lp15@59870
  1429
    using assms
lp15@59870
  1430
    by (metis abs_one complex_i_mult_minus diff_0_right diff_minus_eq_add ii.simps(1) ii.simps(2) 
lp15@59870
  1431
              less_irrefl minus_diff_eq mult.right_neutral right_minus_eq)
lp15@59870
  1432
  have "z \<noteq> -\<i>" using assms
lp15@59870
  1433
    by auto
lp15@59870
  1434
  then have zz: "1 + z * z \<noteq> 0"
lp15@59870
  1435
    by (metis abs_one assms i_squared ii.simps less_irrefl minus_unique square_eq_iff)
lp15@59870
  1436
  have nz1: "1 - \<i>*z \<noteq> 0"
lp15@59870
  1437
    using assms by (force simp add: ii_times_eq_iff)
lp15@59870
  1438
  have nz2: "inverse (1 + \<i>*z) \<noteq> 0"
lp15@59870
  1439
    using assms
lp15@59870
  1440
    by (metis Im_complex_div_lemma Re_complex_div_lemma cmod_eq_Im divide_complex_def
lp15@59870
  1441
              less_irrefl mult_zero_right zero_complex.simps(1) zero_complex.simps(2))
lp15@59870
  1442
  have nzi: "((1 - \<i>*z) * inverse (1 + \<i>*z)) \<noteq> 0"
lp15@59870
  1443
    using nz1 nz2 by auto
lp15@59870
  1444
  have *: "Im ((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<Longrightarrow> 0 < Re ((1 - \<i>*z) / (1 + \<i>*z))"
lp15@59870
  1445
    apply (simp add: divide_complex_def)
lp15@59870
  1446
    apply (simp add: divide_simps split: split_if_asm)
lp15@59870
  1447
    using assms
lp15@59870
  1448
    apply (auto simp: algebra_simps abs_square_less_1 [unfolded power2_eq_square])
lp15@59870
  1449
    done
lp15@59870
  1450
  show "abs(Re(Arctan z)) < pi/2"
lp15@59870
  1451
    unfolding Arctan_def divide_complex_def
lp15@59870
  1452
    using mpi_less_Im_Ln [OF nzi]
lp15@59870
  1453
    by (auto simp: abs_if intro: Im_Ln_less_pi * [unfolded divide_complex_def])
lp15@59870
  1454
  show "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
lp15@59870
  1455
    unfolding Arctan_def scaleR_conv_of_real
lp15@59870
  1456
    apply (rule DERIV_cong)
lp15@59870
  1457
    apply (intro derivative_eq_intros | simp add: nz0 *)+
lp15@59870
  1458
    using nz0 nz1 zz
lp15@59870
  1459
    apply (simp add: divide_simps power2_eq_square)
lp15@59870
  1460
    apply (auto simp: algebra_simps)
lp15@59870
  1461
    done
lp15@59870
  1462
qed
lp15@59870
  1463
lp15@59870
  1464
lemma complex_differentiable_at_Arctan: "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable at z"
lp15@59870
  1465
  using has_field_derivative_Arctan
lp15@59870
  1466
  by (auto simp: complex_differentiable_def)
lp15@59870
  1467
lp15@59870
  1468
lemma complex_differentiable_within_Arctan:
lp15@59870
  1469
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable (at z within s)"
lp15@59870
  1470
  using complex_differentiable_at_Arctan complex_differentiable_at_within by blast
lp15@59870
  1471
lp15@59870
  1472
declare has_field_derivative_Arctan [derivative_intros]
lp15@59870
  1473
declare has_field_derivative_Arctan [THEN DERIV_chain2, derivative_intros]
lp15@59870
  1474
lp15@59870
  1475
lemma continuous_at_Arctan:
lp15@59870
  1476
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z) Arctan"
lp15@59870
  1477
  by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Arctan)
lp15@59870
  1478
lp15@59870
  1479
lemma continuous_within_Arctan:
lp15@59870
  1480
    "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z within s) Arctan"
lp15@59870
  1481
  using continuous_at_Arctan continuous_at_imp_continuous_within by blast
lp15@59870
  1482
lp15@59870
  1483
lemma continuous_on_Arctan [continuous_intros]:
lp15@59870
  1484
    "(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous_on s Arctan"
lp15@59870
  1485
  by (auto simp: continuous_at_imp_continuous_on continuous_within_Arctan)
lp15@59870
  1486
lp15@59870
  1487
lemma holomorphic_on_Arctan:
lp15@59870
  1488
    "(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan holomorphic_on s"
lp15@59870
  1489
  by (simp add: complex_differentiable_within_Arctan holomorphic_on_def)
lp15@59870
  1490
lp15@59870
  1491
lp15@59870
  1492
subsection {*Real arctangent*}
lp15@59870
  1493
lp15@59870
  1494
lemma norm_exp_ii_times [simp]: "norm (exp(\<i> * of_real y)) = 1"
lp15@59870
  1495
  by simp
lp15@59870
  1496
lp15@59870
  1497
lemma norm_exp_imaginary: "norm(exp z) = 1 \<Longrightarrow> Re z = 0"
lp15@59870
  1498
  by (simp add: complex_norm_eq_1_exp)
lp15@59870
  1499
lp15@59870
  1500
lemma Im_Arctan_of_real [simp]: "Im (Arctan (of_real x)) = 0"
lp15@59870
  1501
  unfolding Arctan_def divide_complex_def
lp15@59870
  1502
  apply (simp add: complex_eq_iff)
lp15@59870
  1503
  apply (rule norm_exp_imaginary)
lp15@59870
  1504
  apply (subst exp_Ln, auto)
lp15@59870
  1505
  apply (simp_all add: cmod_def complex_eq_iff)
lp15@59870
  1506
  apply (auto simp: divide_simps)
lp15@59870
  1507
  apply (metis power_one realpow_two_sum_zero_iff zero_neq_one, algebra)
lp15@59870
  1508
  done
lp15@59870
  1509
lp15@59870
  1510
lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
lp15@59870
  1511
proof (rule arctan_unique)
lp15@59870
  1512
  show "- (pi / 2) < Re (Arctan (complex_of_real x))"
lp15@59870
  1513
    apply (simp add: Arctan_def)
lp15@59870
  1514
    apply (rule Im_Ln_less_pi)
lp15@59870
  1515
    apply (auto simp: Im_complex_div_lemma)
lp15@59870
  1516
    done
lp15@59870
  1517
next
lp15@59870
  1518
  have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
lp15@59870
  1519
    by (simp add: divide_simps) ( simp add: complex_eq_iff)
lp15@59870
  1520
  show "Re (Arctan (complex_of_real x)) < pi / 2"
lp15@59870
  1521
    using mpi_less_Im_Ln [OF *]
lp15@59870
  1522
    by (simp add: Arctan_def)
lp15@59870
  1523
next
lp15@59870
  1524
  have "tan (Re (Arctan (of_real x))) = Re (tan (Arctan (of_real x)))"
lp15@59870
  1525
    apply (auto simp: tan_def Complex.Re_divide Re_sin Re_cos Im_sin Im_cos)
lp15@59870
  1526
    apply (simp add: field_simps)
lp15@59870
  1527
    by (simp add: power2_eq_square)
lp15@59870
  1528
  also have "... = x"
lp15@59870
  1529
    apply (subst tan_Arctan, auto)
lp15@59870
  1530
    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
  1531
  finally show "tan (Re (Arctan (complex_of_real x))) = x" .
