(* Author: John Harrison
Ported from "hol_light/Multivariate/transcendentals.ml" by L C Paulson (2015)
*)
section {* Complex Transcendental Functions *}
theory Complex_Transcendental
imports "~~/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics"
begin
lemma cmod_add_real_less:
assumes "Im z \<noteq> 0" "r\<noteq>0"
shows "cmod (z + r) < cmod z + abs r"
proof (cases z)
case (Complex x y)
have "r * x / \<bar>r\<bar> < sqrt (x*x + y*y)"
apply (rule real_less_rsqrt)
using assms
apply (simp add: Complex power2_eq_square)
using not_real_square_gt_zero by blast
then show ?thesis using assms Complex
apply (auto simp: cmod_def)
apply (rule power2_less_imp_less, auto)
apply (simp add: power2_eq_square field_simps)
done
qed
lemma cmod_diff_real_less: "Im z \<noteq> 0 \<Longrightarrow> x\<noteq>0 \<Longrightarrow> cmod (z - x) < cmod z + abs x"
using cmod_add_real_less [of z "-x"]
by simp
lemma cmod_square_less_1_plus:
assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1"
shows "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
using assms
apply (cases "Im z = 0 \<or> Re z = 0")
using abs_square_less_1
apply (force simp add: Re_power2 Im_power2 cmod_def)
using cmod_diff_real_less [of "1 - z\<^sup>2" "1"]
apply (simp add: norm_power Im_power2)
done
subsection{*The Exponential Function is Differentiable and Continuous*}
lemma complex_differentiable_at_exp: "exp complex_differentiable at z"
using DERIV_exp complex_differentiable_def by blast
lemma complex_differentiable_within_exp: "exp complex_differentiable (at z within s)"
by (simp add: complex_differentiable_at_exp complex_differentiable_at_within)
lemma continuous_within_exp:
fixes z::"'a::{real_normed_field,banach}"
shows "continuous (at z within s) exp"
by (simp add: continuous_at_imp_continuous_within)
lemma continuous_on_exp:
fixes s::"'a::{real_normed_field,banach} set"
shows "continuous_on s exp"
by (simp add: continuous_on_exp continuous_on_id)
lemma holomorphic_on_exp: "exp holomorphic_on s"
by (simp add: complex_differentiable_within_exp holomorphic_on_def)
subsection{*Euler and de Moivre formulas.*}
text{*The sine series times @{term i}*}
lemma sin_ii_eq: "(\<lambda>n. (ii * sin_coeff n) * z^n) sums (ii * sin z)"
proof -
have "(\<lambda>n. ii * sin_coeff n *\<^sub>R z^n) sums (ii * sin z)"
using sin_converges sums_mult by blast
then show ?thesis
by (simp add: scaleR_conv_of_real field_simps)
qed
theorem exp_Euler: "exp(ii * z) = cos(z) + ii * sin(z)"
proof -
have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n)
= (\<lambda>n. (ii * z) ^ n /\<^sub>R (fact n))"
proof
fix n
show "(cos_coeff n + ii * sin_coeff n) * z^n = (ii * z) ^ n /\<^sub>R (fact n)"
by (auto simp: cos_coeff_def sin_coeff_def scaleR_conv_of_real field_simps elim!: evenE oddE)
qed
also have "... sums (exp (ii * z))"
by (rule exp_converges)
finally have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n) sums (exp (ii * z))" .
moreover have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n) sums (cos z + ii * sin z)"
using sums_add [OF cos_converges [of z] sin_ii_eq [of z]]
by (simp add: field_simps scaleR_conv_of_real)
ultimately show ?thesis
using sums_unique2 by blast
qed
corollary exp_minus_Euler: "exp(-(ii * z)) = cos(z) - ii * sin(z)"
using exp_Euler [of "-z"]
by simp
lemma sin_exp_eq: "sin z = (exp(ii * z) - exp(-(ii * z))) / (2*ii)"
by (simp add: exp_Euler exp_minus_Euler)
lemma sin_exp_eq': "sin z = ii * (exp(-(ii * z)) - exp(ii * z)) / 2"
by (simp add: exp_Euler exp_minus_Euler)
lemma cos_exp_eq: "cos z = (exp(ii * z) + exp(-(ii * z))) / 2"
by (simp add: exp_Euler exp_minus_Euler)
subsection{*Relationships between real and complex trig functions*}
lemma real_sin_eq [simp]:
fixes x::real
shows "Re(sin(of_real x)) = sin x"
by (simp add: sin_of_real)
lemma real_cos_eq [simp]:
fixes x::real
shows "Re(cos(of_real x)) = cos x"
by (simp add: cos_of_real)
lemma DeMoivre: "(cos z + ii * sin z) ^ n = cos(n * z) + ii * sin(n * z)"
apply (simp add: exp_Euler [symmetric])
by (metis exp_of_nat_mult mult.left_commute)
lemma exp_cnj:
fixes z::complex
shows "cnj (exp z) = exp (cnj z)"
proof -
have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) = (\<lambda>n. (cnj z)^n /\<^sub>R (fact n))"
by auto
also have "... sums (exp (cnj z))"
by (rule exp_converges)
finally have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (exp (cnj z))" .
moreover have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (cnj (exp z))"
by (metis exp_converges sums_cnj)
ultimately show ?thesis
using sums_unique2
by blast
qed
lemma cnj_sin: "cnj(sin z) = sin(cnj z)"
by (simp add: sin_exp_eq exp_cnj field_simps)
lemma cnj_cos: "cnj(cos z) = cos(cnj z)"
by (simp add: cos_exp_eq exp_cnj field_simps)
lemma complex_differentiable_at_sin: "sin complex_differentiable at z"
using DERIV_sin complex_differentiable_def by blast
lemma complex_differentiable_within_sin: "sin complex_differentiable (at z within s)"
by (simp add: complex_differentiable_at_sin complex_differentiable_at_within)
lemma complex_differentiable_at_cos: "cos complex_differentiable at z"
using DERIV_cos complex_differentiable_def by blast
lemma complex_differentiable_within_cos: "cos complex_differentiable (at z within s)"
by (simp add: complex_differentiable_at_cos complex_differentiable_at_within)
lemma holomorphic_on_sin: "sin holomorphic_on s"
by (simp add: complex_differentiable_within_sin holomorphic_on_def)
lemma holomorphic_on_cos: "cos holomorphic_on s"
by (simp add: complex_differentiable_within_cos holomorphic_on_def)
subsection{* Get a nice real/imaginary separation in Euler's formula.*}
lemma Euler: "exp(z) = of_real(exp(Re z)) *
(of_real(cos(Im z)) + ii * of_real(sin(Im z)))"
by (cases z) (simp add: exp_add exp_Euler cos_of_real exp_of_real sin_of_real)
lemma Re_sin: "Re(sin z) = sin(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
by (simp add: sin_exp_eq field_simps Re_divide Im_exp)
lemma Im_sin: "Im(sin z) = cos(Re z) * (exp(Im z) - exp(-(Im z))) / 2"
by (simp add: sin_exp_eq field_simps Im_divide Re_exp)
lemma Re_cos: "Re(cos z) = cos(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
by (simp add: cos_exp_eq field_simps Re_divide Re_exp)
lemma Im_cos: "Im(cos z) = sin(Re z) * (exp(-(Im z)) - exp(Im z)) / 2"
by (simp add: cos_exp_eq field_simps Im_divide Im_exp)
lemma Re_sin_pos: "0 < Re z \<Longrightarrow> Re z < pi \<Longrightarrow> Re (sin z) > 0"
by (auto simp: Re_sin Im_sin add_pos_pos sin_gt_zero)
lemma Im_sin_nonneg: "Re z = 0 \<Longrightarrow> 0 \<le> Im z \<Longrightarrow> 0 \<le> Im (sin z)"
by (simp add: Re_sin Im_sin algebra_simps)
lemma Im_sin_nonneg2: "Re z = pi \<Longrightarrow> Im z \<le> 0 \<Longrightarrow> 0 \<le> Im (sin z)"
by (simp add: Re_sin Im_sin algebra_simps)
subsection{*More on the Polar Representation of Complex Numbers*}
lemma exp_Complex: "exp(Complex r t) = of_real(exp r) * Complex (cos t) (sin t)"
by (simp add: exp_add exp_Euler exp_of_real sin_of_real cos_of_real)
lemma exp_eq_1: "exp z = 1 \<longleftrightarrow> Re(z) = 0 \<and> (\<exists>n::int. Im(z) = of_int (2 * n) * pi)"
apply auto
apply (metis exp_eq_one_iff norm_exp_eq_Re norm_one)
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)
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)
lemma exp_eq: "exp w = exp z \<longleftrightarrow> (\<exists>n::int. w = z + (of_int (2 * n) * pi) * ii)"
(is "?lhs = ?rhs")
proof -
have "exp w = exp z \<longleftrightarrow> exp (w-z) = 1"
by (simp add: exp_diff)
also have "... \<longleftrightarrow> (Re w = Re z \<and> (\<exists>n::int. Im w - Im z = of_int (2 * n) * pi))"
by (simp add: exp_eq_1)
also have "... \<longleftrightarrow> ?rhs"
by (auto simp: algebra_simps intro!: complex_eqI)
finally show ?thesis .
qed
lemma exp_complex_eqI: "abs(Im w - Im z) < 2*pi \<Longrightarrow> exp w = exp z \<Longrightarrow> w = z"
by (auto simp: exp_eq abs_mult)
lemma exp_integer_2pi:
assumes "n \<in> Ints"
shows "exp((2 * n * pi) * ii) = 1"
proof -
have "exp((2 * n * pi) * ii) = exp 0"
using assms
by (simp only: Ints_def exp_eq) auto
also have "... = 1"
by simp
finally show ?thesis .
