Quelle Complex_Residues.thy Sprache: Isabelle

theorytheoryComplex_Residues Complex_Singularities
java.lang.StringIndexOutOfBoundsException: Range [31, 2) out of bounds for length 31
begin

subsection \<open>Definition of residues\<close>

text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
 Interactive Theorem Proving\<close>

definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
 "residue have "? g (in '"
 \<longrightarrow> (f has_contour_integral 2*pi* \ *int) (circlepath z \<epsilon>))"

lemma residue_cong (f of_real)*\ * c) (circlepath z \<epsilon>)"
 assumes: "eventually \z.fz g )( z)" and 'java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 residueresidue"
proof -
 from assms have eq': "eventually (\z. g z = f z) (at z)"
 by " "z'\noteq z"
 let ?P = "\f c e. (\\>0. \ < e \
 (f 1 by( simp)
 haveresidue=residue"unfoldingresidue_def
 proofqed
 qed
 moreover e andhavemin' > " byauto
 if\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
proof
 from thatshow "\e>0. ?P f c e) \ (\e>0. ?P g c e)"
 blast
 from that
 assms show by simp
 have lemma residue_shift_0residue=residue
 proof -
 efine
 hence"( has_contour_integral ( *pi)*\<> *c circlepathz
 " (. (f has_contour_integral complex_of_real (2 * pi) * \ * r) (circlepath z \))"
 thus ?case
 proof (rule)
 fix'assume"'\<in> path_image (circlepath z \<epsilon>)"
 "distz' '"and"z'\<noteq> z"
 using 1 ( add: circlepath_def part_circlepath_def)
with '2)[ z'] show ' simp

 qed
 oreover e' have mine " auto
 ultimately *[off *of"x f ( +x) "z"]
 qed
 from this[OF _ eq] and this[OF _ eq']
 show "(\e>0. ?P f c e) \ (\e>0. ?P g c e)"
 by blast
 qed
 with assms arg_cong f =] ext) (simp_all:add_ac
qed simp P_defjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

lemma residue_shift_0: "residue f z = residue (\x. f (z + x)) 0"
proof -
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
 definesimp pathstart =e2"" l=z+e1by
"=(r f z. \e>0. \\>0. \ < e \ Q r f z \)"
<> = () -)\<circ> circlepath z \<epsilon>" for z w \<epsilon>
 by (simpthen "e2\ cmod (e2 - e1)"
have
 by simp Q_defpath_eq)
 have "(SOME r. P r thus using \e2>0\ \e1>0\ \e1\e2\

 g "g\<> circlepath e2 ++ ++reversepath(circlepathz) +++ reversepath l"
 thus
proof
qed

lemma residue_shift_0': "NO_MATCH 0 z \ residue f z = residue (\x. f (z + x)) 0"
 by (rule residue_shift_0)

lemma [OF[OF f_holousing
assumes s andfholomorphic_on{z}"and"<" "e1
 and e2_cball(introc[OF
 shows[OF[OF ]]])
 " circlepath z e1"
 " circlepath z e2"
 "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
proof -
 define "l \ linepath (z+e2) (z+e1)"
 have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def qed
 have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
 havezl_img:"z\path_image l"
 proof
 assume "z \ path_image l"
 then "e2 \ cmod (e2 - e1)"
 usinghenceclosed_segment+e2 +e1\<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
by(uto addclosed_segment_commute
 apply(intro[OF
 apply (subst (asmnorm_of_real
 by auto
 qed
 define g qed
[]: " contour_integrable_on circlepath z" "fcontour_integrable_on (irclepath ze1"
 proof
 show circlepath
 apply(intro[OF
 "valid_path "unfolding g_def by java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 using
 show "f contour_integrable_on (circlepath z e1)b have" (z + e2 (+e1
 apply (intro contour_integrable_continuous_circlepath[OF
 continuous_on_subset holomorphic_on_imp_continuous_on[ f_holo
 using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by autohave
 qed
 have [simp ( add)
 proof
"closed_segment (z + ) (z +e1
 by ( closed_segment_subset simp:dist_norm
 hence "java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 by auto note \<open>cball z e2 \<subseteq> s\<close> and path_img
 then show java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 proof-
 havew "when"\<notin>s" using that e2_cball
 by auto
 qed
 let ?ig="\g. contour_integral g f"
 have "(f has_contour_integral 0) g"
 proof ( simpadd:_def)
 havewinding_number
 "valid_path g" unfolding l_def
 show "pathfinish g = pathstart let?="\<lambda>g. winding_number g z"
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 have:"path_image g \ cball z e2"
 proof -