lp15@59870
  1532
qed
lp15@59870
  1533
lp15@59870
  1534
lemma Arctan_of_real: "Arctan (of_real x) = of_real (arctan x)"
lp15@59870
  1535
  unfolding arctan_eq_Re_Arctan divide_complex_def
lp15@59870
  1536
  by (simp add: complex_eq_iff)
lp15@59870
  1537
lp15@59870
  1538
lemma Arctan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Arctan z \<in> \<real>"
lp15@59870
  1539
  by (metis Reals_cases Reals_of_real Arctan_of_real)
lp15@59870
  1540
lp15@59870
  1541
declare arctan_one [simp]
lp15@59870
  1542
lp15@59870
  1543
lemma arctan_less_pi4_pos: "x < 1 \<Longrightarrow> arctan x < pi/4"
lp15@59870
  1544
  by (metis arctan_less_iff arctan_one)
lp15@59870
  1545
lp15@59870
  1546
lemma arctan_less_pi4_neg: "-1 < x \<Longrightarrow> -(pi/4) < arctan x"
lp15@59870
  1547
  by (metis arctan_less_iff arctan_minus arctan_one)
lp15@59870
  1548
lp15@59870
  1549
lemma arctan_less_pi4: "abs x < 1 \<Longrightarrow> abs(arctan x) < pi/4"
lp15@59870
  1550
  by (metis abs_less_iff arctan_less_pi4_pos arctan_minus)
lp15@59870
  1551
lp15@59870
  1552
lemma arctan_le_pi4: "abs x \<le> 1 \<Longrightarrow> abs(arctan x) \<le> pi/4"
lp15@59870
  1553
  by (metis abs_le_iff arctan_le_iff arctan_minus arctan_one)
lp15@59870
  1554
lp15@59870
  1555
lemma abs_arctan: "abs(arctan x) = arctan(abs x)"
lp15@59870
  1556
  by (simp add: abs_if arctan_minus)
lp15@59870
  1557
lp15@59870
  1558
lemma arctan_add_raw:
lp15@59870
  1559
  assumes "abs(arctan x + arctan y) < pi/2"
lp15@59870
  1560
    shows "arctan x + arctan y = arctan((x + y) / (1 - x * y))"
lp15@59870
  1561
proof (rule arctan_unique [symmetric])
lp15@59870
  1562
  show 12: "- (pi / 2) < arctan x + arctan y" "arctan x + arctan y < pi / 2"
lp15@59870
  1563
    using assms by linarith+
lp15@59870
  1564
  show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
lp15@59870
  1565
    using cos_gt_zero_pi [OF 12]
lp15@59870
  1566
    by (simp add: arctan tan_add)
lp15@59870
  1567
qed
lp15@59870
  1568
lp15@59870
  1569
lemma arctan_inverse:
lp15@59870
  1570
  assumes "0 < x"
lp15@59870
  1571
    shows "arctan(inverse x) = pi/2 - arctan x"
lp15@59870
  1572
proof -
lp15@59870
  1573
  have "arctan(inverse x) = arctan(inverse(tan(arctan x)))"
lp15@59870
  1574
    by (simp add: arctan)
lp15@59870
  1575
  also have "... = arctan (tan (pi / 2 - arctan x))"
lp15@59870
  1576
    by (simp add: tan_cot)
lp15@59870
  1577
  also have "... = pi/2 - arctan x"
lp15@59870
  1578
  proof -
lp15@59870
  1579
    have "0 < pi - arctan x"
lp15@59870
  1580
    using arctan_ubound [of x] pi_gt_zero by linarith
lp15@59870
  1581
    with assms show ?thesis
lp15@59870
  1582
      by (simp add: Transcendental.arctan_tan)
lp15@59870
  1583
  qed
lp15@59870
  1584
  finally show ?thesis .