qed
lemma sin_cos_eq_iff: "sin y = sin x \<and> cos y = cos x \<longleftrightarrow> (\<exists>n::int. y = x + 2 * n * pi)"
proof -
{ assume "sin y = sin x" "cos y = cos x"
then have "cos (y-x) = 1"
using cos_add [of y "-x"] by simp
then have "\<exists>n::int. y-x = real n * 2 * pi"
using cos_one_2pi_int by blast }
then show ?thesis
apply (auto simp: sin_add cos_add)
apply (metis add.commute diff_add_cancel mult.commute)
done
qed
lemma exp_i_ne_1:
assumes "0 < x" "x < 2*pi"
shows "exp(\<i> * of_real x) \<noteq> 1"
proof
assume "exp (\<i> * of_real x) = 1"
then have "exp (\<i> * of_real x) = exp 0"
by simp
then obtain n where "\<i> * of_real x = (of_int (2 * n) * pi) * \<i>"
by (simp only: Ints_def exp_eq) auto
then have "of_real x = (of_int (2 * n) * pi)"
by (metis complex_i_not_zero mult.commute mult_cancel_left of_real_eq_iff real_scaleR_def scaleR_conv_of_real)
then have "x = (of_int (2 * n) * pi)"
by simp
then show False using assms
by (cases n) (auto simp: zero_less_mult_iff mult_less_0_iff)
qed
lemma sin_eq_0:
fixes z::complex
shows "sin z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi))"
by (simp add: sin_exp_eq exp_eq of_real_numeral)
lemma cos_eq_0:
fixes z::complex
shows "cos z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi) + of_real pi/2)"
using sin_eq_0 [of "z - of_real pi/2"]
by (simp add: sin_diff algebra_simps)
lemma cos_eq_1:
fixes z::complex
shows "cos z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi))"
proof -
have "cos z = cos (2*(z/2))"
by simp
also have "... = 1 - 2 * sin (z/2) ^ 2"
by (simp only: cos_double_sin)
finally have [simp]: "cos z = 1 \<longleftrightarrow> sin (z/2) = 0"
by simp
show ?thesis
by (auto simp: sin_eq_0 of_real_numeral)
qed
lemma csin_eq_1:
fixes z::complex
shows "sin z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + of_real pi/2)"
using cos_eq_1 [of "z - of_real pi/2"]
by (simp add: cos_diff algebra_simps)
lemma csin_eq_minus1:
fixes z::complex
shows "sin z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + 3/2*pi)"
(is "_ = ?rhs")
proof -
have "sin z = -1 \<longleftrightarrow> sin (-z) = 1"
by (simp add: equation_minus_iff)
also have "... \<longleftrightarrow> (\<exists>n::int. -z = of_real(2 * n * pi) + of_real pi/2)"
by (simp only: csin_eq_1)
also have "... \<longleftrightarrow> (\<exists>n::int. z = - of_real(2 * n * pi) - of_real pi/2)"
apply (rule iff_exI)
by (metis (no_types) is_num_normalize(8) minus_minus of_real_def real_vector.scale_minus_left uminus_add_conv_diff)
also have "... = ?rhs"
apply (auto simp: of_real_numeral)
apply (rule_tac [2] x="-(x+1)" in exI)
apply (rule_tac x="-(x+1)" in exI)
apply (simp_all add: algebra_simps)
done
finally show ?thesis .
qed
lemma ccos_eq_minus1:
fixes z::complex
shows "cos z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + pi)"
using csin_eq_1 [of "z - of_real pi/2"]
apply (simp add: sin_diff)
apply (simp add: algebra_simps of_real_numeral equation_minus_iff)
done
lemma sin_eq_1: "sin x = 1 \<longleftrightarrow> (\<exists>n::int. x = (2 * n + 1 / 2) * pi)"
(is "_ = ?rhs")
proof -
have "sin x = 1 \<longleftrightarrow> sin (complex_of_real x) = 1"
by (metis of_real_1 one_complex.simps(1) real_sin_eq sin_of_real)
also have "... \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + of_real pi/2)"
by (simp only: csin_eq_1)
also have "... \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + of_real pi/2)"
apply (rule iff_exI)
apply (auto simp: algebra_simps of_real_numeral)
apply (rule injD [OF inj_of_real [where 'a = complex]])
apply (auto simp: of_real_numeral)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed
lemma sin_eq_minus1: "sin x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 3/2) * pi)" (is "_ = ?rhs")
proof -
have "sin x = -1 \<longleftrightarrow> sin (complex_of_real x) = -1"
by (metis Re_complex_of_real of_real_def scaleR_minus1_left sin_of_real)
also have "... \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + 3/2*pi)"
by (simp only: csin_eq_minus1)
also have "... \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + 3/2*pi)"
apply (rule iff_exI)
apply (auto simp: algebra_simps)
apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed
lemma cos_eq_minus1: "cos x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 1) * pi)"
(is "_ = ?rhs")
proof -
have "cos x = -1 \<longleftrightarrow> cos (complex_of_real x) = -1"
by (metis Re_complex_of_real of_real_def scaleR_minus1_left cos_of_real)
also have "... \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + pi)"
by (simp only: ccos_eq_minus1)
also have "... \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + pi)"
apply (rule iff_exI)
apply (auto simp: algebra_simps)
apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed
lemma dist_exp_ii_1: "norm(exp(ii * of_real t) - 1) = 2 * abs(sin(t / 2))"
apply (simp add: exp_Euler cmod_def power2_diff sin_of_real cos_of_real algebra_simps)
using cos_double_sin [of "t/2"]
apply (simp add: real_sqrt_mult)
done
lemma sinh_complex:
fixes z :: complex
shows "(exp z - inverse (exp z)) / 2 = -ii * sin(ii * z)"
by (simp add: sin_exp_eq divide_simps exp_minus of_real_numeral)
lemma sin_ii_times:
fixes z :: complex
shows "sin(ii * z) = ii * ((exp z - inverse (exp z)) / 2)"
using sinh_complex by auto
lemma sinh_real:
fixes x :: real
shows "of_real((exp x - inverse (exp x)) / 2) = -ii * sin(ii * of_real x)"
by (simp add: exp_of_real sin_ii_times of_real_numeral)
lemma cosh_complex:
fixes z :: complex
shows "(exp z + inverse (exp z)) / 2 = cos(ii * z)"
by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lemma cosh_real:
fixes x :: real
shows "of_real((exp x + inverse (exp x)) / 2) = cos(ii * of_real x)"
by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)
lemmas cos_ii_times = cosh_complex [symmetric]
lemma norm_cos_squared:
"norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
apply (cases z)
apply (simp add: cos_add cmod_power2 cos_of_real sin_of_real)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
apply (simp add: sin_squared_eq)
apply (simp add: power2_eq_square algebra_simps divide_simps)
done
lemma norm_sin_squared:
"norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
apply (cases z)
apply (simp add: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
apply (simp add: cos_squared_eq)
apply (simp add: power2_eq_square algebra_simps divide_simps)
done
lemma exp_uminus_Im: "exp (- Im z) \<le> exp (cmod z)"
using abs_Im_le_cmod linear order_trans by fastforce
lemma norm_cos_le:
fixes z::complex
shows "norm(cos z) \<le> exp(norm z)"
proof -
have "Im z \<le> cmod z"
using abs_Im_le_cmod abs_le_D1 by auto
with exp_uminus_Im show ?thesis
apply (simp add: cos_exp_eq norm_divide)
apply (rule order_trans [OF norm_triangle_ineq], simp)
apply (metis add_mono exp_le_cancel_iff mult_2_right)
done
qed
lemma norm_cos_plus1_le:
fixes z::complex
shows "norm(1 + cos z) \<le> 2 * exp(norm z)"
proof -
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"
by arith
have *: "Im z \<le> cmod z"
using abs_Im_le_cmod abs_le_D1 by auto
have triangle3: "\<And>x y z. norm(x + y + z) \<le> norm(x) + norm(y) + norm(z)"
by (simp add: norm_add_rule_thm)
have "norm(1 + cos z) = cmod (1 + (exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
by (simp add: cos_exp_eq)
also have "... = cmod ((2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
by (simp add: field_simps)
also have "... = cmod (2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2"
by (simp add: norm_divide)
finally show ?thesis
apply (rule ssubst, simp)
apply (rule order_trans [OF triangle3], simp)
using exp_uminus_Im *
apply (auto intro: mono)
done
qed
subsection{* Taylor series for complex exponential, sine and cosine.*}
context
begin
declare power_Suc [simp del]
lemma Taylor_exp:
"norm(exp z - (\<Sum>k\<le>n. z ^ k / (fact k))) \<le> exp\<bar>Re z\<bar> * (norm z) ^ (Suc n) / (fact n)"
proof (rule complex_taylor [of _ n "\<lambda>k. exp" "exp\<bar>Re z\<bar>" 0 z, simplified])
show "convex (closed_segment 0 z)"
by (rule convex_segment [of 0 z])
next
fix k x
assume "x \<in> closed_segment 0 z" "k \<le> n"
show "(exp has_field_derivative exp x) (at x within closed_segment 0 z)"
using DERIV_exp DERIV_subset by blast
next
fix x
assume "x \<in> closed_segment 0 z"
then show "Re x \<le> \<bar>Re z\<bar>"
apply (auto simp: closed_segment_def scaleR_conv_of_real)
by (meson abs_ge_self abs_ge_zero linear mult_left_le_one_le mult_nonneg_nonpos order_trans)
next
show "0 \<in> closed_segment 0 z"
by (auto simp: closed_segment_def)
next
show "z \<in> closed_segment 0 z"
apply (simp add: closed_segment_def scaleR_conv_of_real)
using of_real_1 zero_le_one by blast
qed
lemma
assumes "0 \<le> u" "u \<le> 1"
shows cmod_sin_le_exp: "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
and cmod_cos_le_exp: "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
by arith
show "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
apply (auto simp: scaleR_conv_of_real norm_mult norm_power sin_exp_eq norm_divide)
apply (rule order_trans [OF norm_triangle_ineq4])
apply (rule mono)
apply (auto simp: abs_if mult_left_le_one_le)
apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
done
show "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
apply (auto simp: scaleR_conv_of_real norm_mult norm_power cos_exp_eq norm_divide)
apply (rule order_trans [OF norm_triangle_ineq])
apply (rule mono)
apply (auto simp: abs_if mult_left_le_one_le)
apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
done
qed
lemma Taylor_sin:
"norm(sin z - (\<Sum>k\<le>n. complex_of_real (sin_coeff k) * z ^ k))
\<le> exp\<bar>Im z\<bar> * (norm z) ^ (Suc n) / (fact n)"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
by arith
have *: "cmod (sin z -
(\<Sum>i\<le>n. (-1) ^ (i div 2) * (if even i then sin 0 else cos 0) * z ^ i / (fact i)))
\<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
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,
simplified])
show "convex (closed_segment 0 z)"
by (rule convex_segment [of 0 z])
next
fix k x
show "((\<lambda>x. (- 1) ^ (k div 2) * (if even k then sin x else cos x)) has_field_derivative
(- 1) ^ (Suc k div 2) * (if odd k then sin x else cos x))
(at x within closed_segment 0 z)"
apply (auto simp: power_Suc)
apply (intro derivative_eq_intros | simp)+
done
next
fix x
assume "x \<in> closed_segment 0 z"
then show "cmod ((- 1) ^ (Suc n div 2) * (if odd n then sin x else cos x)) \<le> exp \<bar>Im z\<bar>"
by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