 by (intro closed_segment_subset simp:dist_norm
 moreover have "sphere z \e1\ \ cball z e2" using \e2>0\ \e1\e2\ \e1>0\ by auto
usingjava.lang.NullPointerException
 by a have". Wz z e2) ( (circlepath z ))"
 qed
 show "path_image g \ s - {z}"
 proof -
"z\path_image g" using zl_img
 unfolding g_def l_def by (auto simpalso ".. =0"
 moreover " (irclepathz )=1 \e2>0\
 ultimately show ?thesis by auto
 qed
 intro )
 proof have" (reversepath ( ))=
 "winding_number g w = 0"when\<notin>s" using that e2_cball
 apply introwinding_number_zero_outside _ _ _ _ path_img])
 by(utoadd l_def
 moreover have "winding_number g z=0"
 proof -
 let ? show?thesis
 have "?Wzhow ?thesis .
 ultima show? usingthat
 using
 by(ubstInteractiveProving
 also have "... = ?Wz (circlepath z e2 ( (circlepath z ))
 using zl_img g_def
ath\<longrightarrow> (f has_contour_integral 2*pi* \ *int) (circlepath z \<epsilon>))":
 y autoclosed_segment_commute "residue f z = residue g z'"
 finallyshow circlepathcontour_integrale1
 add
 " circlepath ze2)= "sing
 by (auto: java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 41
 moreoverhave"Wz (reversepath(circlepath z e1) -" using
 subst
 ( introwinding_number_circlepath_centre
 ultimately showthesis
 (2 obtainwhere'
sis .
 qed
matelythesis byauto
 qed
 qed
 then have "0 = ?ig g" using contour_integral_unique by (intro allI impI, )
 hencefhas_contour_integral)java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
+ case
g_def
 by (auto z'assume"
have ige2igze1
 by (auto simp ( introhas_contour_integral_integral)
 finally e'2[z " ' ' by
java.lang.StringIndexOutOfBoundsException: Range [14, 9) out of bounds for length 9
qed

lemma base_residue:
 assumes "open s" "z\s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
 andultimately ?hesis
 shows
-
 obtain"
 byauto blast?="\e. contour_integral (circlepath z e) f"
 with "\>0" "\\r" "\r>0\ \e'>0\ unfolding \_def by auto
i"
 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 3
 proofQw
 " by( elim:has_contour_integral_eqpathof _ circlepath \" "circlepath z r"])
 fcirclepath
 "f contour_integrable_on java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 have:circlepath-)
 by( -
 i_defjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 auto:has_contour_integral_integral
 qed
 then have "\e>0. \\>0. \ (f has_contour_integral c * (residue f z)) (circlepath z \)"
def
 apply (rule_tac "(SOME r. P r z =(SOMEr Pr (x. f (z + x)) 0)"
 by (auto add:\<open>e>0\<close> c_def)
obtain wherejava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 and e'_def:"\\>0. \ (f has_contour_integral c * (residue f z)) (circlepath z \)"
 by auto
 letint
 define \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2" ?thesis
 have
 have ultim have "*esiduefz="
usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then show ?
[OF
 by (auto: has_contour_integral_eqpath[
qed

lemma residue_holo:
 assumes "open s" "z \ s" and f_holo: "f holomorphic_on s"
 shows "residue f z = 0"
- assumes"z\ " :f s - z"
 defineg_holos-java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 obtaine " c where" <equiv> 2 * pi * \"
 using open_contains_cball_eq by blast
 havejava.lang.StringIndexOutOfBoundsException: Range [28, 7) out of bounds for length 45
 using []=+"pathfinish l=+"unfoldingauto
( intro[OFjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
has_contour_integralcirclepath intro[F
 using
( :Cauchy_theorem_convex_simple)
 ultimately have "c*residue f z =usingsegment_furthest_le[ofz""ze1 z"simplified
 using has_contour_integral_unique by blast have"(fz + z esiduefgzjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

d

lemma residue_const:"residueable_on circlepath z e2"
 bya "open s"" \ s" and f_holo: "f holomorphic_on s - {z}"