lp15@59870
  1585
qed
lp15@59870
  1586
lp15@59870
  1587
lemma arctan_add_small:
lp15@59870
  1588
  assumes "abs(x * y) < 1"
lp15@59870
  1589
    shows "(arctan x + arctan y = arctan((x + y) / (1 - x * y)))"
lp15@59870
  1590
proof (cases "x = 0 \<or> y = 0")
lp15@59870
  1591
  case True then show ?thesis
lp15@59870
  1592
    by auto
lp15@59870
  1593
next
lp15@59870
  1594
  case False
lp15@59870
  1595
  then have *: "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
lp15@59870
  1596
    apply (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
lp15@59870
  1597
    apply (simp add: divide_simps abs_mult)
lp15@59870
  1598
    done
lp15@59870
  1599
  show ?thesis
lp15@59870
  1600
    apply (rule arctan_add_raw)
lp15@59870
  1601
    using * by linarith
lp15@59870
  1602
qed
lp15@59870
  1603
lp15@59870
  1604
lemma abs_arctan_le:
lp15@59870
  1605
  fixes x::real shows "abs(arctan x) \<le> abs x"
lp15@59870
  1606
proof -
lp15@59870
  1607
  { fix w::complex and z::complex
lp15@59870
  1608
    assume *: "w \<in> \<real>" "z \<in> \<real>"
lp15@59870
  1609
    have "cmod (Arctan w - Arctan z) \<le> 1 * cmod (w-z)"
lp15@59870
  1610
      apply (rule complex_differentiable_bound [OF convex_Reals, of Arctan _ 1])
lp15@59870
  1611
      apply (rule has_field_derivative_at_within [OF has_field_derivative_Arctan])
lp15@59870
  1612
      apply (force simp add: Reals_def)
lp15@59870
  1613
      apply (simp add: norm_divide divide_simps in_Reals_norm complex_is_Real_iff power2_eq_square)
lp15@59870
  1614
      using * by auto
lp15@59870
  1615
  }
lp15@59870
  1616
  then have "cmod (Arctan (of_real x) - Arctan 0) \<le> 1 * cmod (of_real x -0)"
lp15@59870
  1617
    using Reals_0 Reals_of_real by blast
lp15@59870
  1618
  then show ?thesis
lp15@59870
  1619
    by (simp add: Arctan_of_real)
lp15@59870
  1620
qed
lp15@59870
  1621
lp15@59870
  1622
lemma arctan_le_self: "0 \<le> x \<Longrightarrow> arctan x \<le> x"
lp15@59870
  1623
  by (metis abs_arctan_le abs_of_nonneg zero_le_arctan_iff)
lp15@59870
  1624
lp15@59870
  1625
lemma abs_tan_ge: "abs x < pi/2 \<Longrightarrow> abs x \<le> abs(tan x)"
lp15@59870
  1626
  by (metis abs_arctan_le abs_less_iff arctan_tan minus_less_iff)
lp15@59870
  1627
lp15@59870
  1628
lp15@59870
  1629
subsection{*Inverse Sine*}
lp15@59870
  1630
lp15@59870
  1631
definition Arcsin :: "complex \<Rightarrow> complex" where
lp15@59870
  1632
   "Arcsin \<equiv> \<lambda>z. -\<i> * Ln(\<i> * z + csqrt(1 - z\<^sup>2))"
lp15@59870
  1633
lp15@59870
  1634
lemma Arcsin_body_lemma: "\<i> * z + csqrt(1 - z\<^sup>2) \<noteq> 0"
lp15@59870
  1635
  using power2_csqrt [of "1 - z\<^sup>2"]
lp15@59870
  1636
  apply auto
lp15@59870
  1637
  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
  1638
lp15@59870
  1639
lemma Arcsin_range_lemma: "abs (Re z) < 1 \<Longrightarrow> 0 < Re(\<i> * z + csqrt(1 - z\<^sup>2))"
lp15@59870
  1640
  using Complex.cmod_power2 [of z, symmetric]
lp15@59870
  1641
  by (simp add: real_less_rsqrt algebra_simps Re_power2 cmod_square_less_1_plus)
lp15@59870
  1642
lp15@59870
  1643
lemma Re_Arcsin: "Re(Arcsin z) = Im (Ln (\<i> * z + csqrt(1 - z\<^sup>2)))"
lp15@59870
  1644
  by (simp add: Arcsin_def)
lp15@59870
  1645
lp15@59870
  1646
lemma Im_Arcsin: "Im(Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"
lp15@59870
  1647
  by (simp add: Arcsin_def Arcsin_body_lemma)
lp15@59870
  1648
lp15@59870
  1649
lemma isCont_Arcsin:
lp15@59870
  1650
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  1651
    shows "isCont Arcsin z"
lp15@59870
  1652
proof -
lp15@59870
  1653
  have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
lp15@59870
  1654
    using assms
lp15@59870
  1655
    by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
lp15@59870
  1656
  have cmz: "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
lp15@59870
  1657
    by (blast intro: assms cmod_square_less_1_plus)
lp15@59870
  1658
  show ?thesis
lp15@59870
  1659
    using assms
lp15@59870
  1660
    apply (simp add: Arcsin_def)
lp15@59870
  1661
    apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
lp15@59870
  1662
    apply (erule rez)
lp15@59870
  1663
    apply (auto simp: Re_power2 Im_power2 abs_square_less_1 [symmetric] real_less_rsqrt algebra_simps split: split_if_asm)
lp15@59870
  1664
    apply (simp add: norm_complex_def)
lp15@59870
  1665
    using cmod_power2 [of z, symmetric] cmz
lp15@59870
  1666
    apply (simp add: real_less_rsqrt)
lp15@59870
  1667
    done
lp15@59870
  1668
qed
lp15@59870
  1669
lp15@59870
  1670
lemma isCont_Arcsin' [simp]:
lp15@59870
  1671
  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
  1672
  by (blast intro: isCont_o2 [OF _ isCont_Arcsin])
lp15@59870
  1673
lp15@59870
  1674
lemma sin_Arcsin [simp]: "sin(Arcsin z) = z"
lp15@59870
  1675
proof -  
lp15@59870
  1676
  have "\<i>*z*2 + csqrt (1 - z\<^sup>2)*2 = 0 \<longleftrightarrow> (\<i>*z)*2 + csqrt (1 - z\<^sup>2)*2 = 0"
lp15@59870
  1677
    by (simp add: algebra_simps)  --{*Cancelling a factor of 2*}
lp15@59870
  1678
  moreover have "... \<longleftrightarrow> (\<i>*z) + csqrt (1 - z\<^sup>2) = 0"
lp15@59870
  1679
    by (metis Arcsin_body_lemma distrib_right no_zero_divisors zero_neq_numeral)
lp15@59870
  1680
  ultimately show ?thesis
lp15@59870
  1681
    apply (simp add: sin_exp_eq Arcsin_def Arcsin_body_lemma exp_minus divide_simps)
lp15@59870
  1682
    apply (simp add: algebra_simps)
lp15@59870
  1683
    apply (simp add: power2_eq_square [symmetric] algebra_simps)
lp15@59870
  1684
    done
lp15@59870
  1685
qed
lp15@59870
  1686
lp15@59870
  1687
lemma Re_eq_pihalf_lemma:
lp15@59870
  1688
    "\<bar>Re z\<bar> = pi/2 \<Longrightarrow> Im z = 0 \<Longrightarrow>
lp15@59870
  1689
      Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2) = 0 \<and> 0 \<le> Im ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
lp15@59870
  1690
  apply (simp add: cos_ii_times [symmetric] Re_cos Im_cos abs_if del: eq_divide_eq_numeral1)
lp15@59870
  1691
  by (metis cos_minus cos_pi_half)
lp15@59870
  1692
lp15@59870
  1693
lemma Re_less_pihalf_lemma:
lp15@59870
  1694
  assumes "\<bar>Re z\<bar> < pi / 2"
lp15@59870
  1695
    shows "0 < Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
lp15@59870
  1696
proof -
lp15@59870
  1697
  have "0 < cos (Re z)" using assms
lp15@59870
  1698
    using cos_gt_zero_pi by auto
lp15@59870
  1699
  then show ?thesis
lp15@59870
  1700
    by (simp add: cos_ii_times [symmetric] Re_cos Im_cos add_pos_pos)
lp15@59870
  1701
qed
lp15@59870
  1702
lp15@59870
  1703
lemma Arcsin_sin:
lp15@59870
  1704
    assumes "\<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)"
lp15@59870
  1705
      shows "Arcsin(sin z) = z"
lp15@59870
  1706
proof -
lp15@59870
  1707
  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
  1708
    by (simp add: sin_exp_eq Arcsin_def exp_minus)
lp15@59870
  1709
  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
  1710
    by (simp add: field_simps power2_eq_square)
lp15@59870
  1711
  also have "... = - (\<i> * Ln (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
lp15@59870
  1712
    apply (subst csqrt_square)
lp15@59870
  1713
    using assms Re_eq_pihalf_lemma Re_less_pihalf_lemma
lp15@59870
  1714
    apply auto
lp15@59870
  1715
    done
lp15@59870
  1716
  also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
lp15@59870
  1717
    by (simp add: field_simps power2_eq_square)
lp15@59870
  1718
  also have "... = z"
lp15@59870
  1719
    apply (subst Complex_Transcendental.Ln_exp)
lp15@59870
  1720
    using assms
lp15@59870
  1721
    apply (auto simp: abs_if simp del: eq_divide_eq_numeral1 split: split_if_asm)
lp15@59870
  1722
    done
lp15@59870
  1723
  finally show ?thesis .