next
show "0 \<in> closed_segment 0 z"
by (auto simp: closed_segment_def)
next
show "z \<in> closed_segment 0 z"
apply (simp add: closed_segment_def scaleR_conv_of_real)
using of_real_1 zero_le_one by blast
qed
have **: "\<And>k. complex_of_real (sin_coeff k) * z ^ k
= (-1)^(k div 2) * (if even k then sin 0 else cos 0) * z^k / of_nat (fact k)"
by (auto simp: sin_coeff_def elim!: oddE)
show ?thesis
apply (rule order_trans [OF _ *])
apply (simp add: **)
done
qed
lemma Taylor_cos:
"norm(cos z - (\<Sum>k\<le>n. complex_of_real (cos_coeff k) * z ^ k))
\<le> exp\<bar>Im z\<bar> * (norm z) ^ Suc n / (fact n)"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
by arith
have *: "cmod (cos z -
(\<Sum>i\<le>n. (-1) ^ (Suc i div 2) * (if even i then cos 0 else sin 0) * z ^ i / (fact i)))
\<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
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,
simplified])
show "convex (closed_segment 0 z)"
by (rule convex_segment [of 0 z])
next
fix k x
assume "x \<in> closed_segment 0 z" "k \<le> n"
show "((\<lambda>x. (- 1) ^ (Suc k div 2) * (if even k then cos x else sin x)) has_field_derivative
(- 1) ^ Suc (k div 2) * (if odd k then cos x else sin x))
(at x within closed_segment 0 z)"
apply (auto simp: power_Suc)
apply (intro derivative_eq_intros | simp)+
done
next
fix x
assume "x \<in> closed_segment 0 z"
then show "cmod ((- 1) ^ Suc (n div 2) * (if odd n then cos x else sin x)) \<le> exp \<bar>Im z\<bar>"
by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
next
show "0 \<in> closed_segment 0 z"
by (auto simp: closed_segment_def)
next
show "z \<in> closed_segment 0 z"
apply (simp add: closed_segment_def scaleR_conv_of_real)
using of_real_1 zero_le_one by blast
qed
have **: "\<And>k. complex_of_real (cos_coeff k) * z ^ k
= (-1)^(Suc k div 2) * (if even k then cos 0 else sin 0) * z^k / of_nat (fact k)"
by (auto simp: cos_coeff_def elim!: evenE)
show ?thesis
apply (rule order_trans [OF _ *])
apply (simp add: **)
done
qed
end (* of context *)
text{*32-bit Approximation to e*}
lemma e_approx_32: "abs(exp(1) - 5837465777 / 2147483648) \<le> (inverse(2 ^ 32)::real)"
using Taylor_exp [of 1 14] exp_le
apply (simp add: setsum_left_distrib in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
apply (simp only: pos_le_divide_eq [symmetric], linarith)
done
subsection{*The argument of a complex number*}
definition Arg :: "complex \<Rightarrow> real" where
"Arg z \<equiv> if z = 0 then 0
else THE t. 0 \<le> t \<and> t < 2*pi \<and>
z = of_real(norm z) * exp(ii * of_real t)"
lemma Arg_0 [simp]: "Arg(0) = 0"
by (simp add: Arg_def)
lemma Arg_unique_lemma:
assumes z: "z = of_real(norm z) * exp(ii * of_real t)"
and z': "z = of_real(norm z) * exp(ii * of_real t')"
and t: "0 \<le> t" "t < 2*pi"
and t': "0 \<le> t'" "t' < 2*pi"
and nz: "z \<noteq> 0"
shows "t' = t"
proof -
have [dest]: "\<And>x y z::real. x\<ge>0 \<Longrightarrow> x+y < z \<Longrightarrow> y<z"
by arith
have "of_real (cmod z) * exp (\<i> * of_real t') = of_real (cmod z) * exp (\<i> * of_real t)"
by (metis z z')
then have "exp (\<i> * of_real t') = exp (\<i> * of_real t)"
by (metis nz mult_left_cancel mult_zero_left z)
then have "sin t' = sin t \<and> cos t' = cos t"
apply (simp add: exp_Euler sin_of_real cos_of_real)
by (metis Complex_eq complex.sel)
then obtain n::int where n: "t' = t + 2 * real n * pi"
by (auto simp: sin_cos_eq_iff)
then have "n=0"
apply (rule_tac z=n in int_cases)
using t t'
apply (auto simp: mult_less_0_iff algebra_simps)
done
then show "t' = t"
by (simp add: n)
qed
lemma Arg: "0 \<le> Arg z & Arg z < 2*pi & z = of_real(norm z) * exp(ii * of_real(Arg z))"
proof (cases "z=0")
case True then show ?thesis
by (simp add: Arg_def)
next
case False
obtain t where t: "0 \<le> t" "t < 2*pi"
and ReIm: "Re z / cmod z = cos t" "Im z / cmod z = sin t"
using sincos_total_2pi [OF complex_unit_circle [OF False]]
by blast
have z: "z = of_real(norm z) * exp(ii * of_real t)"
apply (rule complex_eqI)
using t False ReIm
apply (auto simp: exp_Euler sin_of_real cos_of_real divide_simps)
done
show ?thesis
apply (simp add: Arg_def False)
apply (rule theI [where a=t])
using t z False
apply (auto intro: Arg_unique_lemma)
done
qed
corollary
shows Arg_ge_0: "0 \<le> Arg z"
and Arg_lt_2pi: "Arg z < 2*pi"
and Arg_eq: "z = of_real(norm z) * exp(ii * of_real(Arg z))"
using Arg by auto
lemma complex_norm_eq_1_exp: "norm z = 1 \<longleftrightarrow> (\<exists>t. z = exp(ii * of_real t))"
using Arg [of z] by auto
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"
apply (rule Arg_unique_lemma [OF _ Arg_eq])
using Arg [of z]
apply (auto simp: norm_mult)
done
lemma Arg_minus: "z \<noteq> 0 \<Longrightarrow> Arg (-z) = (if Arg z < pi then Arg z + pi else Arg z - pi)"
apply (rule Arg_unique [of "norm z"])
apply (rule complex_eqI)
using Arg_ge_0 [of z] Arg_eq [of z] Arg_lt_2pi [of z] Arg_eq [of z]
apply auto
apply (auto simp: Re_exp Im_exp cos_diff sin_diff cis_conv_exp [symmetric])
apply (metis Re_rcis Im_rcis rcis_def)+
done
lemma Arg_times_of_real [simp]: "0 < r \<Longrightarrow> Arg (of_real r * z) = Arg z"
apply (cases "z=0", simp)
apply (rule Arg_unique [of "r * norm z"])
using Arg
apply auto
done
lemma Arg_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg (z * of_real r) = Arg z"
by (metis Arg_times_of_real mult.commute)
lemma Arg_divide_of_real [simp]: "0 < r \<Longrightarrow> Arg (z / of_real r) = Arg z"
by (metis Arg_times_of_real2 less_numeral_extra(3) nonzero_eq_divide_eq of_real_eq_0_iff)
lemma Arg_le_pi: "Arg z \<le> pi \<longleftrightarrow> 0 \<le> Im z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
have "0 \<le> Im z \<longleftrightarrow> 0 \<le> Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
by (metis Arg_eq)
also have "... = (0 \<le> Im (exp (\<i> * complex_of_real (Arg z))))"
using False
by (simp add: zero_le_mult_iff)
also have "... \<longleftrightarrow> Arg z \<le> pi"
by (simp add: Im_exp) (metis Arg_ge_0 Arg_lt_2pi sin_lt_zero sin_ge_zero not_le)
finally show ?thesis
by blast
qed
lemma Arg_lt_pi: "0 < Arg z \<and> Arg z < pi \<longleftrightarrow> 0 < Im z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
have "0 < Im z \<longleftrightarrow> 0 < Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
by (metis Arg_eq)
also have "... = (0 < Im (exp (\<i> * complex_of_real (Arg z))))"
using False
by (simp add: zero_less_mult_iff)
also have "... \<longleftrightarrow> 0 < Arg z \<and> Arg z < pi"
using Arg_ge_0 Arg_lt_2pi sin_le_zero sin_gt_zero
apply (auto simp: Im_exp)
using le_less apply fastforce
using not_le by blast
finally show ?thesis
by blast
qed
lemma Arg_eq_0: "Arg z = 0 \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
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)))"
by (metis Arg_eq)
also have "... \<longleftrightarrow> z \<in> Reals \<and> 0 \<le> Re (exp (\<i> * complex_of_real (Arg z)))"
using False
by (simp add: zero_le_mult_iff)
also have "... \<longleftrightarrow> Arg z = 0"
apply (auto simp: Re_exp)
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)
using Arg_eq [of z]
apply (auto simp: Reals_def)
done
finally show ?thesis
by blast
qed
lemma Arg_of_real: "Arg(of_real x) = 0 \<longleftrightarrow> 0 \<le> x"
by (simp add: Arg_eq_0)
lemma Arg_eq_pi: "Arg z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
apply (cases "z=0", simp)
using Arg_eq_0 [of "-z"]
apply (auto simp: complex_is_Real_iff Arg_minus)
apply (simp add: complex_Re_Im_cancel_iff)
apply (metis Arg_minus pi_gt_zero add.left_neutral minus_minus minus_zero)
done
lemma Arg_eq_0_pi: "Arg z = 0 \<or> Arg z = pi \<longleftrightarrow> z \<in> \<real>"
using Arg_eq_0 Arg_eq_pi not_le by auto
lemma Arg_inverse: "Arg(inverse z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
apply (cases "z=0", simp)
apply (rule Arg_unique [of "inverse (norm z)"])
using Arg_ge_0 [of z] Arg_lt_2pi [of z] Arg_eq [of z] Arg_eq_0 [of z] Exp_two_pi_i
apply (auto simp: of_real_numeral algebra_simps exp_diff divide_simps)
done
lemma Arg_eq_iff:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w = Arg z \<longleftrightarrow> (\<exists>x. 0 < x & w = of_real x * z)"
using assms Arg_eq [of z] Arg_eq [of w]
apply auto
apply (rule_tac x="norm w / norm z" in exI)
apply (simp add: divide_simps)
by (metis mult.commute mult.left_commute)
lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
using complex_is_Real_iff
apply (simp add: Arg_eq_0)
apply (auto simp: divide_simps not_sum_power2_lt_zero)
done
lemma Arg_divide:
assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
shows "Arg(z / w) = Arg z - Arg w"
apply (rule Arg_unique [of "norm(z / w)"])
using assms Arg_eq [of z] Arg_eq [of w] Arg_ge_0 [of w] Arg_lt_2pi [of z]
apply (auto simp: exp_diff norm_divide algebra_simps divide_simps)
done
lemma Arg_le_div_sum:
assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
shows "Arg z = Arg w + Arg(z / w)"
by (simp add: Arg_divide assms)
lemma Arg_le_div_sum_eq:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w \<le> Arg z \<longleftrightarrow> Arg z = Arg w + Arg(z / w)"
using assms
by (auto simp: Arg_ge_0 intro: Arg_le_div_sum)
lemma Arg_diff:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w - Arg z = (if Arg z \<le> Arg w then Arg(w / z) else Arg(w/z) - 2*pi)"
using assms
apply (auto simp: Arg_ge_0 Arg_divide not_le)
using Arg_divide [of w z] Arg_inverse [of "w/z"]
apply auto
by (metis Arg_eq_0 less_irrefl minus_diff_eq right_minus_eq)
lemma Arg_add:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w + Arg z = (if Arg w + Arg z < 2*pi then Arg(w * z) else Arg(w * z) + 2*pi)"
using assms
using Arg_diff [of "w*z" z] Arg_le_div_sum_eq [of z "w*z"]
apply (auto simp: Arg_ge_0 Arg_divide not_le)
apply (metis Arg_lt_2pi add.commute)
apply (metis (no_types) Arg add.commute diff_0 diff_add_cancel diff_less_eq diff_minus_eq_add not_less)
done
lemma Arg_times:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg (w * z) = (if Arg w + Arg z < 2*pi then Arg w + Arg z
else (Arg w + Arg z) - 2*pi)"
using Arg_add [OF assms]
by auto
lemma Arg_cnj: "Arg(cnj z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
apply (cases "z=0", simp)
apply (rule trans [of _ "Arg(inverse z)"])
apply (simp add: Arg_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
apply (metis norm_eq_zero of_real_power zero_less_power2)
apply (auto simp: of_real_numeral Arg_inverse)
done
lemma Arg_real: "z \<in> \<real> \<Longrightarrow> Arg z = (if 0 \<le> Re z then 0 else pi)"
using Arg_eq_0 Arg_eq_0_pi
