wenzelm@41959: (* Title: HOL/Complex.thy paulson@13957: Author: Jacques D. Fleuriot paulson@13957: Copyright: 2001 University of Edinburgh paulson@14387: Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 paulson@13957: *) paulson@13957: paulson@14377: header {* Complex Numbers: Rectangular and Polar Representations *} paulson@14373: nipkow@15131: theory Complex haftmann@28952: imports Transcendental nipkow@15131: begin paulson@13957: paulson@14373: datatype complex = Complex real real paulson@13957: huffman@44724: primrec Re :: "complex \ real" huffman@44724: where Re: "Re (Complex x y) = x" paulson@14373: huffman@44724: primrec Im :: "complex \ real" huffman@44724: where Im: "Im (Complex x y) = y" paulson@14373: paulson@14373: lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" paulson@14373: by (induct z) simp paulson@13957: huffman@44065: lemma complex_eqI [intro?]: "\Re x = Re y; Im x = Im y\ \ x = y" haftmann@25712: by (induct x, induct y) simp huffman@23125: huffman@44065: lemma complex_eq_iff: "x = y \ Re x = Re y \ Im x = Im y" haftmann@25712: by (induct x, induct y) simp huffman@23125: huffman@23125: huffman@23125: subsection {* Addition and Subtraction *} huffman@23125: haftmann@25599: instantiation complex :: ab_group_add haftmann@25571: begin haftmann@25571: huffman@44724: definition complex_zero_def: huffman@44724: "0 = Complex 0 0" haftmann@25571: huffman@44724: definition complex_add_def: huffman@44724: "x + y = Complex (Re x + Re y) (Im x + Im y)" huffman@23124: huffman@44724: definition complex_minus_def: huffman@44724: "- x = Complex (- Re x) (- Im x)" paulson@14323: huffman@44724: definition complex_diff_def: huffman@44724: "x - (y\complex) = x + - y" haftmann@25571: haftmann@25599: lemma Complex_eq_0 [simp]: "Complex a b = 0 \ a = 0 \ b = 0" haftmann@25599: by (simp add: complex_zero_def) paulson@14323: paulson@14374: lemma complex_Re_zero [simp]: "Re 0 = 0" haftmann@25599: by (simp add: complex_zero_def) paulson@14374: paulson@14374: lemma complex_Im_zero [simp]: "Im 0 = 0" haftmann@25599: by (simp add: complex_zero_def) haftmann@25599: haftmann@25712: lemma complex_add [simp]: haftmann@25712: "Complex a b + Complex c d = Complex (a + c) (b + d)" haftmann@25712: by (simp add: complex_add_def) haftmann@25712: haftmann@25599: lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" haftmann@25599: by (simp add: complex_add_def) haftmann@25599: haftmann@25599: lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" haftmann@25599: by (simp add: complex_add_def) paulson@14323: haftmann@25712: lemma complex_minus [simp]: haftmann@25712: "- (Complex a b) = Complex (- a) (- b)" haftmann@25599: by (simp add: complex_minus_def) huffman@23125: huffman@23125: lemma complex_Re_minus [simp]: "Re (- x) = - Re x" haftmann@25599: by (simp add: complex_minus_def) huffman@23125: huffman@23125: lemma complex_Im_minus [simp]: "Im (- x) = - Im x" haftmann@25599: by (simp add: complex_minus_def) huffman@23125: huffman@23275: lemma complex_diff [simp]: huffman@23125: "Complex a b - Complex c d = Complex (a - c) (b - d)" haftmann@25599: by (simp add: complex_diff_def) huffman@23125: huffman@23125: lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" haftmann@25599: by (simp add: complex_diff_def) huffman@23125: huffman@23125: lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" haftmann@25599: by (simp add: complex_diff_def) huffman@23125: haftmann@25712: instance haftmann@25712: by intro_classes (simp_all add: complex_add_def complex_diff_def) haftmann@25712: haftmann@25712: end haftmann@25712: haftmann@25712: huffman@23125: subsection {* Multiplication and Division *} huffman@23125: haftmann@36409: instantiation complex :: field_inverse_zero haftmann@25571: begin haftmann@25571: huffman@44724: definition complex_one_def: huffman@44724: "1 = Complex 1 0" haftmann@25571: huffman@44724: definition complex_mult_def: huffman@44724: "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)" huffman@23125: huffman@44724: definition complex_inverse_def: huffman@44724: "inverse x = wenzelm@53015: Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))" huffman@23125: huffman@44724: definition complex_divide_def: huffman@44724: "x / (y\complex) = x * inverse y" haftmann@25571: haftmann@54489: lemma Complex_eq_1 [simp]: haftmann@54489: "Complex a b = 1 \ a = 1 \ b = 0" haftmann@54489: by (simp add: complex_one_def) haftmann@54489: haftmann@54489: lemma Complex_eq_neg_1 [simp]: haftmann@54489: "Complex a b = - 1 \ a = - 1 \ b = 0" haftmann@25712: by (simp add: complex_one_def) huffman@22861: paulson@14374: lemma complex_Re_one [simp]: "Re 1 = 1" haftmann@25712: by (simp add: complex_one_def) paulson@14323: paulson@14374: lemma complex_Im_one [simp]: "Im 1 = 0" haftmann@25712: by (simp add: complex_one_def) paulson@14323: huffman@23125: lemma complex_mult [simp]: huffman@23125: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" haftmann@25712: by (simp add: complex_mult_def) paulson@14323: huffman@23125: lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" haftmann@25712: by (simp add: complex_mult_def) paulson@14323: huffman@23125: lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" haftmann@25712: by (simp add: complex_mult_def) paulson@14323: paulson@14377: lemma complex_inverse [simp]: wenzelm@53015: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))" haftmann@25712: by (simp add: complex_inverse_def) paulson@14335: huffman@23125: lemma complex_Re_inverse: wenzelm@53015: "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" haftmann@25712: by (simp add: complex_inverse_def) paulson@14323: huffman@23125: lemma complex_Im_inverse: wenzelm@53015: "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" haftmann@25712: by (simp add: complex_inverse_def) paulson@14335: haftmann@25712: instance haftmann@25712: by intro_classes (simp_all add: complex_mult_def webertj@49962: distrib_left distrib_right right_diff_distrib left_diff_distrib huffman@44724: complex_inverse_def complex_divide_def huffman@44724: power2_eq_square add_divide_distrib [symmetric] huffman@44724: complex_eq_iff) paulson@14335: haftmann@25712: end huffman@23125: huffman@23125: huffman@23125: subsection {* Numerals and Arithmetic *} huffman@23125: huffman@23125: lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" huffman@44724: by (induct n) simp_all huffman@20556: huffman@23125: lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" huffman@44724: by (induct n) simp_all huffman@23125: huffman@23125: lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" huffman@44724: by (cases z rule: int_diff_cases) simp huffman@23125: huffman@23125: lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" huffman@44724: by (cases z rule: int_diff_cases) simp huffman@23125: huffman@47108: lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v" huffman@47108: using complex_Re_of_int [of "numeral v"] by simp huffman@47108: haftmann@54489: lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v" haftmann@54489: using complex_Re_of_int [of "- numeral v"] by simp huffman@47108: huffman@47108: lemma complex_Im_numeral [simp]: "Im (numeral v) = 0" huffman@47108: using complex_Im_of_int [of "numeral v"] by simp huffman@20556: haftmann@54489: lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0" haftmann@54489: using complex_Im_of_int [of "- numeral v"] by simp huffman@23125: huffman@47108: lemma Complex_eq_numeral [simp]: haftmann@54489: "Complex a b = numeral w \ a = numeral w \ b = 0" huffman@47108: by (simp add: complex_eq_iff) huffman@47108: huffman@47108: lemma Complex_eq_neg_numeral [simp]: haftmann@54489: "Complex a b = - numeral w \ a = - numeral w \ b = 0" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: huffman@23125: subsection {* Scalar Multiplication *} huffman@20556: haftmann@25712: instantiation complex :: real_field haftmann@25571: begin haftmann@25571: huffman@44724: definition complex_scaleR_def: huffman@44724: "scaleR r x = Complex (r * Re x) (r * Im x)" haftmann@25571: huffman@23125: lemma complex_scaleR [simp]: huffman@23125: "scaleR r (Complex a b) = Complex (r * a) (r * b)" haftmann@25712: unfolding complex_scaleR_def by simp huffman@23125: huffman@23125: lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" haftmann@25712: unfolding complex_scaleR_def by simp huffman@23125: huffman@23125: lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" haftmann@25712: unfolding complex_scaleR_def by simp huffman@22972: haftmann@25712: instance huffman@20556: proof huffman@23125: fix a b :: real and x y :: complex huffman@23125: show "scaleR a (x + y) = scaleR a x + scaleR a y" webertj@49962: by (simp add: complex_eq_iff distrib_left) huffman@23125: show "scaleR (a + b) x = scaleR a x + scaleR b x" webertj@49962: by (simp add: complex_eq_iff distrib_right) huffman@23125: show "scaleR a (scaleR b x) = scaleR (a * b) x" huffman@44065: by (simp add: complex_eq_iff mult_assoc) huffman@23125: show "scaleR 1 x = x" huffman@44065: by (simp add: complex_eq_iff) huffman@23125: show "scaleR a x * y = scaleR a (x * y)" huffman@44065: by (simp add: complex_eq_iff algebra_simps) huffman@23125: show "x * scaleR a y = scaleR a (x * y)" huffman@44065: by (simp add: complex_eq_iff algebra_simps) huffman@20556: qed huffman@20556: haftmann@25712: end haftmann@25712: huffman@20556: huffman@23125: subsection{* Properties of Embedding from Reals *} paulson@14323: huffman@44724: abbreviation complex_of_real :: "real \ complex" huffman@44724: where "complex_of_real \ of_real" huffman@20557: hoelzl@56331: declare [[coercion complex_of_real]] hoelzl@56331: huffman@20557: lemma complex_of_real_def: "complex_of_real r = Complex r 0" huffman@44724: by (simp add: of_real_def complex_scaleR_def) huffman@20557: huffman@20557: lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" huffman@44724: by (simp add: complex_of_real_def) huffman@20557: huffman@20557: lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" huffman@44724: by (simp add: complex_of_real_def) huffman@20557: paulson@14377: lemma Complex_add_complex_of_real [simp]: huffman@44724: shows "Complex x y + complex_of_real r = Complex (x+r) y" huffman@44724: by (simp add: complex_of_real_def) paulson@14377: paulson@14377: lemma complex_of_real_add_Complex [simp]: huffman@44724: shows "complex_of_real r + Complex x y = Complex (r+x) y" huffman@44724: by (simp add: complex_of_real_def) paulson@14377: paulson@14377: lemma Complex_mult_complex_of_real: huffman@44724: shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)" huffman@44724: by (simp add: complex_of_real_def) paulson@14377: paulson@14377: lemma complex_of_real_mult_Complex: huffman@44724: shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)" huffman@44724: by (simp add: complex_of_real_def) huffman@20557: huffman@44841: lemma complex_eq_cancel_iff2 [simp]: huffman@44841: shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" huffman@44841: by (simp add: complex_of_real_def) huffman@44841: huffman@44827: lemma complex_split_polar: huffman@44827: "\r a. z = complex_of_real r * (Complex (cos a) (sin a))" huffman@44827: by (simp add: complex_eq_iff polar_Ex) huffman@44827: paulson@14377: huffman@23125: subsection {* Vector Norm *} paulson@14323: haftmann@25712: instantiation complex :: real_normed_field haftmann@25571: begin haftmann@25571: huffman@31413: definition complex_norm_def: wenzelm@53015: "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" haftmann@25571: huffman@44724: abbreviation cmod :: "complex \ real" huffman@44724: where "cmod \ norm" haftmann@25571: huffman@31413: definition complex_sgn_def: huffman@31413: "sgn x = x /\<^sub>R cmod x" haftmann@25571: huffman@31413: definition dist_complex_def: huffman@31413: "dist x y = cmod (x - y)" huffman@31413: haftmann@37767: definition open_complex_def: huffman@31492: "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" huffman@31292: huffman@20557: lemmas cmod_def = complex_norm_def huffman@20557: wenzelm@53015: lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" haftmann@25712: by (simp add: complex_norm_def) huffman@22852: huffman@31413: instance proof huffman@31492: fix r :: real and x y :: complex and S :: "complex set" huffman@23125: show "(norm x = 0) = (x = 0)" huffman@22861: by (induct x) simp huffman@23125: show "norm (x + y) \ norm x + norm y" huffman@23125: by (induct x, induct y) huffman@23125: (simp add: real_sqrt_sum_squares_triangle_ineq) huffman@23125: show "norm (scaleR r x) = \r\ * norm x" huffman@23125: by (induct x) webertj@49962: (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult) huffman@23125: show "norm (x * y) = norm x * norm y" huffman@23125: by (induct x, induct y) nipkow@29667: (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) huffman@31292: show "sgn x = x /\<^sub>R cmod x" huffman@31292: by (rule complex_sgn_def) huffman@31292: show "dist x y = cmod (x - y)" huffman@31292: by (rule dist_complex_def) huffman@31492: show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" huffman@31492: by (rule open_complex_def) huffman@24520: qed huffman@20557: haftmann@25712: end haftmann@25712: huffman@44761: lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1" huffman@44724: by simp paulson@14323: huffman@44761: lemma cmod_complex_polar: huffman@44724: "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" huffman@44724: by (simp add: norm_mult) huffman@22861: huffman@22861: lemma complex_Re_le_cmod: "Re x \ cmod x" huffman@44724: unfolding complex_norm_def huffman@44724: by (rule real_sqrt_sum_squares_ge1) huffman@22861: huffman@44761: lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x" huffman@44724: by (rule order_trans [OF _ norm_ge_zero], simp) huffman@22861: huffman@44761: lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a" huffman@44724: by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) paulson@14323: chaieb@26117: lemma abs_Re_le_cmod: "\Re x\ \ cmod x" huffman@44724: by (cases x) simp chaieb@26117: chaieb@26117: lemma abs_Im_le_cmod: "\Im x\ \ cmod x" huffman@44724: by (cases x) simp huffman@44724: hoelzl@56369: hoelzl@56369: lemma abs_sqrt_wlog: hoelzl@56369: fixes x::"'a::linordered_idom" hoelzl@56369: assumes "\x::'a. x \ 0 \ P x (x\<^sup>2)" shows "P \x\ (x\<^sup>2)" hoelzl@56369: by (metis abs_ge_zero assms power2_abs) hoelzl@56369: hoelzl@56369: lemma complex_abs_le_norm: "\Re z\ + \Im z\ \ sqrt 2 * norm z" hoelzl@56369: unfolding complex_norm_def hoelzl@56369: apply (rule abs_sqrt_wlog [where x="Re z"]) hoelzl@56369: apply (rule abs_sqrt_wlog [where x="Im z"]) hoelzl@56369: apply (rule power2_le_imp_le) hoelzl@56369: apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric]) hoelzl@56369: done hoelzl@56369: hoelzl@56369: huffman@44843: text {* Properties of complex signum. *} huffman@44843: huffman@44843: lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" huffman@44843: by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute) huffman@44843: huffman@44843: lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" huffman@44843: by (simp add: complex_sgn_def divide_inverse) huffman@44843: huffman@44843: lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" huffman@44843: by (simp add: complex_sgn_def divide_inverse) huffman@44843: paulson@14354: huffman@23123: subsection {* Completeness of the Complexes *} huffman@23123: huffman@44290: lemma bounded_linear_Re: "bounded_linear Re" huffman@44290: by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) huffman@44290: huffman@44290: lemma