lp15@59870
  1724
qed
lp15@59870
  1725
lp15@59870
  1726
lemma Arcsin_unique:
lp15@59870
  1727
    "\<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
  1728
  by (metis Arcsin_sin)
lp15@59870
  1729
lp15@59870
  1730
lemma Arcsin_0 [simp]: "Arcsin 0 = 0"
lp15@59870
  1731
  by (metis Arcsin_sin norm_zero pi_half_gt_zero real_norm_def sin_zero zero_complex.simps(1))
lp15@59870
  1732
lp15@59870
  1733
lemma Arcsin_1 [simp]: "Arcsin 1 = pi/2"
lp15@59870
  1734
  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
  1735
lp15@59870
  1736
lemma Arcsin_minus_1 [simp]: "Arcsin(-1) = - (pi/2)"
lp15@59870
  1737
  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
  1738
lp15@59870
  1739
lemma has_field_derivative_Arcsin:
lp15@59870
  1740
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  1741
    shows "(Arcsin has_field_derivative inverse(cos(Arcsin z))) (at z)"
lp15@59870
  1742
proof -
lp15@59870
  1743
  have "(sin (Arcsin z))\<^sup>2 \<noteq> 1"
lp15@59870
  1744
    using assms
lp15@59870
  1745
    apply atomize
lp15@59870
  1746
    apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
lp15@59870
  1747
    apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
lp15@59870
  1748
    by (metis abs_minus_cancel abs_one abs_power2 one_neq_neg_one)
lp15@59870
  1749
  then have "cos (Arcsin z) \<noteq> 0"
lp15@59870
  1750
    by (metis diff_0_right power_zero_numeral sin_squared_eq)
lp15@59870
  1751
  then show ?thesis
lp15@59870
  1752
    apply (rule has_complex_derivative_inverse_basic [OF DERIV_sin])
lp15@59870
  1753
    apply (auto intro: isCont_Arcsin open_ball [of z 1] assms)
lp15@59870
  1754
    done
lp15@59870
  1755
qed
lp15@59870
  1756
lp15@59870
  1757
declare has_field_derivative_Arcsin [derivative_intros]
lp15@59870
  1758
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]
lp15@59870
  1759
lp15@59870
  1760
lemma complex_differentiable_at_Arcsin:
lp15@59870
  1761
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable at z"
lp15@59870
  1762
  using complex_differentiable_def has_field_derivative_Arcsin by blast
lp15@59870
  1763
lp15@59870
  1764
lemma complex_differentiable_within_Arcsin:
lp15@59870
  1765
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable (at z within s)"
lp15@59870
  1766
  using complex_differentiable_at_Arcsin complex_differentiable_within_subset by blast
lp15@59870
  1767
lp15@59870
  1768
lemma continuous_within_Arcsin:
lp15@59870
  1769
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arcsin"
lp15@59870
  1770
  using continuous_at_imp_continuous_within isCont_Arcsin by blast
lp15@59870
  1771
lp15@59870
  1772
lemma continuous_on_Arcsin [continuous_intros]:
lp15@59870
  1773
    "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arcsin"
lp15@59870
  1774
  by (simp add: continuous_at_imp_continuous_on)
lp15@59870
  1775
lp15@59870
  1776
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
  1777
  by (simp add: complex_differentiable_within_Arcsin holomorphic_on_def)
lp15@59870
  1778
lp15@59870
  1779
lp15@59870
  1780
subsection{*Inverse Cosine*}
lp15@59870
  1781
lp15@59870
  1782
definition Arccos :: "complex \<Rightarrow> complex" where
lp15@59870
  1783
   "Arccos \<equiv> \<lambda>z. -\<i> * Ln(z + \<i> * csqrt(1 - z\<^sup>2))"
lp15@59870
  1784
lp15@59870
  1785
lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"
lp15@59870
  1786
  using Arcsin_range_lemma [of "-z"]
lp15@59870
  1787
  by simp
lp15@59870
  1788
lp15@59870
  1789
lemma Arccos_body_lemma: "z + \<i> * csqrt(1 - z\<^sup>2) \<noteq> 0"
lp15@59870
  1790
  using Arcsin_body_lemma [of z]
lp15@59870
  1791
  by (metis complex_i_mult_minus diff_add_cancel minus_diff_eq minus_unique mult.assoc mult.left_commute
lp15@59870
  1792
           power2_csqrt power2_eq_square zero_neq_one)
lp15@59870
  1793
lp15@59870
  1794
lemma Re_Arccos: "Re(Arccos z) = Im (Ln (z + \<i> * csqrt(1 - z\<^sup>2)))"
lp15@59870
  1795
  by (simp add: Arccos_def)
lp15@59870
  1796
lp15@59870
  1797
lemma Im_Arccos: "Im(Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"
lp15@59870
  1798
  by (simp add: Arccos_def Arccos_body_lemma)
lp15@59870
  1799
lp15@59870
  1800
text{*A very tricky argument to find!*}
lp15@59870
  1801
lemma abs_Re_less_1_preserve:
lp15@59870
  1802
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"  "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0"
lp15@59870
  1803
    shows "0 < Re (z + \<i> * csqrt (1 - z\<^sup>2))"
lp15@59870
  1804
proof (cases "Im z = 0")
lp15@59870
  1805
  case True
lp15@59870
  1806
  then show ?thesis
lp15@59870
  1807
    using assms 
lp15@59870
  1808
    by (fastforce simp add: cmod_def Re_power2 Im_power2 algebra_simps abs_square_less_1 [symmetric])