by auto
lemma Arg_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg(exp z) = Im z"
by (rule Arg_unique [of "exp(Re z)"]) (auto simp: Exp_eq_polar)
subsection{*Analytic properties of tangent function*}
lemma cnj_tan: "cnj(tan z) = tan(cnj z)"
by (simp add: cnj_cos cnj_sin tan_def)
lemma complex_differentiable_at_tan: "~(cos z = 0) \<Longrightarrow> tan complex_differentiable at z"
unfolding complex_differentiable_def
using DERIV_tan by blast
lemma complex_differentiable_within_tan: "~(cos z = 0)
\<Longrightarrow> tan complex_differentiable (at z within s)"
using complex_differentiable_at_tan complex_differentiable_at_within by blast
lemma continuous_within_tan: "~(cos z = 0) \<Longrightarrow> continuous (at z within s) tan"
using continuous_at_imp_continuous_within isCont_tan by blast
lemma continuous_on_tan [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> continuous_on s tan"
by (simp add: continuous_at_imp_continuous_on)
lemma holomorphic_on_tan: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> tan holomorphic_on s"
by (simp add: complex_differentiable_within_tan holomorphic_on_def)
subsection{*Complex logarithms (the conventional principal value)*}
definition Ln where
"Ln \<equiv> \<lambda>z. THE w. exp w = z & -pi < Im(w) & Im(w) \<le> pi"
lemma
assumes "z \<noteq> 0"
shows exp_Ln [simp]: "exp(Ln z) = z"
and mpi_less_Im_Ln: "-pi < Im(Ln z)"
and Im_Ln_le_pi: "Im(Ln z) \<le> pi"
proof -
obtain \<psi> where z: "z / (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
using complex_unimodular_polar [of "z / (norm z)"] assms
by (auto simp: norm_divide divide_simps)
obtain \<phi> where \<phi>: "- pi < \<phi>" "\<phi> \<le> pi" "sin \<phi> = sin \<psi>" "cos \<phi> = cos \<psi>"
using sincos_principal_value [of "\<psi>"] assms
by (auto simp: norm_divide divide_simps)
have "exp(Ln z) = z & -pi < Im(Ln z) & Im(Ln z) \<le> pi" unfolding Ln_def
apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
using z assms \<phi>
apply (auto simp: field_simps exp_complex_eqI Exp_eq_polar cis.code)
done
then show "exp(Ln z) = z" "-pi < Im(Ln z)" "Im(Ln z) \<le> pi"
by auto
qed
lemma Ln_exp [simp]:
assumes "-pi < Im(z)" "Im(z) \<le> pi"
shows "Ln(exp z) = z"
apply (rule exp_complex_eqI)
using assms mpi_less_Im_Ln [of "exp z"] Im_Ln_le_pi [of "exp z"]
apply auto
done
lemma Ln_eq_iff: "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (Ln w = Ln z \<longleftrightarrow> w = z)"
by (metis exp_Ln)
lemma Ln_unique: "exp(z) = w \<Longrightarrow> -pi < Im(z) \<Longrightarrow> Im(z) \<le> pi \<Longrightarrow> Ln w = z"
using Ln_exp by blast
lemma Re_Ln [simp]: "z \<noteq> 0 \<Longrightarrow> Re(Ln z) = ln(norm z)"
by (metis exp_Ln assms ln_exp norm_exp_eq_Re)
lemma exists_complex_root:
fixes a :: complex
shows "n \<noteq> 0 \<Longrightarrow> \<exists>z. z ^ n = a"
apply (cases "a=0", simp)
apply (rule_tac x= "exp(Ln(a) / n)" in exI)
apply (auto simp: exp_of_nat_mult [symmetric])
done
subsection{*The Unwinding Number and the Ln-product Formula*}
text{*Note that in this special case the unwinding number is -1, 0 or 1.*}
definition unwinding :: "complex \<Rightarrow> complex" where
"unwinding(z) = (z - Ln(exp z)) / (of_real(2*pi) * ii)"
lemma unwinding_2pi: "(2*pi) * ii * unwinding(z) = z - Ln(exp z)"
by (simp add: unwinding_def)
lemma Ln_times_unwinding:
"w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z) - (2*pi) * ii * unwinding(Ln w + Ln z)"
using unwinding_2pi by (simp add: exp_add)
subsection{*Derivative of Ln away from the branch cut*}
lemma
assumes "Im(z) = 0 \<Longrightarrow> 0 < Re(z)"
shows has_field_derivative_Ln: "(Ln has_field_derivative inverse(z)) (at z)"
and Im_Ln_less_pi: "Im (Ln z) < pi"
proof -
have znz: "z \<noteq> 0"
using assms by auto
then show *: "Im (Ln z) < pi" using assms
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)
show "(Ln has_field_derivative inverse(z)) (at z)"
apply (rule has_complex_derivative_inverse_strong_x
[where f = exp and s = "{w. -pi < Im(w) & Im(w) < pi}"])
using znz *
apply (auto simp: continuous_on_exp open_Collect_conj open_halfspace_Im_gt open_halfspace_Im_lt)
apply (metis DERIV_exp exp_Ln)
apply (metis mpi_less_Im_Ln)
done
qed
declare has_field_derivative_Ln [derivative_intros]
declare has_field_derivative_Ln [THEN DERIV_chain2, derivative_intros]
lemma complex_differentiable_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln complex_differentiable at z"
using complex_differentiable_def has_field_derivative_Ln by blast
lemma complex_differentiable_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z))
\<Longrightarrow> Ln complex_differentiable (at z within s)"
using complex_differentiable_at_Ln complex_differentiable_within_subset by blast
lemma continuous_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) Ln"
by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Ln)
lemma isCont_Ln' [simp]:
"\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. Ln (f x)) z"
by (blast intro: isCont_o2 [OF _ continuous_at_Ln])
lemma continuous_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) Ln"
using continuous_at_Ln continuous_at_imp_continuous_within by blast
lemma continuous_on_Ln [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous_on s Ln"
by (simp add: continuous_at_imp_continuous_on continuous_within_Ln)
lemma holomorphic_on_Ln: "(\<And>z. z \<in> s \<Longrightarrow> Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> Ln holomorphic_on s"
by (simp add: complex_differentiable_within_Ln holomorphic_on_def)
subsection{*Relation to Real Logarithm*}
lemma Ln_of_real:
assumes "0 < z"
shows "Ln(of_real z) = of_real(ln z)"
proof -
have "Ln(of_real (exp (ln z))) = Ln (exp (of_real (ln z)))"
by (simp add: exp_of_real)
also have "... = of_real(ln z)"
using assms
by (subst Ln_exp) auto
finally show ?thesis
using assms by simp
qed
corollary Ln_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Re z > 0 \<Longrightarrow> Ln z \<in> \<real>"
by (auto simp: Ln_of_real elim: Reals_cases)
subsection{*Quadrant-type results for Ln*}
lemma cos_lt_zero_pi: "pi/2 < x \<Longrightarrow> x < 3*pi/2 \<Longrightarrow> cos x < 0"
using cos_minus_pi cos_gt_zero_pi [of "x-pi"]
by simp
lemma Re_Ln_pos_lt:
assumes "z \<noteq> 0"
shows "abs(Im(Ln z)) < pi/2 \<longleftrightarrow> 0 < Re(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
by auto
then have "abs(Im w) < pi/2 \<longleftrightarrow> 0 < Re(exp w)"
apply (auto simp: Re_exp zero_less_mult_iff cos_gt_zero_pi)
using cos_lt_zero_pi [of "-(Im w)"] cos_lt_zero_pi [of "(Im w)"]
apply (simp add: abs_if split: split_if_asm)
apply (metis (no_types) cos_minus cos_pi_half eq_divide_eq_numeral1(1) eq_numeral_simps(4)
less_numeral_extra(3) linorder_neqE_linordered_idom minus_mult_minus minus_mult_right
mult_numeral_1_right)
done
}
then show ?thesis using assms
by auto
qed
lemma Re_Ln_pos_le:
assumes "z \<noteq> 0"
shows "abs(Im(Ln z)) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
by auto
then have "abs(Im w) \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(exp w)"
apply (auto simp: Re_exp zero_le_mult_iff cos_ge_zero)
using cos_lt_zero_pi [of "- (Im w)"] cos_lt_zero_pi [of "(Im w)"] not_le
apply (auto simp: abs_if split: split_if_asm)
done
}
then show ?thesis using assms
by auto
qed
lemma Im_Ln_pos_lt:
assumes "z \<noteq> 0"
shows "0 < Im(Ln z) \<and> Im(Ln z) < pi \<longleftrightarrow> 0 < Im(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
by auto
then have "0 < Im w \<and> Im w < pi \<longleftrightarrow> 0 < Im(exp w)"
using sin_gt_zero [of "- (Im w)"] sin_gt_zero [of "(Im w)"]
apply (auto simp: Im_exp zero_less_mult_iff)
using less_linear apply fastforce
using less_linear apply fastforce
done
}
then show ?thesis using assms
by auto
qed
lemma Im_Ln_pos_le:
assumes "z \<noteq> 0"
shows "0 \<le> Im(Ln z) \<and> Im(Ln z) \<le> pi \<longleftrightarrow> 0 \<le> Im(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
by auto
then have "0 \<le> Im w \<and> Im w \<le> pi \<longleftrightarrow> 0 \<le> Im(exp w)"
using sin_ge_zero [of "- (Im w)"] sin_ge_zero [of "(Im w)"]
apply (auto simp: Im_exp zero_le_mult_iff sin_ge_zero)
apply (metis not_le not_less_iff_gr_or_eq pi_not_less_zero sin_eq_0_pi)
done }
then show ?thesis using assms
by auto
qed
lemma Re_Ln_pos_lt_imp: "0 < Re(z) \<Longrightarrow> abs(Im(Ln z)) < pi/2"
by (metis Re_Ln_pos_lt less_irrefl zero_complex.simps(1))
lemma Im_Ln_pos_lt_imp: "0 < Im(z) \<Longrightarrow> 0 < Im(Ln z) \<and> Im(Ln z) < pi"
by (metis Im_Ln_pos_lt not_le order_refl zero_complex.simps(2))
lemma Im_Ln_eq_0: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = 0 \<longleftrightarrow> 0 < Re(z) \<and> Im(z) = 0)"
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
complex_eq exp_of_real le_less mult_zero_right norm_exp_eq_Re norm_le_zero_iff not_le of_real_0)
lemma Im_Ln_eq_pi: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = pi \<longleftrightarrow> Re(z) < 0 \<and> Im(z) = 0)"
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)
subsection{*More Properties of Ln*}
lemma cnj_Ln: "(Im z = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> cnj(Ln z) = Ln(cnj z)"
apply (cases "z=0", auto)
apply (rule exp_complex_eqI)
apply (auto simp: abs_if split: split_if_asm)
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)
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)
by (metis exp_Ln exp_cnj)
lemma Ln_inverse: "(Im(z) = 0 \<Longrightarrow> 0 < Re z) \<Longrightarrow> Ln(inverse z) = -(Ln z)"
apply (cases "z=0", auto)
apply (rule exp_complex_eqI)
using mpi_less_Im_Ln [of z] mpi_less_Im_Ln [of "inverse z"]
apply (auto simp: abs_if exp_minus split: split_if_asm)
apply (metis Im_Ln_less_pi Im_Ln_pos_le add_less_cancel_left add_strict_mono
inverse_inverse_eq inverse_zero le_less mult.commute mult_2_right)
done
lemma Ln_1 [simp]: "Ln(1) = 0"
proof -
have "Ln (exp 0) = 0"
by (metis exp_zero ln_exp Ln_of_real of_real_0 of_real_1 zero_less_one)
then show ?thesis
by simp
qed
lemma Ln_minus1 [simp]: "Ln(-1) = ii * pi"
apply (rule exp_complex_eqI)
using Im_Ln_le_pi [of "-1"] mpi_less_Im_Ln [of "-1"] cis_conv_exp cis_pi
apply (auto simp: abs_if)
done
lemma Ln_ii [simp]: "Ln ii = ii * of_real pi/2"
using Ln_exp [of "ii * (of_real pi/2)"]
unfolding exp_Euler
by simp
lemma Ln_minus_ii [simp]: "Ln(-ii) = - (ii * pi/2)"
proof -
have "Ln(-ii) = Ln(1/ii)"
by simp
also have "... = - (Ln ii)"
by (metis Ln_inverse ii.sel(2) inverse_eq_divide zero_neq_one)
also have "... = - (ii * pi/2)"
by (simp add: Ln_ii)
finally show ?thesis .