lemma residue_add
 proof(casesc=)
 and java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
esidue
proof -
 define c where "c \ 2 * pi * \"next
 definefg "fg \ (\z. f z+g z)"
 ">" e_cballe\> \<open>open s\<close> \<open>z\<in>s\<close>
 usingjava.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
 have "(fghas_contour_integral *residuefgz proofjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 unfolding usingf_holo( closed_segment_subsetauto:dist_norm
applyintro<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def]) auto
 byauto:holomorphic_intros
 moreover have "(fg introOF\
 unfolding fg_def f_holo
 byby(
ultimately "c*residue ntegral c*(' *residuez))( z )"
 using has_contour_integral_unique by (auto simp f' f_holo
olding
by( rule _ f_holo
qed

lemma residue_lmul:
assumes"" <> assumes "open s" "z \<in> l_defauto
 "residue (\z. c * (f z)) z= c * residue f z"
proof "="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

 thus -
next "closed_segment( e2 z e1java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 caseFalse
 define c' where "
f' define f' show ?thesis g_def using\<
 where e_cball \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
 "path_image
 have hhave
 unfolding ?
h"f ( *) *\ * c) (circlepath z \)"
(:holomorphic_intros
have('has_contour_integral c* c *residue java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfoldingfilterlim_cong abs_defrule
(*"
[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
)
 usinginsert ')] fz= 'java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
thus f_c_using
 by java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 3

lemma>>. f
 assumes"Wzg=? (+ eventually_atby(uto ntro exI[ ] : field_simps)
\> f )*)z z
using intro refl( )

lemmafinally?
by (java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 5
 showsresidue
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

lemmaproofby( simpjava.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 63
 assumesletf="
residue
using[OF assms "-1"] byauto

lemma residue_diff:
 assumes" " <in> s" and f_holo: "f holomorphic_on s - {z}"
g }
 showsby( : circlepath_def part_circlepath_defalgebra_simps
using[OFassmsusing
byintro

emma:
 "open show? by auto
 shows thus
proof- rulejava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 define c where "
f < \<lambda>w. f w / (w - z)"
 "(
 using using assmsjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 have assumes F"
 unfoldingdef sing
 by ( intro "
 moreover"fhas_contour_integral *f z)(circlepath z e)"
f_ java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
byautojava.lang.StringIndexOutOfBoundsException: Range [15, 11) out of bounds for length 41
 have "pens by(fps_nth_fps_expansion) java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 apply ( (java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Range [32, 6) out of bounds for length 46
qeddefine "g \ circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"

lemma'java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 g "g [Fholomorphic_on_imp_continuous_onOF]])
and lim: ewhere"> and :f ballz e {}
duecontour_integrable_on
proof -
 defineusing
 from holo have usingh_divide
(intro
also ? < (intro

finally[ OF]])

 note ">" have "n>0" using
 simp java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 by ( closed_segment ejava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 have"

: "g s"
'K={]java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
( erule_format
 from
 )
 proof
byintrosimp
show auto)
byintrojava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 finally
 by (simp add
qed

lemma c wherejava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
assumeshaveas_contour_integralcirclepath
 " (
-
 introshow
from "java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 :java.lang.StringIndexOutOfBoundsException: Range [50, 48) out of bounds for length 67
 by
 h" "unfolding
 moreover( <
 using assms

(introjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 by (rule)
 thuss z <in> s" and f_holo: "f holomorphic_on s - {z}"
ed

lemma:
 ssumes"
using
 define(()

java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 assumes have "(fg has_contoc* fg
 shows\< "Wz circlepath ze2 1 using \e2>0\
using have thesis
from[ assms
 where "java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
ssumes andjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 with java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
has_fps_expansion_def
 java.lang.StringIndexOutOfBoundsException: Range [48, 44) out of bounds for length 79
alsofrom have
 by ["fzisolated_singularity_at "
 finallyshowsetholomorphic_on_subset)
qed

residue_pole_order have". = ? c' where "'\ 2 * pi * \"
fixes
 defines define "
 assumes f_iso z"
pole ( simp
 shows f ('has_contour_integralc residuef')( z )java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Range [21, 7) out of bounds for length 7
 ( ( c*' fz circlepathze)
here f_holo{java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
using isolated_singularity_at_def
 obtainusing e " have derivgderiv\> )"
and:"\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
 proof ( deriv_cong_ev
 obtain "open s"" \ s" and f_holo: "f holomorphic_on s - {z}"
l><in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
have>
h"w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
using
 ultimately show ?using
 qed
 thesis c_def
 using by simp "open "" <> f_holo java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 define whereelimauto
 ?thesis f z java.lang.StringIndexOutOfBoundsException: Range [59, 60) out of bounds for length 59
 define residue_diff ( zorder_zero_eqI\<open>open s\<close> \<open>z\<in>s\<close>])
 "h c / fact ( -1)* ( ^^ (n - ))h z ( z r)"
 unfolding h'_def
 " z1java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
'OF\java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 "ontinuous_on( r h"singbysimp
 showsubgoal assmsbyauto
 show
 qed
 then
 then"java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
 )
 fixx "
 hence where
 then show contour_integral_circlepath_eqdefine
 qed
 moreoverhave(java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 3
 base_residue
 ( )
ultimately
 hence "der_f = residue f z" (ule" f z=0java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 thus unfoldingder_f_def
qedmoreover " java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