lp15@59870
  1809
next
lp15@59870
  1810
  case False
lp15@59870
  1811
  have Imz: "Im z = - sqrt ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
lp15@59870
  1812
    using assms abs_Re_le_cmod [of "1-z\<^sup>2"]
lp15@59870
  1813
    by (simp add: Re_power2 algebra_simps)
lp15@59870
  1814
  have "(cmod z)\<^sup>2 - 1 \<noteq> cmod (1 - z\<^sup>2)"
lp15@59870
  1815
  proof (clarsimp simp add: cmod_def)
lp15@59870
  1816
    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
  1817
    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
  1818
      by simp
lp15@59870
  1819
    then show False using False
lp15@59870
  1820
      by (simp add: power2_eq_square algebra_simps)
lp15@59870
  1821
  qed
lp15@59870
  1822
  moreover have "(Im z)\<^sup>2 = ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
lp15@59870
  1823
    apply (subst Imz, simp)
lp15@59870
  1824
    apply (subst real_sqrt_pow2)
lp15@59870
  1825
    using abs_Re_le_cmod [of "1-z\<^sup>2"]
lp15@59870
  1826
    apply (auto simp: Re_power2 field_simps)
lp15@59870
  1827
    done
lp15@59870
  1828
  ultimately show ?thesis
lp15@59870
  1829
    by (simp add: Re_power2 Im_power2 cmod_power2)
lp15@59870
  1830
qed
lp15@59870
  1831
lp15@59870
  1832
lemma isCont_Arccos:
lp15@59870
  1833
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  1834
    shows "isCont Arccos z"
lp15@59870
  1835
proof -
lp15@59870
  1836
  have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
lp15@59870
  1837
    using assms
lp15@59870
  1838
    by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
lp15@59870
  1839
  show ?thesis
lp15@59870
  1840
    using assms
lp15@59870
  1841
    apply (simp add: Arccos_def)
lp15@59870
  1842
    apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
lp15@59870
  1843
    apply (erule rez)
lp15@59870
  1844
    apply (blast intro: abs_Re_less_1_preserve)
lp15@59870
  1845
    done
lp15@59870
  1846
qed
lp15@59870
  1847
lp15@59870
  1848
lemma isCont_Arccos' [simp]:
lp15@59870
  1849
  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
  1850
  by (blast intro: isCont_o2 [OF _ isCont_Arccos])
lp15@59870
  1851
lp15@59870
  1852
lemma cos_Arccos [simp]: "cos(Arccos z) = z"
lp15@59870
  1853
proof -
lp15@59870
  1854
  have "z*2 + \<i> * (2 * csqrt (1 - z\<^sup>2)) = 0 \<longleftrightarrow> z*2 + \<i> * csqrt (1 - z\<^sup>2)*2 = 0"
lp15@59870
  1855
    by (simp add: algebra_simps)  --{*Cancelling a factor of 2*}
lp15@59870
  1856
  moreover have "... \<longleftrightarrow> z + \<i> * csqrt (1 - z\<^sup>2) = 0"
lp15@59870
  1857
    by (metis distrib_right mult_eq_0_iff zero_neq_numeral)
lp15@59870
  1858
  ultimately show ?thesis
lp15@59870
  1859
    apply (simp add: cos_exp_eq Arccos_def Arccos_body_lemma exp_minus field_simps)
lp15@59870
  1860
    apply (simp add: power2_eq_square [symmetric])
lp15@59870
  1861
    done
lp15@59870
  1862
qed
lp15@59870
  1863
lp15@59870
  1864
lemma Arccos_cos:
lp15@59870
  1865
    assumes "0 < Re z & Re z < pi \<or>
lp15@59870
  1866
             Re z = 0 & 0 \<le> Im z \<or>
lp15@59870
  1867
             Re z = pi & Im z \<le> 0"
lp15@59870
  1868
      shows "Arccos(cos z) = z"
lp15@59870
  1869
proof -
lp15@59870
  1870
  have *: "((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z))) = sin z"
lp15@59870
  1871
    by (simp add: sin_exp_eq exp_minus field_simps power2_eq_square)
lp15@59870
  1872
  have "1 - (exp (\<i> * z) + inverse (exp (\<i> * z)))\<^sup>2 / 4 = ((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z)))\<^sup>2"
lp15@59870
  1873
    by (simp add: field_simps power2_eq_square)
lp15@59870
  1874
  then have "Arccos(cos z) = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
lp15@59870
  1875
                           \<i> * csqrt (((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z)))\<^sup>2)))"
lp15@59870
  1876
    by (simp add: cos_exp_eq Arccos_def exp_minus)
lp15@59870
  1877
  also have "... = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
lp15@59870
  1878
                              \<i> * ((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z)))))"
lp15@59870
  1879
    apply (subst csqrt_square)
lp15@59870
  1880
    using assms Re_sin_pos [of z] Im_sin_nonneg [of z] Im_sin_nonneg2 [of z]
lp15@59870
  1881
    apply (auto simp: * Re_sin Im_sin)
lp15@59870
  1882
    done
lp15@59870
  1883
  also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
lp15@59870
  1884
    by (simp add: field_simps power2_eq_square)
lp15@59870
  1885
  also have "... = z"
lp15@59870
  1886
    using assms
lp15@59870
  1887
    apply (subst Complex_Transcendental.Ln_exp, auto)
lp15@59870
  1888
    done
lp15@59870
  1889
  finally show ?thesis .