bounded_linear_Im: "bounded_linear Im" huffman@44127: by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) huffman@23123: huffman@44290: lemmas tendsto_Re [tendsto_intros] = huffman@44290: bounded_linear.tendsto [OF bounded_linear_Re] huffman@44290: huffman@44290: lemmas tendsto_Im [tendsto_intros] = huffman@44290: bounded_linear.tendsto [OF bounded_linear_Im] huffman@44290: huffman@44290: lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] huffman@44290: lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] huffman@44290: lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] huffman@44290: lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] huffman@23123: hoelzl@56369: lemma continuous_Re: "continuous_on X Re" hoelzl@56369: by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Re hoelzl@56369: continuous_on_cong continuous_on_id) hoelzl@56369: hoelzl@56369: lemma continuous_Im: "continuous_on X Im" hoelzl@56369: by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Im hoelzl@56369: continuous_on_cong continuous_on_id) hoelzl@56369: hoelzl@56369: lemma has_derivative_Re [has_derivative_intros] : "(Re has_derivative Re) F" hoelzl@56369: by (auto simp add: has_derivative_def bounded_linear_Re tendsto_const) hoelzl@56369: hoelzl@56369: lemma has_derivative_Im [has_derivative_intros] : "(Im has_derivative Im) F" hoelzl@56369: by (auto simp add: has_derivative_def bounded_linear_Im tendsto_const) hoelzl@56369: huffman@36825: lemma tendsto_Complex [tendsto_intros]: huffman@44724: assumes "(f ---> a) F" and "(g ---> b) F" huffman@44724: shows "((\x. Complex (f x) (g x)) ---> Complex a b) F" huffman@36825: proof (rule tendstoI) huffman@36825: fix r :: real assume "0 < r" huffman@36825: hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) huffman@44724: have "eventually (\x. dist (f x) a < r / sqrt 2) F" huffman@44724: using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD) huffman@36825: moreover huffman@44724: have "eventually (\x. dist (g x) b < r / sqrt 2) F" huffman@44724: using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD) huffman@36825: ultimately huffman@44724: show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F" huffman@36825: by (rule eventually_elim2) huffman@36825: (simp add: dist_norm real_sqrt_sum_squares_less) huffman@36825: qed huffman@36825: hoelzl@56369: hoelzl@56369: lemma tendsto_complex_iff: hoelzl@56369: "(f ---> x) F \ (((\x. Re (f x)) ---> Re x) F \ ((\x. Im (f x)) ---> Im x) F)" hoelzl@56369: proof - hoelzl@56369: have f: "f = (\x. Complex (Re (f x)) (Im (f x)))" and x: "x = Complex (Re x) (Im x)" hoelzl@56369: by simp_all hoelzl@56369: show ?thesis hoelzl@56369: apply (subst f) hoelzl@56369: apply (subst x) hoelzl@56369: apply (intro iffI tendsto_Complex conjI) hoelzl@56369: apply (simp_all add: tendsto_Re tendsto_Im) hoelzl@56369: done hoelzl@56369: qed hoelzl@56369: huffman@23123: instance complex :: banach huffman@23123: proof huffman@23123: fix X :: "nat \ complex" huffman@23123: assume X: "Cauchy X" huffman@44290: from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))" huffman@23123: by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) huffman@44290: from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))" huffman@23123: by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) huffman@23123: have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))" huffman@44748: using tendsto_Complex [OF 1 2] by simp huffman@23123: thus "convergent X" huffman@23123: by (rule convergentI) huffman@23123: qed huffman@23123: lp15@56238: declare lp15@56238: DERIV_power[where 'a=complex, THEN DERIV_cong, lp15@56238: unfolded of_nat_def[symmetric], DERIV_intros] lp15@56238: huffman@23123: huffman@44827: subsection {* The Complex Number $i$ *} huffman@23125: huffman@44724: definition "ii" :: complex ("\") huffman@44724: where i_def: "ii \ Complex 0 1" huffman@23125: huffman@23125: lemma complex_Re_i [simp]: "Re ii = 0" huffman@44724: by (simp add: i_def) paulson@14354: huffman@23125: lemma complex_Im_i [simp]: "Im ii = 1" huffman@44724: by (simp add: i_def) huffman@23125: huffman@23125: lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)" huffman@44724: by (simp add: i_def) huffman@23125: huffman@44902: lemma norm_ii [simp]: "norm ii = 1" huffman@44902: by (simp add: i_def) huffman@44902: huffman@23125: lemma complex_i_not_zero [simp]: "ii \ 0" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_i_not_one [simp]: "ii \ 1" huffman@44724: by (simp add: complex_eq_iff) huffman@23124: huffman@47108: lemma complex_i_not_numeral [simp]: "ii \ numeral w" huffman@47108: by (simp add: complex_eq_iff) huffman@47108: haftmann@54489: lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" huffman@44724: by (simp add: i_def complex_of_real_def) huffman@23125: huffman@23125: lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" huffman@44724: by (simp add: i_def complex_of_real_def) huffman@23125: huffman@23125: lemma i_squared [simp]: "ii * ii = -1" huffman@44724: by (simp add: i_def) huffman@23125: wenzelm@53015: lemma power2_i [simp]: "ii\<^sup>2 = -1" huffman@44724: by (simp add: power2_eq_square) huffman@23125: huffman@23125: lemma inverse_i [simp]: "inverse ii = - ii" huffman@44724: by (rule inverse_unique, simp) paulson@14354: huffman@44827: lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" huffman@44827: by (simp add: mult_assoc [symmetric]) huffman@44827: paulson@14354: huffman@23125: subsection {* Complex Conjugation *} huffman@23125: huffman@44724: definition cnj :: "complex \ complex" where huffman@23125: "cnj z = Complex (Re z) (- Im z)" huffman@23125: huffman@23125: lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" huffman@44724: by (simp add: cnj_def) huffman@23125: huffman@23125: lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" huffman@44724: by (simp add: cnj_def) huffman@23125: huffman@23125: lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" huffman@44724: by (simp add: cnj_def) huffman@23125: huffman@23125: lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" huffman@44724: by (simp add: cnj_def) huffman@23125: huffman@23125: lemma complex_cnj_zero [simp]: "cnj 0 = 0" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: hoelzl@56369: lemma cnj_setsum: "cnj (setsum f s) = (\x\s. cnj (f x))" hoelzl@56369: by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_add) hoelzl@56369: huffman@23125: lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_minus: "cnj (- x) = - cnj x" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_one [simp]: "cnj 1 = 1" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: hoelzl@56369: lemma cnj_setprod: "cnj (setprod f s) = (\x\s. cnj (f x))" hoelzl@56369: by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_mult) hoelzl@56369: huffman@23125: lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" huffman@44724: by (simp add: complex_inverse_def) paulson@14323: huffman@23125: lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" huffman@44724: by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) huffman@23125: huffman@23125: lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" huffman@44724: by (induct n, simp_all add: complex_cnj_mult) huffman@23125: huffman@23125: lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@47108: lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" huffman@47108: by (simp add: complex_eq_iff) huffman@47108: haftmann@54489: lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" huffman@44724: by (simp add: complex_norm_def) paulson@14323: huffman@23125: lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_cnj_i [simp]: "cnj ii = - ii" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" huffman@44724: by (simp add: complex_eq_iff) huffman@23125: huffman@23125: lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" huffman@44724: by (simp add: complex_eq_iff) paulson@14354: wenzelm@53015: lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" huffman@44724: by (simp add: complex_eq_iff power2_eq_square) huffman@23125: wenzelm@53015: lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" huffman@44724: by (simp add: norm_mult power2_eq_square) huffman@23125: huffman@44827: lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" huffman@44827: by (simp add: cmod_def power2_eq_square) huffman@44827: huffman@44827: lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" huffman@44827: by simp huffman@44827: huffman@44290: lemma bounded_linear_cnj: "bounded_linear cnj" huffman@44127: using complex_cnj_add complex_cnj_scaleR huffman@44127: by (rule bounded_linear_intro [where K=1], simp) paulson@14354: huffman@44290: lemmas tendsto_cnj [tendsto_intros] = huffman@44290: bounded_linear.tendsto [OF bounded_linear_cnj] huffman@44290: huffman@44290: lemmas isCont_cnj [simp] = huffman@44290: bounded_linear.isCont [OF bounded_linear_cnj] huffman@44290: hoelzl@56369: lemma lim_cnj: "((\x. cnj(f x)) ---> cnj l) F \ (f ---> l) F" hoelzl@56369: by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric]) hoelzl@56369: hoelzl@56369: lemma sums_cnj: "((\x. cnj(f x)) sums cnj l) \ (f sums l)" hoelzl@56369: by (simp add: sums_def lim_cnj cnj_setsum [symmetric]) hoelzl@56369: paulson@14354: lp15@55734: subsection{*Basic Lemmas*} lp15@55734: lp15@55734: lemma complex_eq_0: "z=0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" lp15@55734: by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff) lp15@55734: lp15@55734: lemma complex_neq_0: "z\0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" lp15@55734: by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) lp15@55734: lp15@55734: lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" lp15@55734: apply (cases z, auto) lp15@55734: by (metis complex_of_real_def of_real_add of_real_power power2_eq_square) lp15@55734: lp15@55734: lemma complex_div_eq_0: lp15@55734: "(Re(a / b) = 0 \ Re(a * cnj b) = 0) & (Im(a / b) = 0 \ Im(a * cnj b) = 0)" lp15@55734: proof (cases "b=0") lp15@55734: case True then show ?thesis by auto lp15@55734: next lp15@55734: case False lp15@55734: show ?thesis lp15@55734: proof (cases b) lp15@55734: case (Complex x y) lp15@55734: then have "x\<^sup>2 + y\<^sup>2 > 0" lp15@55734: by (metis Complex_eq_0 False sum_power2_gt_zero_iff) lp15@55734: then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" lp15@55734: by (metis add_divide_distrib) lp15@55734: with Complex False show ?thesis lp15@55734: by (auto simp: complex_divide_def) lp15@55734: qed lp15@55734: qed lp15@55734: lp15@55734: lemma re_complex_div_eq_0: "Re(a / b) = 0 \ Re(a * cnj b) = 0" lp15@55734: and im_complex_div_eq_0: "Im(a / b) = 0 \ Im(a * cnj b) = 0" lp15@55734: using complex_div_eq_0 by auto lp15@55734: lp15@55734: lp15@55734: lemma complex_div_gt_0: lp15@55734: "(Re(a / b) > 0 \ Re(a * cnj b) > 0) & (Im(a / b) > 0 \ Im(a * cnj b) > 0)" lp15@55734: proof (cases "b=0") lp15@55734: case True then show ?thesis by auto lp15@55734: next lp15@55734: case False lp15@55734: show ?thesis lp15@55734: proof (cases b) lp15@55734: case (Complex x y) lp15@55734: then have "x\<^sup>2 + y\<^sup>2 > 0" lp15@55734: by (metis Complex_eq_0 False sum_power2_gt_zero_iff) lp15@55734: moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" lp15@55734: by (metis add_divide_distrib) lp15@55734: ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2` lp15@55734: apply (simp add: complex_divide_def zero_less_divide_iff less_divide_eq) lp15@55734: apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left) lp15@55734: done lp15@55734: qed lp15@55734: qed lp15@55734: lp15@55734: lemma re_complex_div_gt_0: "Re(a / b) > 0 \ Re(a * cnj b) > 0" lp15@55734: and im_complex_div_gt_0: "Im(a / b) > 0 \ Im(a * cnj b) > 0" lp15@55734: using complex_div_gt_0 by auto lp15@55734: lp15@55734: lemma re_complex_div_ge_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0" lp15@55734: by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0) lp15@55734: lp15@55734: lemma im_complex_div_ge_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0" lp15@55734: by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less) lp15@55734: lp15@55734: lemma re_complex_div_lt_0: "Re(a / b) < 0 \ Re(a * cnj b) < 0" boehmes@55759: by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0) lp15@55734: lp15@55734: lemma im_complex_div_lt_0: "Im(a / b) < 0 \ Im(a * cnj b) < 0" lp15@55734: by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff) lp15@55734: lp15@55734: lemma re_complex_div_le_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0" lp15@55734: by (metis not_le re_complex_div_gt_0) lp15@55734: lp15@55734: lemma im_complex_div_le_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0" lp15@55734: by (metis im_complex_div_gt_0 not_le) lp15@55734: lp15@56217: lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s" hoelzl@56369: by (induct s rule: infinite_finite_induct) auto lp15@55734: lp15@56217: lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s" hoelzl@56369: by (induct s rule: infinite_finite_induct) auto hoelzl@56369: hoelzl@56369: lemma sums_complex_iff: "f sums x \ ((\x. Re (f x)) sums Re x) \ ((\x. Im (f x)) sums Im x)" hoelzl@56369: unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum .. hoelzl@56369: hoelzl@56369: lemma sums_Re: "f sums a \ (\x. Re (f x)) sums Re a" hoelzl@56369: unfolding sums_complex_iff by blast hoelzl@56369: hoelzl@56369: lemma sums_Im: "f sums a \ (\x. Im (f x)) sums Im a" hoelzl@56369: unfolding sums_complex_iff by blast hoelzl@56369: hoelzl@56369: lemma summable_complex_iff: "summable f \ summable (\x. Re (f x)) \ summable (\x. Im (f x))" hoelzl@56369: unfolding summable_def sums_complex_iff[abs_def] by (metis Im.simps Re.simps) hoelzl@56369: hoelzl@56369: lemma summable_complex_of_real [simp]: "summable (\n. complex_of_real (f n)) \ summable f" hoelzl@56369: unfolding summable_complex_iff by simp hoelzl@56369: hoelzl@56369: lemma summable_Re: "summable f \ summable (\x. Re (f x))" hoelzl@56369: unfolding summable_complex_iff by blast hoelzl@56369: hoelzl@56369: lemma summable_Im: "summable f \ summable (\x. Im (f x))" hoelzl@56369: unfolding summable_complex_iff by blast lp15@56217: lp15@56217: lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0" hoelzl@56369: by (induct s rule: infinite_finite_induct) auto lp15@55734: lp15@56217: lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s" lp15@56217: by (metis Complex_setsum') lp15@56217: lp15@56217: lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s" hoelzl@56369: by (induct s rule: infinite_finite_induct) auto lp15@55734: lp15@56217: lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s" hoelzl@56369: by (induct s rule: infinite_finite_induct) auto lp15@55734: lp15@55734: lemma Reals_cnj_iff: "z \ \ \ cnj z = z" lp15@55734: by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj lp15@55734: complex_of_real_def equal_neg_zero) lp15@55734: lp15@55734: lemma Complex_in_Reals: "Complex x 0 \ \" lp15@55734: by (metis Reals_of_real complex_of_real_def) lp15@55734: lp15@55734: lemma in_Reals_norm: "z \ \ \ norm(z) = abs(Re z)" lp15@55734: by (metis Re_complex_of_real Reals_cases norm_of_real) lp15@55734: hoelzl@56369: lemma complex_is_Real_iff: "z \ \ \ Im z = 0" hoelzl@56369: by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj) hoelzl@56369: hoelzl@56369: lemma series_comparison_complex: hoelzl@56369: fixes f:: "nat \ 'a::banach" hoelzl@56369: assumes sg: "summable g" hoelzl@56369: and "\n. g n \ \" "\n. Re (g n) \ 0" hoelzl@56369: and fg: "\n. n \ N \ norm(f n) \ norm(g n)" hoelzl@56369: shows "summable f" hoelzl@56369: proof - hoelzl@56369: have g: "\n. cmod (g n) = Re (g n)" using assms hoelzl@56369: by (metis abs_of_nonneg in_Reals_norm) hoelzl@56369: show ?thesis hoelzl@56369: apply (rule summable_comparison_test' [where g = "\n. norm (g n)" and N=N]) hoelzl@56369: using sg hoelzl@56369: apply (auto simp: summable_def) hoelzl@56369: apply (rule_tac x="Re s" in exI) hoelzl@56369: apply (auto simp: g sums_Re) hoelzl@56369: apply (metis fg g) hoelzl@56369: done hoelzl@56369: qed lp15@55734: paulson@14323: subsection{*Finally! Polar Form for Complex Numbers*} paulson@14323: huffman@44827: subsubsection {* $\cos \theta + i \sin \theta$ *} huffman@20557: huffman@44715: definition cis :: "real \ complex" where huffman@20557: "cis a = Complex (cos a) (sin a)" huffman@20557: huffman@44827: lemma Re_cis [simp]: "Re (cis a) = cos a" huffman@44827: by (simp add: cis_def) huffman@44827: huffman@44827: lemma Im_cis [simp]: "Im (cis a) = sin a" huffman@44827: by (simp add: cis_def) huffman@44827: huffman@44827: lemma cis_zero [simp]: "cis 0 = 1" huffman@44827: by (simp add: cis_def) huffman@44827: huffman@44828: lemma norm_cis [simp]: "norm (cis a) = 1" huffman@44828: by (simp add: cis_def) huffman@44828: huffman@44828: lemma sgn_cis [simp]: "sgn (cis a) = cis a" huffman@44828: by (simp add: sgn_div_norm) huffman@44828: huffman@44828: lemma cis_neq_zero [simp]: "cis a \ 0" huffman@44828: by (metis norm_cis norm_zero zero_neq_one) huffman@44828: huffman@44827: lemma cis_mult: "cis a * cis b = cis (a + b)" huffman@44827: by (simp add: cis_def cos_add sin_add) huffman@44827: huffman@44827: lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" huffman@44827: by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult) huffman@44827: huffman@44827: lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" huffman@44827: by (simp add: cis_def) huffman@44827: huffman@44827: lemma cis_divide: "cis a / cis b = cis (a - b)" haftmann@54230: by (simp add: complex_divide_def cis_mult) huffman@44827: huffman@44827: lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" huffman@44827: by (auto simp add: DeMoivre) huffman@44827: huffman@44827: lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" huffman@44827: by (auto simp add: DeMoivre) huffman@44827: huffman@44827: subsubsection {* $r(\cos \theta + i \sin \theta)$ *} huffman@44715: huffman@44715: definition rcis :: "[real, real] \ complex" where huffman@20557: "rcis r a = complex_of_real r * cis a" huffman@20557: huffman@44827: lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" huffman@44828: by (simp add: rcis_def) huffman@44827: huffman@44827: lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" huffman@44828: by (simp add: rcis_def) huffman@44827: huffman@44827: lemma rcis_Ex: "\r a. z = rcis r a" huffman@44828: by (simp add: complex_eq_iff polar_Ex) huffman@44827: huffman@44827: lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" huffman@44828: by (simp add: rcis_def norm_mult) huffman@44827: huffman@44827: lemma cis_rcis_eq: "cis a = rcis 1 a" huffman@44827: by (simp add: rcis_def) huffman@44827: huffman@44827: lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" huffman@44828: by (simp add: rcis_def cis_mult) huffman@44827: huffman@44827: lemma rcis_zero_mod [simp]: "rcis 0 a = 0" huffman@44827: by (simp add: rcis_def) huffman@44827: huffman@44827: lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" huffman@44827: by (simp add: rcis_def) huffman@44827: huffman@44828: lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0" huffman@44828: by (simp add: rcis_def) huffman@44828: huffman@44827: lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" huffman@44827: by (simp add: rcis_def power_mult_distrib DeMoivre) huffman@44827: huffman@44827: lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" huffman@44827: by (simp add: divide_inverse rcis_def) huffman@44827: huffman@44827: lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" huffman@44828: by (simp add: rcis_def cis_divide [symmetric]) huffman@44827: huffman@44827: subsubsection {* Complex exponential *} huffman@44827: huffman@44291: abbreviation expi :: "complex \ complex" huffman@44291: where "expi \ exp" huffman@44291: huffman@44712: lemma cis_conv_exp: "cis b = exp (Complex 0 b)" huffman@44291: proof (rule complex_eqI) huffman@44291: { fix n have "Complex 0 b ^ n = huffman@44291: real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)" huffman@44291: apply (induct n) huffman@44291: apply (simp add: cos_coeff_def sin_coeff_def) huffman@44291: apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc) huffman@44291: done } note * = this huffman@44712: show "Re (cis b) = Re (exp (Complex 0 b))" huffman@44291: unfolding exp_def cis_def cos_def huffman@44291: by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic], huffman@44291: simp add: * mult_assoc [symmetric]) huffman@44712: show "Im (cis b) = Im (exp (Complex 0 b))" huffman@44291: unfolding exp_def cis_def sin_def huffman@44291: by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic], huffman@44291: simp add: * mult_assoc [symmetric]) huffman@44291: qed huffman@44291: huffman@44291: lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)" huffman@44712: unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp huffman@20557: huffman@44828: lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" huffman@44828: unfolding expi_def by simp huffman@44828: huffman@44828: lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" huffman@44828: unfolding expi_def by simp huffman@44828: paulson@14374: lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a" paulson@14373: apply (insert rcis_Ex [of z]) huffman@23125: apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) paulson@14334: apply (rule_tac x = "ii * complex_of_real a" in exI, auto) paulson@14323: done paulson@14323: paulson@14387: lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" huffman@44724: by (simp add: expi_def cis_def) paulson@14387: huffman@44844: subsubsection {* Complex argument *} huffman@44844: huffman@44844: definition arg :: "complex \ real" where huffman@44844: "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))" huffman@44844: huffman@44844: lemma arg_zero: "arg 0 = 0" huffman@44844: by (simp add: arg_def) huffman@44844: huffman@44844: lemma of_nat_less_of_int_iff: (* TODO: move *) huffman@44844: "(of_nat n :: 'a::linordered_idom) < of_int x \ int n < x" huffman@44844: by (metis of_int_of_nat_eq of_int_less_iff) huffman@44844: huffman@47108: lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *) huffman@47108: "real (n::nat) < numeral w \ n < numeral w" huffman@47108: using of_nat_less_of_int_iff [of n "numeral w", where 'a=real] huffman@47108: by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric]) huffman@44844: huffman@44844: lemma arg_unique: huffman@44844: assumes "sgn z = cis x" and "-pi < x" and "x \ pi" huffman@44844: shows "arg z = x" huffman@44844: proof - huffman@44844: from assms have "z \ 0" by auto huffman@44844: have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x" huffman@44844: proof huffman@44844: fix a def d \ "a - x" huffman@44844: assume a: "sgn z = cis a \ - pi < a \ a \ pi" huffman@44844: from a assms have "- (2*pi) < d \ d < 2*pi" huffman@44844: unfolding d_def by simp huffman@44844: moreover from a assms have "cos a = cos x" and "sin a = sin x" huffman@44844: by (simp_all add: complex_eq_iff) wenzelm@53374: hence cos: "cos d = 1" unfolding d_def cos_diff by simp wenzelm@53374: moreover from cos have "sin d = 0" by (rule cos_one_sin_zero) huffman@44844: ultimately have "d = 0" huffman@44844: unfolding sin_zero_iff even_mult_two_ex wenzelm@53374: by (auto simp add: numeral_2_eq_2 less_Suc_eq) huffman@44844: thus "a = x" unfolding d_def by simp huffman@44844: qed (simp add: assms del: Re_sgn Im_sgn) huffman@44844: with `z \ 0` show "arg z = x" huffman@44844: unfolding arg_def by simp huffman@44844: qed huffman@44844: huffman@44844: lemma arg_correct: huffman@44844: assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi" huffman@44844: proof (simp add: arg_def assms, rule someI_ex) huffman@44844: obtain r a where z: "z = rcis r a" using rcis_Ex by fast huffman@44844: with assms have "r \ 0" by auto huffman@44844: def b \ "if 0 < r then a else a + pi" huffman@44844: have b: "sgn z = cis b" huffman@44844: unfolding z b_def rcis_def using `r \ 0` huffman@44844: by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def) huffman@44844: have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1" webertj@49962: by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric], huffman@44844: simp add: cis_def) huffman@44844: have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1" huffman@44844: by (case_tac x rule: int_diff_cases, huffman@44844: simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) huffman@44844: def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\" huffman@44844: have "sgn z = cis c" huffman@44844: unfolding b c_def huffman@44844: by (simp add: cis_divide [symmetric] cis_2pi_int) huffman@44844: moreover have "- pi < c \ c \ pi" huffman@44844: using ceiling_correct [of "(b - pi) / (2*pi)"] huffman@44844: by (simp add: c_def less_divide_eq divide_le_eq algebra_simps) huffman@44844: ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast huffman@44844: qed huffman@44844: huffman@44844: lemma arg_bounded: "- pi < arg z \ arg z \ pi" huffman@44844: by (cases "z = 0", simp_all add: arg_zero arg_correct) huffman@44844: huffman@44844: lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z" huffman@44844: by (simp add: arg_correct) huffman@44844: huffman@44844: lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" huffman@44844: by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) huffman@44844: huffman@44844: lemma cos_arg_i_mult_zero [simp]: huffman@44844: "y \ 0 ==> cos (arg(Complex 0 y)) = 0" huffman@44844: using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff) huffman@44844: huffman@44065: text {* Legacy theorem names *} huffman@44065: huffman@44065: lemmas expand_complex_eq = complex_eq_iff huffman@44065: lemmas complex_Re_Im_cancel_iff = complex_eq_iff huffman@44065: lemmas complex_equality = complex_eqI huffman@44065: hoelzl@56369: paulson@13957: end