qed
lemma Ln_times:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Ln(w * z) =
(if Im(Ln w + Ln z) \<le> -pi then
(Ln(w) + Ln(z)) + ii * of_real(2*pi)
else if Im(Ln w + Ln z) > pi then
(Ln(w) + Ln(z)) - ii * of_real(2*pi)
else Ln(w) + Ln(z))"
using pi_ge_zero Im_Ln_le_pi [of w] Im_Ln_le_pi [of z]
using assms mpi_less_Im_Ln [of w] mpi_less_Im_Ln [of z]
by (auto simp: of_real_numeral exp_add exp_diff sin_double cos_double exp_Euler intro!: Ln_unique)
lemma Ln_times_simple:
"\<lbrakk>w \<noteq> 0; z \<noteq> 0; -pi < Im(Ln w) + Im(Ln z); Im(Ln w) + Im(Ln z) \<le> pi\<rbrakk>
\<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z)"
by (simp add: Ln_times)
lemma Ln_minus:
assumes "z \<noteq> 0"
shows "Ln(-z) = (if Im(z) \<le> 0 \<and> ~(Re(z) < 0 \<and> Im(z) = 0)
then Ln(z) + ii * pi
else Ln(z) - ii * pi)" (is "_ = ?rhs")
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z]
by (auto simp: of_real_numeral exp_add exp_diff exp_Euler intro!: Ln_unique)
lemma Ln_inverse_if:
assumes "z \<noteq> 0"
shows "Ln (inverse z) =
(if (Im(z) = 0 \<longrightarrow> 0 < Re z)
then -(Ln z)
else -(Ln z) + \<i> * 2 * complex_of_real pi)"
proof (cases "(Im(z) = 0 \<longrightarrow> 0 < Re z)")
case True then show ?thesis
by (simp add: Ln_inverse)
next
case False
then have z: "Im z = 0" "Re z < 0"
using assms
apply auto
by (metis cnj.code complex_cnj_cnj not_less_iff_gr_or_eq zero_complex.simps(1) zero_complex.simps(2))
have "Ln(inverse z) = Ln(- (inverse (-z)))"
by simp
also have "... = Ln (inverse (-z)) + \<i> * complex_of_real pi"
using assms z
apply (simp add: Ln_minus)
apply (simp add: field_simps)
done
also have "... = - Ln (- z) + \<i> * complex_of_real pi"
apply (subst Ln_inverse)
using z assms by auto
also have "... = - (Ln z) + \<i> * 2 * complex_of_real pi"
apply (subst Ln_minus [OF assms])
using assms z
apply simp
done
finally show ?thesis
using assms z
by simp
qed
lemma Ln_times_ii:
assumes "z \<noteq> 0"
shows "Ln(ii * z) = (if 0 \<le> Re(z) | Im(z) < 0
then Ln(z) + ii * of_real pi/2
else Ln(z) - ii * of_real(3 * pi/2))"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z] Re_Ln_pos_le [of z]
by (auto simp: of_real_numeral Ln_times)
subsection{*Relation between Square Root and exp/ln, hence its derivative*}
lemma csqrt_exp_Ln:
assumes "z \<noteq> 0"
shows "csqrt z = exp(Ln(z) / 2)"
proof -
have "(exp (Ln z / 2))\<^sup>2 = (exp (Ln z))"
by (metis exp_double nonzero_mult_divide_cancel_left times_divide_eq_right zero_neq_numeral)
also have "... = z"
using assms exp_Ln by blast
finally have "csqrt z = csqrt ((exp (Ln z / 2))\<^sup>2)"
by simp
also have "... = exp (Ln z / 2)"
apply (subst csqrt_square)
using cos_gt_zero_pi [of "(Im (Ln z) / 2)"] Im_Ln_le_pi mpi_less_Im_Ln assms
apply (auto simp: Re_exp Im_exp zero_less_mult_iff zero_le_mult_iff, fastforce+)
done
finally show ?thesis using assms csqrt_square
by simp
qed
lemma csqrt_inverse:
assumes "Im(z) = 0 \<Longrightarrow> 0 < Re z"
shows "csqrt (inverse z) = inverse (csqrt z)"
proof (cases "z=0", simp)
assume "z \<noteq> 0 "
then show ?thesis
using assms
by (simp add: csqrt_exp_Ln Ln_inverse exp_minus)
qed
lemma cnj_csqrt:
assumes "Im z = 0 \<Longrightarrow> 0 \<le> Re(z)"
shows "cnj(csqrt z) = csqrt(cnj z)"
proof (cases "z=0", simp)
assume z: "z \<noteq> 0"
then have "Im z = 0 \<Longrightarrow> 0 < Re(z)"
using assms cnj.code complex_cnj_zero_iff by fastforce
then show ?thesis
using z by (simp add: csqrt_exp_Ln cnj_Ln exp_cnj)
qed
lemma has_field_derivative_csqrt:
assumes "Im z = 0 \<Longrightarrow> 0 < Re(z)"
shows "(csqrt has_field_derivative inverse(2 * csqrt z)) (at z)"
proof -
have z: "z \<noteq> 0"
using assms by auto
then have *: "inverse z = inverse (2*z) * 2"
by (simp add: divide_simps)
show ?thesis
apply (rule DERIV_transform_at [where f = "\<lambda>z. exp(Ln(z) / 2)" and d = "norm z"])
apply (intro derivative_eq_intros | simp add: assms)+
apply (rule *)
using z
apply (auto simp: field_simps csqrt_exp_Ln [symmetric])
apply (metis power2_csqrt power2_eq_square)
apply (metis csqrt_exp_Ln dist_0_norm less_irrefl)
done
qed
lemma complex_differentiable_at_csqrt:
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable at z"
using complex_differentiable_def has_field_derivative_csqrt by blast
lemma complex_differentiable_within_csqrt:
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt complex_differentiable (at z within s)"
using complex_differentiable_at_csqrt complex_differentiable_within_subset by blast
lemma continuous_at_csqrt:
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) csqrt"
by (simp add: complex_differentiable_within_csqrt complex_differentiable_imp_continuous_at)
corollary isCont_csqrt' [simp]:
"\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. csqrt (f x)) z"
by (blast intro: isCont_o2 [OF _ continuous_at_csqrt])
lemma continuous_within_csqrt:
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) csqrt"
by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_csqrt)
lemma continuous_on_csqrt [continuous_intros]:
"(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous_on s csqrt"
by (simp add: continuous_at_imp_continuous_on continuous_within_csqrt)
lemma holomorphic_on_csqrt:
"(\<And>z. z \<in> s \<and> Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> csqrt holomorphic_on s"
by (simp add: complex_differentiable_within_csqrt holomorphic_on_def)
lemma continuous_within_closed_nontrivial:
"closed s \<Longrightarrow> a \<notin> s ==> continuous (at a within s) f"
using open_Compl
by (force simp add: continuous_def eventually_at_topological filterlim_iff open_Collect_neg)
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
using closed_halfspace_Re_ge
by (simp add: closed_Int closed_complex_Reals)
lemma continuous_within_csqrt_posreal:
"continuous (at z within (\<real> \<inter> {w. 0 \<le> Re(w)})) csqrt"
proof (cases "Im z = 0 --> 0 < Re(z)")
case True then show ?thesis
by (blast intro: continuous_within_csqrt)
next
case False
then have "Im z = 0" "Re z < 0 \<or> z = 0"
using False cnj.code complex_cnj_zero_iff by auto force
then show ?thesis
apply (auto simp: continuous_within_closed_nontrivial [OF closed_Real_halfspace_Re_ge])
apply (auto simp: continuous_within_eps_delta norm_conv_dist [symmetric])
apply (rule_tac x="e^2" in exI)
apply (auto simp: Reals_def)
by (metis linear not_less real_sqrt_less_iff real_sqrt_pow2_iff real_sqrt_power)
qed
subsection{*Complex arctangent*}
text{*branch cut gives standard bounds in real case.*}
definition Arctan :: "complex \<Rightarrow> complex" where
"Arctan \<equiv> \<lambda>z. (\<i>/2) * Ln((1 - \<i>*z) / (1 + \<i>*z))"
lemma Arctan_0 [simp]: "Arctan 0 = 0"
by (simp add: Arctan_def)
lemma Im_complex_div_lemma: "Im((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<longleftrightarrow> Re z = 0"
by (auto simp: Im_complex_div_eq_0 algebra_simps)
lemma Re_complex_div_lemma: "0 < Re((1 - \<i>*z) / (1 + \<i>*z)) \<longleftrightarrow> norm z < 1"
by (simp add: Re_complex_div_gt_0 algebra_simps cmod_def power2_eq_square)
lemma tan_Arctan:
assumes "z\<^sup>2 \<noteq> -1"
shows [simp]:"tan(Arctan z) = z"
proof -
have "1 + \<i>*z \<noteq> 0"
by (metis assms complex_i_mult_minus i_squared minus_unique power2_eq_square power2_minus)
moreover
have "1 - \<i>*z \<noteq> 0"
by (metis assms complex_i_mult_minus i_squared power2_eq_square power2_minus right_minus_eq)
ultimately
show ?thesis
by (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus csqrt_exp_Ln [symmetric]
divide_simps power2_eq_square [symmetric])
qed
lemma Arctan_tan [simp]:
assumes "\<bar>Re z\<bar> < pi/2"
shows "Arctan(tan z) = z"
proof -
have ge_pi2: "\<And>n::int. abs (of_int (2*n + 1) * pi/2) \<ge> pi/2"
by (case_tac n rule: int_cases) (auto simp: abs_mult)
have "exp (\<i>*z)*exp (\<i>*z) = -1 \<longleftrightarrow> exp (2*\<i>*z) = -1"
by (metis distrib_right exp_add mult_2)
also have "... \<longleftrightarrow> exp (2*\<i>*z) = exp (\<i>*pi)"
using cis_conv_exp cis_pi by auto
also have "... \<longleftrightarrow> exp (2*\<i>*z - \<i>*pi) = 1"
by (metis (no_types) diff_add_cancel diff_minus_eq_add exp_add exp_minus_inverse mult.commute)
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)"
by (simp add: exp_eq_1)
also have "... \<longleftrightarrow> Im z = 0 \<and> (\<exists>n::int. 2 * Re z = of_int (2*n + 1) * pi)"
by (simp add: algebra_simps)
also have "... \<longleftrightarrow> False"
using assms ge_pi2
apply (auto simp: algebra_simps)
by (metis abs_mult_pos not_less not_real_of_nat_less_zero real_of_nat_numeral)
finally have *: "exp (\<i>*z)*exp (\<i>*z) + 1 \<noteq> 0"
by (auto simp: add.commute minus_unique)
show ?thesis
using assms *
apply (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus divide_simps
ii_times_eq_iff power2_eq_square [symmetric])
apply (rule Ln_unique)
apply (auto simp: divide_simps exp_minus)
apply (simp add: algebra_simps exp_double [symmetric])
done
qed
lemma
assumes "Re z = 0 \<Longrightarrow> abs(Im z) < 1"
shows Re_Arctan_bounds: "abs(Re(Arctan z)) < pi/2"
and has_field_derivative_Arctan: "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
proof -
have nz0: "1 + \<i>*z \<noteq> 0"
using assms
by (metis abs_one complex_i_mult_minus diff_0_right diff_minus_eq_add ii.simps(1) ii.simps(2)
less_irrefl minus_diff_eq mult.right_neutral right_minus_eq)
have "z \<noteq> -\<i>" using assms
by auto
then have zz: "1 + z * z \<noteq> 0"
by (metis abs_one assms i_squared ii.simps less_irrefl minus_unique square_eq_iff)
have nz1: "1 - \<i>*z \<noteq> 0"
using assms by (force simp add: ii_times_eq_iff)
have nz2: "inverse (1 + \<i>*z) \<noteq> 0"
using assms
by (metis Im_complex_div_lemma Re_complex_div_lemma cmod_eq_Im divide_complex_def
less_irrefl mult_zero_right zero_complex.simps(1) zero_complex.simps(2))
have nzi: "((1 - \<i>*z) * inverse (1 + \<i>*z)) \<noteq> 0"
using nz1 nz2 by auto
have *: "Im ((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<Longrightarrow> 0 < Re ((1 - \<i>*z) / (1 + \<i>*z))"
apply (simp add: divide_complex_def)
apply (simp add: divide_simps split: split_if_asm)
using assms
apply (auto simp: algebra_simps abs_square_less_1 [unfolded power2_eq_square])
done
show "abs(Re(Arctan z)) < pi/2"
unfolding Arctan_def divide_complex_def
using mpi_less_Im_Ln [OF nzi]
by (auto simp: abs_if intro: Im_Ln_less_pi * [unfolded divide_complex_def])
show "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
unfolding Arctan_def scaleR_conv_of_real
apply (rule DERIV_cong)
apply (intro derivative_eq_intros | simp add: nz0 *)+
using nz0 nz1 zz
apply (simp add: divide_simps power2_eq_square)
apply (auto simp: algebra_simps)
done
qed
lemma complex_differentiable_at_Arctan: "(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable at z"
using has_field_derivative_Arctan
by (auto simp: complex_differentiable_def)
lemma complex_differentiable_within_Arctan:
"(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> Arctan complex_differentiable (at z within s)"
using complex_differentiable_at_Arctan complex_differentiable_at_within by blast
declare has_field_derivative_Arctan [derivative_intros]
declare has_field_derivative_Arctan [THEN DERIV_chain2, derivative_intros]
lemma continuous_at_Arctan:
"(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z) Arctan"
by (simp add: complex_differentiable_imp_continuous_at complex_differentiable_within_Arctan)
lemma continuous_within_Arctan:
"(Re z = 0 \<Longrightarrow> abs(Im z) < 1) \<Longrightarrow> continuous (at z within s) Arctan"
using continuous_at_Arctan continuous_at_imp_continuous_within by blast
lemma continuous_on_Arctan [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous_on s Arctan"
by (auto simp: continuous_at_imp_continuous_on continuous_within_Arctan)
lemma holomorphic_on_Arctan:
"(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> abs \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan holomorphic_on s"
by (simp add: complex_differentiable_within_Arctan holomorphic_on_def)
subsection {*Real arctangent*}
lemma norm_exp_ii_times [simp]: "norm (exp(\<i> * of_real y)) = 1"
by simp
lemma norm_exp_imaginary: "norm(exp z) = 1 \<Longrightarrow> Re z = 0"
by (simp add: complex_norm_eq_1_exp)
lemma Im_Arctan_of_real [simp]: "Im (Arctan (of_real x)) = 0"
unfolding Arctan_def divide_complex_def
apply (simp add: complex_eq_iff)
apply (rule norm_exp_imaginary)
apply (subst exp_Ln, auto)
apply (simp_all add: cmod_def complex_eq_iff)
apply (auto simp: divide_simps)
apply (metis power_one realpow_two_sum_zero_iff zero_neq_one, algebra)
done
lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
proof (rule arctan_unique)
show "- (pi / 2) < Re (Arctan (complex_of_real x))"
apply (simp add: Arctan_def)
apply (rule Im_Ln_less_pi)
apply (auto simp: Im_complex_div_lemma)
done
next
have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
by (simp add: divide_simps) ( simp add: complex_eq_iff)
show "Re (Arctan (complex_of_real x)) < pi / 2"
using mpi_less_Im_Ln [OF *]
by (simp add: Arctan_def)
next
have "tan (Re (Arctan (of_real x))) = Re (tan (Arctan (of_real x)))"
apply (auto simp: tan_def Complex.Re_divide Re_sin Re_cos Im_sin Im_cos)
apply (simp add: field_simps)
by (simp add: power2_eq_square)
also have "... = x"
apply (subst tan_Arctan, auto)
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)
finally show "tan (Re (Arctan (complex_of_real x))) = x" .
qed
lemma Arctan_of_real: "Arctan (of_real x) = of_real (arctan x)"
unfolding arctan_eq_Re_Arctan divide_complex_def
by (simp add: complex_eq_iff)
lemma Arctan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Arctan z \<in> \<real>"
by (metis Reals_cases Reals_of_real Arctan_of_real)
declare arctan_one [simp]
lemma arctan_less_pi4_pos: "x < 1 \<Longrightarrow> arctan x < pi/4"
by (metis arctan_less_iff arctan_one)
lemma arctan_less_pi4_neg: "-1 < x \<Longrightarrow> -(pi/4) < arctan x"
by (metis arctan_less_iff arctan_minus arctan_one)
lemma arctan_less_pi4: "abs x < 1 \<Longrightarrow> abs(arctan x) < pi/4"
by (metis abs_less_iff arctan_less_pi4_pos arctan_minus)
lemma arctan_le_pi4: "abs x \<le> 1 \<Longrightarrow> abs(arctan x) \<le> pi/4"
by (metis abs_le_iff arctan_le_iff arctan_minus arctan_one)
lemma abs_arctan: "abs(arctan x) = arctan(abs x)"
by (simp add: abs_if arctan_minus)
lemma arctan_add_raw:
assumes "abs(arctan x + arctan y) < pi/2"
shows "arctan x + arctan y = arctan((x + y) / (1 - x * y))"
proof (rule arctan_unique [symmetric])
show 12: "- (pi / 2) < arctan x + arctan y" "arctan x + arctan y < pi / 2"
using assms by linarith+
show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
using cos_gt_zero_pi [OF 12]
by (simp add: arctan tan_add)
qed
lemma arctan_inverse:
assumes "0 < x"
shows "arctan(inverse x) = pi/2 - arctan x"
proof -
have "arctan(inverse x) = arctan(inverse(tan(arctan x)))"
by (simp add: arctan)
also have "... = arctan (tan (pi / 2 - arctan x))"
by (simp add: tan_cot)
also have "... = pi/2 - arctan x"
proof -
have "0 < pi - arctan x"
using arctan_ubound [of x] pi_gt_zero by linarith
with assms show ?thesis
by (simp add: Transcendental.arctan_tan)
qed
finally show ?thesis .