lp15@59870
  1890
qed
lp15@59870
  1891
lp15@59870
  1892
lemma Arccos_unique:
lp15@59870
  1893
    "\<lbrakk>cos z = w;
lp15@59870
  1894
      0 < Re z \<and> Re z < pi \<or>
lp15@59870
  1895
      Re z = 0 \<and> 0 \<le> Im z \<or>
lp15@59870
  1896
      Re z = pi \<and> Im z \<le> 0\<rbrakk> \<Longrightarrow> Arccos w = z"
lp15@59870
  1897
  using Arccos_cos by blast
lp15@59870
  1898
lp15@59870
  1899
lemma Arccos_0 [simp]: "Arccos 0 = pi/2"
lp15@59870
  1900
  by (rule Arccos_unique) (auto simp: of_real_numeral)
lp15@59870
  1901
lp15@59870
  1902
lemma Arccos_1 [simp]: "Arccos 1 = 0"
lp15@59870
  1903
  by (rule Arccos_unique) auto
lp15@59870
  1904
lp15@59870
  1905
lemma Arccos_minus1: "Arccos(-1) = pi"
lp15@59870
  1906
  by (rule Arccos_unique) auto
lp15@59870
  1907
lp15@59870
  1908
lemma has_field_derivative_Arccos:
lp15@59870
  1909
  assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
lp15@59870
  1910
    shows "(Arccos has_field_derivative - inverse(sin(Arccos z))) (at z)"
lp15@59870
  1911
proof -
lp15@59870
  1912
  have "(cos (Arccos z))\<^sup>2 \<noteq> 1"
lp15@59870
  1913
    using assms
lp15@59870
  1914
    apply atomize
lp15@59870
  1915
    apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
lp15@59870
  1916
    apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
lp15@59870
  1917
    apply (metis left_minus less_irrefl power_one sum_power2_gt_zero_iff zero_neq_neg_one)
lp15@59870
  1918
    done
lp15@59870
  1919
  then have "- sin (Arccos z) \<noteq> 0"
lp15@59870
  1920
    by (metis cos_squared_eq diff_0_right mult_zero_left neg_0_equal_iff_equal power2_eq_square)
lp15@59870
  1921
  then have "(Arccos has_field_derivative inverse(- sin(Arccos z))) (at z)"
lp15@59870
  1922
    apply (rule has_complex_derivative_inverse_basic [OF DERIV_cos])
lp15@59870
  1923
    apply (auto intro: isCont_Arccos open_ball [of z 1] assms)
lp15@59870
  1924
    done
lp15@59870
  1925
  then show ?thesis
lp15@59870
  1926
    by simp
lp15@59870
  1927
qed
lp15@59870
  1928
lp15@59870
  1929
declare has_field_derivative_Arcsin [derivative_intros]
lp15@59870
  1930
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]
lp15@59870
  1931
lp15@59870
  1932
lemma complex_differentiable_at_Arccos:
lp15@59870
  1933
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos complex_differentiable at z"
lp15@59870
  1934
  using complex_differentiable_def has_field_derivative_Arccos by blast
lp15@59870
  1935
lp15@59870
  1936
lemma complex_differentiable_within_Arccos:
lp15@59870
  1937
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos complex_differentiable (at z within s)"
lp15@59870
  1938
  using complex_differentiable_at_Arccos complex_differentiable_within_subset by blast
lp15@59870
  1939
lp15@59870
  1940
lemma continuous_within_Arccos:
lp15@59870
  1941
    "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arccos"
lp15@59870
  1942
  using continuous_at_imp_continuous_within isCont_Arccos by blast
lp15@59870
  1943
lp15@59870
  1944
lemma continuous_on_Arccos [continuous_intros]:
lp15@59870
  1945
    "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arccos"
lp15@59870
  1946
  by (simp add: continuous_at_imp_continuous_on)
lp15@59870
  1947
lp15@59870
  1948
lemma holomorphic_on_Arccos: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos holomorphic_on s"
lp15@59870
  1949
  by (simp add: complex_differentiable_within_Arccos holomorphic_on_def)
lp15@59870
  1950
lp15@59870
  1951
lp15@59870
  1952
subsection{*Upper and Lower Bounds for Inverse Sine and Cosine*}
lp15@59870
  1953
lp15@59870
  1954
lemma Arcsin_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> abs(Re(Arcsin z)) < pi/2"
lp15@59870
  1955
  unfolding Re_Arcsin
lp15@59870
  1956
  by (blast intro: Re_Ln_pos_lt_imp Arcsin_range_lemma)
lp15@59870
  1957
lp15@59870
  1958
lemma Arccos_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Re(Arccos z) \<and> Re(Arccos z) < pi"
lp15@59870
  1959
  unfolding Re_Arccos
lp15@59870
  1960
  by (blast intro!: Im_Ln_pos_lt_imp Arccos_range_lemma)
lp15@59870
  1961
lp15@59870
  1962
lemma Re_Arccos_bounds: "-pi < Re(Arccos z) \<and> Re(Arccos z) \<le> pi"
lp15@59870
  1963
  unfolding Re_Arccos
lp15@59870
  1964
  by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arccos_body_lemma)
lp15@59870
  1965
lp15@59870
  1966
lemma Re_Arccos_bound: "abs(Re(Arccos z)) \<le> pi"
lp15@59870
  1967
  using Re_Arccos_bounds abs_le_interval_iff less_eq_real_def by blast
lp15@59870
  1968
lp15@59870
  1969
lemma Re_Arcsin_bounds: "-pi < Re(Arcsin z) & Re(Arcsin z) \<le> pi"
lp15@59870
  1970
  unfolding Re_Arcsin
lp15@59870
  1971
  by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arcsin_body_lemma)
lp15@59870
  1972
lp15@59870
  1973
lemma Re_Arcsin_bound: "abs(Re(Arcsin z)) \<le> pi"
lp15@59870
  1974
  using Re_Arcsin_bounds abs_le_interval_iff less_eq_real_def by blast
lp15@59870
  1975
lp15@59870
  1976
lp15@59870
  1977
subsection{*Interrelations between Arcsin and Arccos*}
lp15@59870
  1978
lp15@59870
  1979
lemma cos_Arcsin_nonzero:
lp15@59870
  1980
  assumes "z\<^sup>2 \<noteq> 1" shows "cos(Arcsin z) \<noteq> 0"
lp15@59870
  1981
proof -
lp15@59870
  1982
  have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = z\<^sup>2 * (z\<^sup>2 - 1)"
lp15@59870
  1983
    by (simp add: power_mult_distrib algebra_simps)
lp15@59870
  1984
  have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> z\<^sup>2 - 1"
lp15@59870
  1985
  proof
lp15@59870
  1986
    assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = z\<^sup>2 - 1"