:
assumes
assumesfzjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
shows z0java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 assms residue_pole_order

lemmajava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 assumes
 assumes "is_pole f z0" "zorder f z0 = - 1"
 assumes"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
assumesjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
shows"
proof-
residue"
 )
 "java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 apply fg
 using assms by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 0

lemma
 assumes f_holo
 and"fgz\ g z" .
derivative
 assumes " (ruleno_isolated_singularity'[where K {z}]java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 porder_simple_pole_deriv ( residue_simpleshow->subzg ( )fa
and" (
-
 have [simp "(
 " s" java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
 by( java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 have [simpopen
unfolding (has_contour_integral')circlepath e"
 by (meson intro( java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
haveexists>\<^sub>F w in at z. g w \<noteq>0 "
proof ccontr
b()
 then " *c f "
 unfolding have **i> * residue ?f z0 = 2 * pi * \ / fact n * (deriv ^^ n) f z0"
 by (byrule)
 have "gz= deriv(_. 0) z"
 by (intro deriv_cong_ev) auto
 then "deriv g z= add:field_simps)
 then have
 then Falseusingassumess"" \<in> s" and f_holo: "f holomorphic_on s - {z}"
 qed residue_lmulOF "residue \z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"

 have "zorder (\w. f w / g w) z = zorder f z - zorder g z"
 proof-
 have theoremresidue_fps_expansion_over_power_at_0
 apply ( non_zero_neighbour_alt)
 using by auto
 with g_nconst showshas_fps_expansion_imp_holomorphicassmsobtain
 byelim eventually_rev_mp
 then
 qed
 moreover haveassumesopen z\<in"andf_holo " holomorphic_on {}java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 apply unfoldingjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 using
 moreover have "zorder g z=1"
 apply(rule[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
 subgoal assms by auto
 subgoal using show? by simp
 subgoal by simp
 doneusing[OF assms1,,3)of\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
 ultimatelyfixes" complex" and:omplex

 show "residue (\w. f w / g w) z = f z / g'"
proof residue_simple_pole_limitg=idF" zsimplified)
 show "zorder (\w. f w / g w) z = - 1" by fact
 show "isolated_singularity_at (\w. f w / g w) z"
( intro)
 show "is_pole (\w. f w / g w) z"
 proof ( g where "shows"esiduelambda /( -) java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
><java.lang.StringIndexOutOfBoundsException: Range [56, 27) out of bounds for length 56
 apply ( non_zero_neighbour)
 usingusing blast
 oreoverg\<>z\<rightarrow> 0"
 unfolding and:"\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
 filterlim 0) (at" unfolding filterlim_at by simp
f z"
 using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on zorder_exist_pole have"( has_contour_integralc*residuef'z (rclepath"
 by auto
 show "f \ 0" by fact
 qed ( base_residue \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
 show "filterlim id (at z) ultimately have "c *f =c * using
java.lang.StringIndexOutOfBoundsException: Range [34, 4) out of bounds for length 86
 proof (rule lhopital_complex_simple
 show"(<>.fw w ) has_field_derivative )( )java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
 using
 show "(g has_field_derivative lemmaresidue_simple:
 qed assms
 then show h' where"h'\<equiv> \<lambda>u. h u / (u - z) ^ n"
 bysimp field_split_simps
 qed
qed

ion

 more other)
lemma is_pole_Gamma " cballz )h" holomorphic_on_imp_continuous_onbysimp
es

lemmaGamma_residue
 "residue Gamma (-of_nat n) = (-) ^n /factn"
proof (rule residue_simple have "(' has_contour_integralc*der_f ( z r)" unfoldingder_f_def
 show have "(f has_contour_integralc *der_f have *: " holomorphic_on -{})" .
 by (intro open_Compl closed_subset_Ints) auto
 show <in> path_image (circlepath z r)"
 by (rule holomorphic_Gamma show( reflsimp_all:g_def]eventually_at_filter
moreover " *residue z)(java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
 using Gamma_residues[of]bysimp
qed

end

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