qed
lemma arctan_add_small:
assumes "abs(x * y) < 1"
shows "(arctan x + arctan y = arctan((x + y) / (1 - x * y)))"
proof (cases "x = 0 \<or> y = 0")
case True then show ?thesis
by auto
next
case False
then have *: "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
apply (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
apply (simp add: divide_simps abs_mult)
done
show ?thesis
apply (rule arctan_add_raw)
using * by linarith
qed
lemma abs_arctan_le:
fixes x::real shows "abs(arctan x) \<le> abs x"
proof -
{ fix w::complex and z::complex
assume *: "w \<in> \<real>" "z \<in> \<real>"
have "cmod (Arctan w - Arctan z) \<le> 1 * cmod (w-z)"
apply (rule complex_differentiable_bound [OF convex_Reals, of Arctan _ 1])
apply (rule has_field_derivative_at_within [OF has_field_derivative_Arctan])
apply (force simp add: Reals_def)
apply (simp add: norm_divide divide_simps in_Reals_norm complex_is_Real_iff power2_eq_square)
using * by auto
}
then have "cmod (Arctan (of_real x) - Arctan 0) \<le> 1 * cmod (of_real x -0)"
using Reals_0 Reals_of_real by blast
then show ?thesis
by (simp add: Arctan_of_real)
qed
lemma arctan_le_self: "0 \<le> x \<Longrightarrow> arctan x \<le> x"
by (metis abs_arctan_le abs_of_nonneg zero_le_arctan_iff)
lemma abs_tan_ge: "abs x < pi/2 \<Longrightarrow> abs x \<le> abs(tan x)"
by (metis abs_arctan_le abs_less_iff arctan_tan minus_less_iff)
subsection{*Inverse Sine*}
definition Arcsin :: "complex \<Rightarrow> complex" where
"Arcsin \<equiv> \<lambda>z. -\<i> * Ln(\<i> * z + csqrt(1 - z\<^sup>2))"
lemma Arcsin_body_lemma: "\<i> * z + csqrt(1 - z\<^sup>2) \<noteq> 0"
using power2_csqrt [of "1 - z\<^sup>2"]
apply auto
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)
lemma Arcsin_range_lemma: "abs (Re z) < 1 \<Longrightarrow> 0 < Re(\<i> * z + csqrt(1 - z\<^sup>2))"
using Complex.cmod_power2 [of z, symmetric]
by (simp add: real_less_rsqrt algebra_simps Re_power2 cmod_square_less_1_plus)
lemma Re_Arcsin: "Re(Arcsin z) = Im (Ln (\<i> * z + csqrt(1 - z\<^sup>2)))"
by (simp add: Arcsin_def)
lemma Im_Arcsin: "Im(Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"
by (simp add: Arcsin_def Arcsin_body_lemma)
lemma isCont_Arcsin:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "isCont Arcsin z"
proof -
have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
using assms
by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
have cmz: "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
by (blast intro: assms cmod_square_less_1_plus)
show ?thesis
using assms
apply (simp add: Arcsin_def)
apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
apply (erule rez)
apply (auto simp: Re_power2 Im_power2 abs_square_less_1 [symmetric] real_less_rsqrt algebra_simps split: split_if_asm)
apply (simp add: norm_complex_def)
using cmod_power2 [of z, symmetric] cmz
apply (simp add: real_less_rsqrt)
done
qed
lemma isCont_Arcsin' [simp]:
shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arcsin (f x)) z"
by (blast intro: isCont_o2 [OF _ isCont_Arcsin])
lemma sin_Arcsin [simp]: "sin(Arcsin z) = z"
proof -
have "\<i>*z*2 + csqrt (1 - z\<^sup>2)*2 = 0 \<longleftrightarrow> (\<i>*z)*2 + csqrt (1 - z\<^sup>2)*2 = 0"
by (simp add: algebra_simps) --{*Cancelling a factor of 2*}
moreover have "... \<longleftrightarrow> (\<i>*z) + csqrt (1 - z\<^sup>2) = 0"
by (metis Arcsin_body_lemma distrib_right no_zero_divisors zero_neq_numeral)
ultimately show ?thesis
apply (simp add: sin_exp_eq Arcsin_def Arcsin_body_lemma exp_minus divide_simps)
apply (simp add: algebra_simps)
apply (simp add: power2_eq_square [symmetric] algebra_simps)
done
qed
lemma Re_eq_pihalf_lemma:
"\<bar>Re z\<bar> = pi/2 \<Longrightarrow> Im z = 0 \<Longrightarrow>
Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2) = 0 \<and> 0 \<le> Im ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
apply (simp add: cos_ii_times [symmetric] Re_cos Im_cos abs_if del: eq_divide_eq_numeral1)
by (metis cos_minus cos_pi_half)
lemma Re_less_pihalf_lemma:
assumes "\<bar>Re z\<bar> < pi / 2"
shows "0 < Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
proof -
have "0 < cos (Re z)" using assms
using cos_gt_zero_pi by auto
then show ?thesis
by (simp add: cos_ii_times [symmetric] Re_cos Im_cos add_pos_pos)
qed
lemma Arcsin_sin:
assumes "\<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)"
shows "Arcsin(sin z) = z"
proof -
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))"
by (simp add: sin_exp_eq Arcsin_def exp_minus)
also have "... = - (\<i> * Ln (csqrt (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2)\<^sup>2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
by (simp add: field_simps power2_eq_square)
also have "... = - (\<i> * Ln (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
apply (subst csqrt_square)
using assms Re_eq_pihalf_lemma Re_less_pihalf_lemma
apply auto
done
also have "... = - (\<i> * Ln (exp (\<i>*z)))"
by (simp add: field_simps power2_eq_square)
also have "... = z"
apply (subst Complex_Transcendental.Ln_exp)
using assms
apply (auto simp: abs_if simp del: eq_divide_eq_numeral1 split: split_if_asm)
done
finally show ?thesis .
qed
lemma Arcsin_unique:
"\<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"
by (metis Arcsin_sin)
lemma Arcsin_0 [simp]: "Arcsin 0 = 0"
by (metis Arcsin_sin norm_zero pi_half_gt_zero real_norm_def sin_zero zero_complex.simps(1))
lemma Arcsin_1 [simp]: "Arcsin 1 = pi/2"
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)
lemma Arcsin_minus_1 [simp]: "Arcsin(-1) = - (pi/2)"
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)
lemma has_field_derivative_Arcsin:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "(Arcsin has_field_derivative inverse(cos(Arcsin z))) (at z)"
proof -
have "(sin (Arcsin z))\<^sup>2 \<noteq> 1"
using assms
apply atomize
apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
by (metis abs_minus_cancel abs_one abs_power2 one_neq_neg_one)
then have "cos (Arcsin z) \<noteq> 0"
by (metis diff_0_right power_zero_numeral sin_squared_eq)
then show ?thesis
apply (rule has_complex_derivative_inverse_basic [OF DERIV_sin])
apply (auto intro: isCont_Arcsin open_ball [of z 1] assms)
done
qed
declare has_field_derivative_Arcsin [derivative_intros]
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]
lemma complex_differentiable_at_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable at z"
using complex_differentiable_def has_field_derivative_Arcsin by blast
lemma complex_differentiable_within_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin complex_differentiable (at z within s)"
using complex_differentiable_at_Arcsin complex_differentiable_within_subset by blast
lemma continuous_within_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arcsin"
using continuous_at_imp_continuous_within isCont_Arcsin by blast
lemma continuous_on_Arcsin [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arcsin"
by (simp add: continuous_at_imp_continuous_on)
lemma holomorphic_on_Arcsin: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin holomorphic_on s"
by (simp add: complex_differentiable_within_Arcsin holomorphic_on_def)
subsection{*Inverse Cosine*}
definition Arccos :: "complex \<Rightarrow> complex" where
"Arccos \<equiv> \<lambda>z. -\<i> * Ln(z + \<i> * csqrt(1 - z\<^sup>2))"
lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"
using Arcsin_range_lemma [of "-z"]
by simp
lemma Arccos_body_lemma: "z + \<i> * csqrt(1 - z\<^sup>2) \<noteq> 0"
using Arcsin_body_lemma [of z]
by (metis complex_i_mult_minus diff_add_cancel minus_diff_eq minus_unique mult.assoc mult.left_commute
power2_csqrt power2_eq_square zero_neq_one)
lemma Re_Arccos: "Re(Arccos z) = Im (Ln (z + \<i> * csqrt(1 - z\<^sup>2)))"
by (simp add: Arccos_def)
lemma Im_Arccos: "Im(Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"
by (simp add: Arccos_def Arccos_body_lemma)
text{*A very tricky argument to find!*}
lemma abs_Re_less_1_preserve:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)" "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0"
shows "0 < Re (z + \<i> * csqrt (1 - z\<^sup>2))"
proof (cases "Im z = 0")
case True
then show ?thesis
using assms
by (fastforce simp add: cmod_def Re_power2 Im_power2 algebra_simps abs_square_less_1 [symmetric])
next
case False
have Imz: "Im z = - sqrt ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
using assms abs_Re_le_cmod [of "1-z\<^sup>2"]
by (simp add: Re_power2 algebra_simps)
have "(cmod z)\<^sup>2 - 1 \<noteq> cmod (1 - z\<^sup>2)"
proof (clarsimp simp add: cmod_def)
assume "(Re z)\<^sup>2 + (Im z)\<^sup>2 - 1 = sqrt ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
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)"
by simp
then show False using False
by (simp add: power2_eq_square algebra_simps)
qed
moreover have "(Im z)\<^sup>2 = ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
apply (subst Imz, simp)
apply (subst real_sqrt_pow2)
using abs_Re_le_cmod [of "1-z\<^sup>2"]
apply (auto simp: Re_power2 field_simps)
done
ultimately show ?thesis
by (simp add: Re_power2 Im_power2 cmod_power2)
qed
lemma isCont_Arccos:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "isCont Arccos z"
proof -
have rez: "Im (1 - z\<^sup>2) = 0 \<Longrightarrow> 0 < Re (1 - z\<^sup>2)"
using assms
by (auto simp: Re_power2 Im_power2 abs_square_less_1 add_pos_nonneg algebra_simps)
show ?thesis
using assms
apply (simp add: Arccos_def)
apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
apply (erule rez)
apply (blast intro: abs_Re_less_1_preserve)
done
qed
lemma isCont_Arccos' [simp]:
shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arccos (f x)) z"
by (blast intro: isCont_o2 [OF _ isCont_Arccos])
lemma cos_Arccos [simp]: "cos(Arccos z) = z"
proof -
have "z*2 + \<i> * (2 * csqrt (1 - z\<^sup>2)) = 0 \<longleftrightarrow> z*2 + \<i> * csqrt (1 - z\<^sup>2)*2 = 0"
by (simp add: algebra_simps) --{*Cancelling a factor of 2*}
moreover have "... \<longleftrightarrow> z + \<i> * csqrt (1 - z\<^sup>2) = 0"
by (metis distrib_right mult_eq_0_iff zero_neq_numeral)
ultimately show ?thesis
apply (simp add: cos_exp_eq Arccos_def Arccos_body_lemma exp_minus field_simps)
apply (simp add: power2_eq_square [symmetric])
done
qed
lemma Arccos_cos:
assumes "0 < Re z & Re z < pi \<or>
Re z = 0 & 0 \<le> Im z \<or>
Re z = pi & Im z \<le> 0"
shows "Arccos(cos z) = z"
proof -
have *: "((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z))) = sin z"
by (simp add: sin_exp_eq exp_minus field_simps power2_eq_square)
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"
by (simp add: field_simps power2_eq_square)
then have "Arccos(cos z) = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
\<i> * csqrt (((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z)))\<^sup>2)))"
by (simp add: cos_exp_eq Arccos_def exp_minus)
also have "... = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
\<i> * ((\<i> - (Exp (\<i> * z))\<^sup>2 * \<i>) / (2 * Exp (\<i> * z)))))"
apply (subst csqrt_square)
using assms Re_sin_pos [of z] Im_sin_nonneg [of z] Im_sin_nonneg2 [of z]
apply (auto simp: * Re_sin Im_sin)
done
also have "... = - (\<i> * Ln (exp (\<i>*z)))"
by (simp add: field_simps power2_eq_square)
also have "... = z"
using assms
apply (subst Complex_Transcendental.Ln_exp, auto)
done
finally show ?thesis .