lp15@59870
  1987
    then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (z\<^sup>2 - 1)\<^sup>2"
lp15@59870
  1988
      by simp
lp15@59870
  1989
    then have "z\<^sup>2 * (z\<^sup>2 - 1) = (z\<^sup>2 - 1)*(z\<^sup>2 - 1)"
lp15@59870
  1990
      using eq power2_eq_square by auto
lp15@59870
  1991
    then show False
lp15@59870
  1992
      using assms by simp
lp15@59870
  1993
  qed
lp15@59870
  1994
  then have "1 + \<i> * z * (csqrt (1 - z * z)) \<noteq> z\<^sup>2"
lp15@59870
  1995
    by (metis add_minus_cancel power2_eq_square uminus_add_conv_diff)
lp15@59870
  1996
  then have "2*(1 + \<i> * z * (csqrt (1 - z * z))) \<noteq> 2*z\<^sup>2"  (*FIXME cancel_numeral_factor*)
lp15@59870
  1997
    by (metis mult_cancel_left zero_neq_numeral)
lp15@59870
  1998
  then have "(\<i> * z + csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> -1"
lp15@59870
  1999
    using assms
lp15@59870
  2000
    apply (auto simp: power2_sum)
lp15@59870
  2001
    apply (simp add: power2_eq_square algebra_simps)
lp15@59870
  2002
    done
lp15@59870
  2003
  then show ?thesis
lp15@59870
  2004
    apply (simp add: cos_exp_eq Arcsin_def exp_minus)
lp15@59870
  2005
    apply (simp add: divide_simps Arcsin_body_lemma)
lp15@59870
  2006
    apply (metis add.commute minus_unique power2_eq_square)
lp15@59870
  2007
    done
lp15@59870
  2008
qed
lp15@59870
  2009
lp15@59870
  2010
lemma sin_Arccos_nonzero:
lp15@59870
  2011
  assumes "z\<^sup>2 \<noteq> 1" shows "sin(Arccos z) \<noteq> 0"
lp15@59870
  2012
proof -
lp15@59870
  2013
  have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = -(z\<^sup>2) * (1 - z\<^sup>2)"
lp15@59870
  2014
    by (simp add: power_mult_distrib algebra_simps)
lp15@59870
  2015
  have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1 - z\<^sup>2"
lp15@59870
  2016
  proof
lp15@59870
  2017
    assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = 1 - z\<^sup>2"
lp15@59870
  2018
    then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (1 - z\<^sup>2)\<^sup>2"
lp15@59870
  2019
      by simp
lp15@59870
  2020
    then have "-(z\<^sup>2) * (1 - z\<^sup>2) = (1 - z\<^sup>2)*(1 - z\<^sup>2)"
lp15@59870
  2021
      using eq power2_eq_square by auto
lp15@59870
  2022
    then have "-(z\<^sup>2) = (1 - z\<^sup>2)"
lp15@59870
  2023
      using assms
lp15@59870
  2024
      by (metis add.commute add.right_neutral diff_add_cancel mult_right_cancel)
lp15@59870
  2025
    then show False
lp15@59870
  2026
      using assms by simp
lp15@59870
  2027
  qed
lp15@59870
  2028
  then have "z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1"
lp15@59870
  2029
    by (simp add: algebra_simps)
lp15@59870
  2030
  then have "2*(z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2))) \<noteq> 2*1"
lp15@59870
  2031
    by (metis mult_cancel_left2 zero_neq_numeral)  (*FIXME cancel_numeral_factor*)
lp15@59870
  2032
  then have "(z + \<i> * csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> 1"
lp15@59870
  2033
    using assms
lp15@59870
  2034
    apply (auto simp: Power.comm_semiring_1_class.power2_sum power_mult_distrib)
lp15@59870
  2035
    apply (simp add: power2_eq_square algebra_simps)
lp15@59870
  2036
    done
lp15@59870
  2037
  then show ?thesis
lp15@59870
  2038
    apply (simp add: sin_exp_eq Arccos_def exp_minus)
lp15@59870
  2039
    apply (simp add: divide_simps Arccos_body_lemma)
lp15@59870
  2040
    apply (simp add: power2_eq_square)
lp15@59870
  2041
    done
lp15@59870
  2042
qed
lp15@59870
  2043
lp15@59870
  2044
lemma cos_sin_csqrt:
lp15@59870
  2045
  assumes "0 < cos(Re z)  \<or>  cos(Re z) = 0 \<and> Im z * sin(Re z) \<le> 0"
lp15@59870
  2046
    shows "cos z = csqrt(1 - (sin z)\<^sup>2)"
lp15@59870
  2047
  apply (rule csqrt_unique [THEN sym])
lp15@59870
  2048
  apply (simp add: cos_squared_eq)
lp15@59870
  2049
  using assms
lp15@59870
  2050
  apply (auto simp: Re_cos Im_cos add_pos_pos mult_le_0_iff zero_le_mult_iff)
lp15@59870
  2051
  apply (auto simp: algebra_simps)
lp15@59870
  2052
  done
lp15@59870
  2053
lp15@59870
  2054
lemma sin_cos_csqrt:
lp15@59870
  2055
  assumes "0 < sin(Re z)  \<or>  sin(Re z) = 0 \<and> 0 \<le> Im z * cos(Re z)"
lp15@59870
  2056
    shows "sin z = csqrt(1 - (cos z)\<^sup>2)"
lp15@59870
  2057
  apply (rule csqrt_unique [THEN sym])
lp15@59870
  2058
  apply (simp add: sin_squared_eq)
lp15@59870
  2059
  using assms
lp15@59870
  2060
  apply (auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff)
lp15@59870
  2061
  apply (auto simp: algebra_simps)
lp15@59870
  2062
  done
lp15@59870
  2063
lp15@59870
  2064
lemma Arcsin_Arccos_csqrt_pos:
lp15@59870
  2065
    "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arcsin z = Arccos(csqrt(1 - z\<^sup>2))"
lp15@59870
  2066
  by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
lp15@59870
  2067
lp15@59870
  2068
lemma Arccos_Arcsin_csqrt_pos:
lp15@59870
  2069
    "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arccos z = Arcsin(csqrt(1 - z\<^sup>2))"
lp15@59870
  2070
  by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
lp15@59870
  2071
lp15@59870
  2072
lemma sin_Arccos:
lp15@59870
  2073
    "0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> sin(Arccos z) = csqrt(1 - z\<^sup>2)"
lp15@59870
  2074
  by (simp add: Arccos_Arcsin_csqrt_pos)
lp15@59870
  2075
lp15@59870
  2076
lemma cos_Arcsin:
lp15@59870
  2077
    "0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> cos(Arcsin z) = csqrt(1 - z\<^sup>2)"
lp15@59870
  2078
  by (simp add: Arcsin_Arccos_csqrt_pos)
lp15@59870
  2079
lp15@59870
  2080
lp15@59870
  2081
subsection{*Relationship with Arcsin on the Real Numbers*}
lp15@59870
  2082
lp15@59870
  2083
lemma Im_Arcsin_of_real:
lp15@59870
  2084
  assumes "abs x \<le> 1"
lp15@59870
  2085
    shows "Im (Arcsin (of_real x)) = 0"
lp15@59870
  2086
proof -
lp15@59870
  2087
  have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
lp15@59870
  2088
    by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
lp15@59870
  2089
  then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
lp15@59870
  2090
    using assms abs_square_le_1
lp15@59870
  2091
    by (force simp add: Complex.cmod_power2)
lp15@59870
  2092
  then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2)) = 1"
lp15@59870
  2093
    by (simp add: norm_complex_def)
lp15@59870
  2094
  then show ?thesis
lp15@59870
  2095
    by (simp add: Im_Arcsin exp_minus)
lp15@59870
  2096
qed
lp15@59870
  2097
lp15@59870
  2098
corollary Arcsin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arcsin z \<in> \<real>"
lp15@59870
  2099
  by (metis Im_Arcsin_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)
lp15@59870
  2100
lp15@59870
  2101
lemma arcsin_eq_Re_Arcsin:
lp15@59870
  2102
  assumes "abs x \<le> 1"
lp15@59870
  2103
    shows "arcsin x = Re (Arcsin (of_real x))"
lp15@59870
  2104
unfolding arcsin_def
lp15@59870
  2105
proof (rule the_equality, safe)
lp15@59870
  2106
  show "- (pi / 2) \<le> Re (Arcsin (complex_of_real x))"
lp15@59870
  2107
    using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
lp15@59870
  2108
    by (auto simp: Complex.in_Reals_norm Re_Arcsin)
lp15@59870
  2109
next
lp15@59870
  2110
  show "Re (Arcsin (complex_of_real x)) \<le> pi / 2"
lp15@59870
  2111
    using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
lp15@59870
  2112
    by (auto simp: Complex.in_Reals_norm Re_Arcsin)
lp15@59870
  2113
next
lp15@59870
  2114
  show "sin (Re (Arcsin (complex_of_real x))) = x"
lp15@59870
  2115
    using Re_sin [of "Arcsin (of_real x)"] Arcsin_body_lemma [of "of_real x"]
lp15@59870
  2116
    by (simp add: Im_Arcsin_of_real assms)
lp15@59870
  2117
next
lp15@59870
  2118
  fix x'
lp15@59870
  2119
  assume "- (pi / 2) \<le> x'" "x' \<le> pi / 2" "x = sin x'"
lp15@59870
  2120
  then show "x' = Re (Arcsin (complex_of_real (sin x')))"
lp15@59870
  2121
    apply (simp add: sin_of_real [symmetric])
lp15@59870
  2122
    apply (subst Arcsin_sin)
lp15@59870
  2123
    apply (auto simp: )
lp15@59870
  2124
    done
lp15@59870
  2125
qed
lp15@59870
  2126
lp15@59870
  2127
lemma of_real_arcsin: "abs x \<le> 1 \<Longrightarrow> of_real(arcsin x) = Arcsin(of_real x)"
lp15@59870
  2128
  by (metis Im_Arcsin_of_real add.right_neutral arcsin_eq_Re_Arcsin complex_eq mult_zero_right of_real_0)
lp15@59870
  2129
lp15@59870
  2130
lp15@59870
  2131
subsection{*Relationship with Arccos on the Real Numbers*}
lp15@59870
  2132
lp15@59870
  2133
lemma Im_Arccos_of_real:
lp15@59870
  2134
  assumes "abs x \<le> 1"
lp15@59870
  2135
    shows "Im (Arccos (of_real x)) = 0"
lp15@59870
  2136
proof -
lp15@59870
  2137
  have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
lp15@59870
  2138
    by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
lp15@59870
  2139
  then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
lp15@59870
  2140
    using assms abs_square_le_1
lp15@59870
  2141
    by (force simp add: Complex.cmod_power2)
lp15@59870
  2142
  then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2)) = 1"
lp15@59870
  2143
    by (simp add: norm_complex_def)
lp15@59870
  2144
  then show ?thesis
lp15@59870
  2145
    by (simp add: Im_Arccos exp_minus)
lp15@59870
  2146
qed
lp15@59870
  2147
lp15@59870
  2148
corollary Arccos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arccos z \<in> \<real>"
lp15@59870
  2149
  by (metis Im_Arccos_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)
lp15@59870
  2150
lp15@59870
  2151
lemma arccos_eq_Re_Arccos:
lp15@59870
  2152
  assumes "abs x \<le> 1"
lp15@59870
  2153
    shows "arccos x = Re (Arccos (of_real x))"
lp15@59870
  2154
unfolding arccos_def
lp15@59870
  2155
proof (rule the_equality, safe)
lp15@59870
  2156
  show "0 \<le> Re (Arccos (complex_of_real x))"
lp15@59870
  2157
    using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
lp15@59870
  2158
    by (auto simp: Complex.in_Reals_norm Re_Arccos)
lp15@59870
  2159
next
lp15@59870
  2160
  show "Re (Arccos (complex_of_real x)) \<le> pi"
lp15@59870
  2161
    using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
lp15@59870
  2162
    by (auto simp: Complex.in_Reals_norm Re_Arccos)
lp15@59870
  2163
next
lp15@59870
  2164
  show "cos (Re (Arccos (complex_of_real x))) = x"
lp15@59870
  2165
    using Re_cos [of "Arccos (of_real x)"] Arccos_body_lemma [of "of_real x"]
lp15@59870
  2166
    by (simp add: Im_Arccos_of_real assms)
lp15@59870
  2167
next
lp15@59870
  2168
  fix x'
lp15@59870
  2169
  assume "0 \<le> x'" "x' \<le> pi" "x = cos x'"
lp15@59870
  2170
  then show "x' = Re (Arccos (complex_of_real (cos x')))"
lp15@59870
  2171
    apply (simp add: cos_of_real [symmetric])
lp15@59870
  2172
    apply (subst Arccos_cos)
lp15@59870
  2173
    apply (auto simp: )
lp15@59870
  2174
    done
lp15@59870
  2175
qed
lp15@59870
  2176
lp15@59870
  2177
lemma of_real_arccos: "abs x \<le> 1 \<Longrightarrow> of_real(arccos x) = Arccos(of_real x)"
lp15@59870
  2178
  by (metis Im_Arccos_of_real add.right_neutral arccos_eq_Re_Arccos complex_eq mult_zero_right of_real_0)
lp15@59751
  2179
lp15@59745
  2180
end