qed
lemma Arccos_unique:
"\<lbrakk>cos z = w;
0 < Re z \<and> Re z < pi \<or>
Re z = 0 \<and> 0 \<le> Im z \<or>
Re z = pi \<and> Im z \<le> 0\<rbrakk> \<Longrightarrow> Arccos w = z"
using Arccos_cos by blast
lemma Arccos_0 [simp]: "Arccos 0 = pi/2"
by (rule Arccos_unique) (auto simp: of_real_numeral)
lemma Arccos_1 [simp]: "Arccos 1 = 0"
by (rule Arccos_unique) auto
lemma Arccos_minus1: "Arccos(-1) = pi"
by (rule Arccos_unique) auto
lemma has_field_derivative_Arccos:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "(Arccos has_field_derivative - inverse(sin(Arccos z))) (at z)"
proof -
have "(cos (Arccos z))\<^sup>2 \<noteq> 1"
using assms
apply atomize
apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
apply (metis left_minus less_irrefl power_one sum_power2_gt_zero_iff zero_neq_neg_one)
done
then have "- sin (Arccos z) \<noteq> 0"
by (metis cos_squared_eq diff_0_right mult_zero_left neg_0_equal_iff_equal power2_eq_square)
then have "(Arccos has_field_derivative inverse(- sin(Arccos z))) (at z)"
apply (rule has_complex_derivative_inverse_basic [OF DERIV_cos])
apply (auto intro: isCont_Arccos open_ball [of z 1] assms)
done
then show ?thesis
by simp
qed
declare has_field_derivative_Arcsin [derivative_intros]
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]
lemma complex_differentiable_at_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos complex_differentiable at z"
using complex_differentiable_def has_field_derivative_Arccos by blast
lemma complex_differentiable_within_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos complex_differentiable (at z within s)"
using complex_differentiable_at_Arccos complex_differentiable_within_subset by blast
lemma continuous_within_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arccos"
using continuous_at_imp_continuous_within isCont_Arccos by blast
lemma continuous_on_Arccos [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arccos"
by (simp add: continuous_at_imp_continuous_on)
lemma holomorphic_on_Arccos: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos holomorphic_on s"
by (simp add: complex_differentiable_within_Arccos holomorphic_on_def)
subsection{*Upper and Lower Bounds for Inverse Sine and Cosine*}
lemma Arcsin_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> abs(Re(Arcsin z)) < pi/2"
unfolding Re_Arcsin
by (blast intro: Re_Ln_pos_lt_imp Arcsin_range_lemma)
lemma Arccos_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Re(Arccos z) \<and> Re(Arccos z) < pi"
unfolding Re_Arccos
by (blast intro!: Im_Ln_pos_lt_imp Arccos_range_lemma)
lemma Re_Arccos_bounds: "-pi < Re(Arccos z) \<and> Re(Arccos z) \<le> pi"
unfolding Re_Arccos
by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arccos_body_lemma)
lemma Re_Arccos_bound: "abs(Re(Arccos z)) \<le> pi"
using Re_Arccos_bounds abs_le_interval_iff less_eq_real_def by blast
lemma Re_Arcsin_bounds: "-pi < Re(Arcsin z) & Re(Arcsin z) \<le> pi"
unfolding Re_Arcsin
by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arcsin_body_lemma)
lemma Re_Arcsin_bound: "abs(Re(Arcsin z)) \<le> pi"
using Re_Arcsin_bounds abs_le_interval_iff less_eq_real_def by blast
subsection{*Interrelations between Arcsin and Arccos*}
lemma cos_Arcsin_nonzero:
assumes "z\<^sup>2 \<noteq> 1" shows "cos(Arcsin z) \<noteq> 0"
proof -
have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = z\<^sup>2 * (z\<^sup>2 - 1)"
by (simp add: power_mult_distrib algebra_simps)
have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> z\<^sup>2 - 1"
proof
assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = z\<^sup>2 - 1"
then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (z\<^sup>2 - 1)\<^sup>2"
by simp
then have "z\<^sup>2 * (z\<^sup>2 - 1) = (z\<^sup>2 - 1)*(z\<^sup>2 - 1)"
using eq power2_eq_square by auto
then show False
using assms by simp
qed
then have "1 + \<i> * z * (csqrt (1 - z * z)) \<noteq> z\<^sup>2"
by (metis add_minus_cancel power2_eq_square uminus_add_conv_diff)
then have "2*(1 + \<i> * z * (csqrt (1 - z * z))) \<noteq> 2*z\<^sup>2" (*FIXME cancel_numeral_factor*)
by (metis mult_cancel_left zero_neq_numeral)
then have "(\<i> * z + csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> -1"
using assms
apply (auto simp: power2_sum)
apply (simp add: power2_eq_square algebra_simps)
done
then show ?thesis
apply (simp add: cos_exp_eq Arcsin_def exp_minus)
apply (simp add: divide_simps Arcsin_body_lemma)
apply (metis add.commute minus_unique power2_eq_square)
done
qed
lemma sin_Arccos_nonzero:
assumes "z\<^sup>2 \<noteq> 1" shows "sin(Arccos z) \<noteq> 0"
proof -
have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = -(z\<^sup>2) * (1 - z\<^sup>2)"
by (simp add: power_mult_distrib algebra_simps)
have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1 - z\<^sup>2"
proof
assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = 1 - z\<^sup>2"
then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (1 - z\<^sup>2)\<^sup>2"
by simp
then have "-(z\<^sup>2) * (1 - z\<^sup>2) = (1 - z\<^sup>2)*(1 - z\<^sup>2)"
using eq power2_eq_square by auto
then have "-(z\<^sup>2) = (1 - z\<^sup>2)"
using assms
by (metis add.commute add.right_neutral diff_add_cancel mult_right_cancel)
then show False
using assms by simp
qed
then have "z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1"
by (simp add: algebra_simps)
then have "2*(z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2))) \<noteq> 2*1"
by (metis mult_cancel_left2 zero_neq_numeral) (*FIXME cancel_numeral_factor*)
then have "(z + \<i> * csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> 1"
using assms
apply (auto simp: Power.comm_semiring_1_class.power2_sum power_mult_distrib)
apply (simp add: power2_eq_square algebra_simps)
done
then show ?thesis
apply (simp add: sin_exp_eq Arccos_def exp_minus)
apply (simp add: divide_simps Arccos_body_lemma)
apply (simp add: power2_eq_square)
done
qed
lemma cos_sin_csqrt:
assumes "0 < cos(Re z) \<or> cos(Re z) = 0 \<and> Im z * sin(Re z) \<le> 0"
shows "cos z = csqrt(1 - (sin z)\<^sup>2)"
apply (rule csqrt_unique [THEN sym])
apply (simp add: cos_squared_eq)
using assms
apply (auto simp: Re_cos Im_cos add_pos_pos mult_le_0_iff zero_le_mult_iff)
apply (auto simp: algebra_simps)
done
lemma sin_cos_csqrt:
assumes "0 < sin(Re z) \<or> sin(Re z) = 0 \<and> 0 \<le> Im z * cos(Re z)"
shows "sin z = csqrt(1 - (cos z)\<^sup>2)"
apply (rule csqrt_unique [THEN sym])
apply (simp add: sin_squared_eq)
using assms
apply (auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff)
apply (auto simp: algebra_simps)
done
lemma Arcsin_Arccos_csqrt_pos:
"(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arcsin z = Arccos(csqrt(1 - z\<^sup>2))"
by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
lemma Arccos_Arcsin_csqrt_pos:
"(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arccos z = Arcsin(csqrt(1 - z\<^sup>2))"
by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
lemma sin_Arccos:
"0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> sin(Arccos z) = csqrt(1 - z\<^sup>2)"
by (simp add: Arccos_Arcsin_csqrt_pos)
lemma cos_Arcsin:
"0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> cos(Arcsin z) = csqrt(1 - z\<^sup>2)"
by (simp add: Arcsin_Arccos_csqrt_pos)
subsection{*Relationship with Arcsin on the Real Numbers*}
lemma Im_Arcsin_of_real:
assumes "abs x \<le> 1"
shows "Im (Arcsin (of_real x)) = 0"
proof -
have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
using assms abs_square_le_1
by (force simp add: Complex.cmod_power2)
then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2)) = 1"
by (simp add: norm_complex_def)
then show ?thesis
by (simp add: Im_Arcsin exp_minus)
qed
corollary Arcsin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arcsin z \<in> \<real>"
by (metis Im_Arcsin_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)
lemma arcsin_eq_Re_Arcsin:
assumes "abs x \<le> 1"
shows "arcsin x = Re (Arcsin (of_real x))"
unfolding arcsin_def
proof (rule the_equality, safe)
show "- (pi / 2) \<le> Re (Arcsin (complex_of_real x))"
using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arcsin)
next
show "Re (Arcsin (complex_of_real x)) \<le> pi / 2"
using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arcsin)
next
show "sin (Re (Arcsin (complex_of_real x))) = x"
using Re_sin [of "Arcsin (of_real x)"] Arcsin_body_lemma [of "of_real x"]
by (simp add: Im_Arcsin_of_real assms)
next
fix x'
assume "- (pi / 2) \<le> x'" "x' \<le> pi / 2" "x = sin x'"
then show "x' = Re (Arcsin (complex_of_real (sin x')))"
apply (simp add: sin_of_real [symmetric])
apply (subst Arcsin_sin)
apply (auto simp: )
done
qed
lemma of_real_arcsin: "abs x \<le> 1 \<Longrightarrow> of_real(arcsin x) = Arcsin(of_real x)"
by (metis Im_Arcsin_of_real add.right_neutral arcsin_eq_Re_Arcsin complex_eq mult_zero_right of_real_0)
subsection{*Relationship with Arccos on the Real Numbers*}
lemma Im_Arccos_of_real:
assumes "abs x \<le> 1"
shows "Im (Arccos (of_real x)) = 0"
proof -
have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
using assms abs_square_le_1
by (force simp add: Complex.cmod_power2)
then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2)) = 1"
by (simp add: norm_complex_def)
then show ?thesis
by (simp add: Im_Arccos exp_minus)
qed
corollary Arccos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arccos z \<in> \<real>"
by (metis Im_Arccos_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)
lemma arccos_eq_Re_Arccos:
assumes "abs x \<le> 1"
shows "arccos x = Re (Arccos (of_real x))"
unfolding arccos_def
proof (rule the_equality, safe)
show "0 \<le> Re (Arccos (complex_of_real x))"
using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arccos)
next
show "Re (Arccos (complex_of_real x)) \<le> pi"
using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arccos)
next
show "cos (Re (Arccos (complex_of_real x))) = x"
using Re_cos [of "Arccos (of_real x)"] Arccos_body_lemma [of "of_real x"]
by (simp add: Im_Arccos_of_real assms)
next
fix x'
assume "0 \<le> x'" "x' \<le> pi" "x = cos x'"
then show "x' = Re (Arccos (complex_of_real (cos x')))"
apply (simp add: cos_of_real [symmetric])
apply (subst Arccos_cos)
apply (auto simp: )
done
qed
lemma of_real_arccos: "abs x \<le> 1 \<Longrightarrow> of_real(arccos x) = Arccos(of_real x)"
by (metis Im_Arccos_of_real add.right_neutral arccos_eq_Re_Arccos complex_eq mult_zero_right of_real_0)
end