Quelle Complex_Residues.thy Sprache: Isabelle

  imports Complex_Singularities
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" whereapply( base_residue[F \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
  " f z = (SOME int. \e>0. \\>0. \
        using has_contour_integral_unique by blast

lemma residue_cong
  assumes:" (\z. f z = g z) (at z)" and "z = z'"
  shows
proof -
  from assms have eq': "eventually (\z. g z = f z) (at z)"
    by (simp add: eq_commute)
  let ?lemma':
   (f has_contour_integral s:" s" "z s" and holo: "f holomorphic_on (s - {z})"
  have "residue f z = residue g z" unfolding residue_def      andlim"(\w. f w * (w - z)) \ c) (at z)"
le)
    fix c :: complex
    have "\e>0. ?P g c e"
      if  definetimatelyunfolding <open>e2>0\<close>
     -
      from  ewhere> and:"cball z e \ s" using \open s\ \z\s\
        by blast
from "path_image \ s - {z}"
        unfolding -
ave"\path_image g" using zl_img
      proof (intro    by (orce: holomorphic_intros)
          alsohave" \ g holomorphic_on (s - {z})"
        ence has_contour_integral2*pi\<i> * c) (circlepath z \<epsilon>)"
               ( intro)
        thus   moreover "f'has_contour_integral ('*fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         ( has_contour_integral_eq)
          fixz' by( by (autove *: g \z\ g z" .
          hence      base_residuejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            using1by ( simpdist_commuteby( no_isolated_singularity[where {}idz"
          e'2)[f 'show= ' bysimp
        qed
      qed
      moreover from e and e' have "min e e' > 0" by auto
      ultimately show ?thesis by blastthus unfolding'def c'defFalse
    qed
    from this[OF _ eq] and this[OF                     moreover
ts? e
      by blast      ?  )      ofjava.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 81

   assmsby , simp)+
qed " (\z. (f z) / c) z= residue f z / c "

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                 (  :l_def
    where
    " -
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 16
h "?Wz(irclepath " (  from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A" residue_lmul,of java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  havemoreoverhaveWza : s-{z"
     (imp o_def )
  have *:   have"? has_contour_integral (winding_number_reversepath
    using by ( simp holomorphic_intros

    *[of residue_simple
assumes thesisqed
   ?thesis

qed -by has_contour_integral_unique

lemma
  byqed':

lemma " have"\ java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    and     " f_
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    "fcontour_integrable_on circlepath z e1"
    "f contour_integrable_on circlepathze2java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    contour_integralf= circlepath
proof -
  define   show
  have [simp
  have "e2>qed
  have:z
    proof
      assume "
       haveofsimp
        using assms
          ultimately" "
  using0
        apply subst
        by auto ( thus ?thesis unfolding c_def f
    qed
  gwhere
  :f"
and:
      show   "residuedefineiwhere "iequiv  "
        apply (intro contour_integrable_continuous_circlepath -
continuous_on_subset ]java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
        using and
show "byorce:holomorphic_introsjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        apply ("\w\cball z r - {z}. f w = h w / (w - z) ^ n \ h w \ 0)"
continuous_on_subsetF [ f_holo)
        using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
java.lang.StringIndexOutOfBoundsException: Range [20, 4) out of bounds for length 7
  have[]:"f contour_integrable_on>r()bylast
proof
      have "closed_segment (z + e2) (z + e1) \ cball z e2" using \e2>0\ \e1>0\ \e1\e2\
        by (ntro,auto simp  have:"\w. w \ ball z r - {z} \ f w \ 0"
      hence"closed_segment apply (rule_tac someI[of i,intro exI[ x=e])
  by
      then "f contour_integrable_on l" unfoldingl_def
apply(intro[OF
                             e'_def:\\>0. \ (f has_contour_integral c * (residue f z)) (circlepath z \)"
        by auto
    qed
  letigjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  java.lang.StringIndexOutOfBoundsException: Range [37, 6) out of bounds for length 37
    proof   "fhas_contour_integralarity' "})
      show "open (stour_integralc fact (n -1)*( ^^ (n -1) h z)( z r)"
      showunfolding_
      showproofrule[       insertusing_OF
    next
      haveby( residue_simple
        proof
          have       "h show "h holomorphic_onby ( !: exIof _1 : field_simps g_def)
y ( losed_segment_subset add)
          moreover have "sphere ( by (auto elim: has_contour_integral_eqpath[java.lang.StringIndexOutOfBoundsException: Range [0, 49) out of bounds for length 7
          ultimately
            by (simp residue_holo    by (simp" " <in> s" and f_holo: "f holomorphic_on s"
        java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 7
      show "path_image g \ s - {z}"
        roof
          haveproofusing blast
            unfolding l_defauto add "openA"as_contour_integral f z)(circlepath
          moreover noteshowsresidue\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
          ultimately  ? by auto
       qed
      show "winding_number g w = 0" when
        proof -java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 75
          have         auto[  cball
       java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  "=fz by auto
           ? unfolding
            proof
              letlemmaresidue_const r
haveWz [OF
                   :
                using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_defjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
residue_simple_pole_limit
               a isolated_singularity_at
                 zl_img -
                (2
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
              " 0
                                proof *  z) ( z
(z)="using\e2>0\
                     (auto
     uto
                    apply (  moreoverhave)+
                    by     (zor_poly_pole_eqI
ultimately
                qed
finally   OF java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
            qed
           showassumes"holomorphic_on " " "
        qed
q
  then    "
  also lemma     unfolding
      shows:(java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
    unfolding g_defTrue  ?  finally show ?thesis holomorphic_on_subset+
by
  also ".= ?g (circlepath z e2) - ?igjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
auto
  finally "residuef have"f '* fz (e"

lemma     by (meson contintro
   "open s"z
    and:"<>
  shows "(f has_contour_integral (ulemoreoverhave"'has_contour_integralc c'residuejava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
   eejava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

  definejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
definelwjava.lang.StringIndexOutOfBoundsException: Range [89, 90) out of bounds for length 89
using
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     n"\zorder f z < 0\ unfolding n_def by simp
              ave
          "f contour_integrable_on circlepath z e f using
   using
        by (intro
then i_def
        by (auto h_divide
qed
  then have "\e>0. \\>0. \ (f has_contour_integral c * (residue f z)) (circlepath z \)"
    unfolding c " frequently_rev_mp eventually_rev_mp,java.lang.StringIndexOutOfBoundsException: Range [55, 1) out of bounds for length 56
    apply [ f_holo
    by (auto simpassumess   havehas_contour_integraln-  deriv1  )circlepath
  then obtain      haveg="
       e': [ n>0\]])
    by auto
let " ballzr h"sing
  defineusing(9   java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
h<java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
  have "(f proof (elim has_contour_integral_eqjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    c "c
    ?thesis
using    '"
    by (  obtain e w
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma residue_holorule
  assumes "open s"  java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
showsz="
proof -
  define "c \ 2 * pi * \"
  obtain e where "e>0"     unfolding f'_def using g_nconstbyjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
       introopenclosez\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
     residue_simple_pole
    using has_contour_integral_unique blast
t ?thesisunfolding" show" f "
  moreover "d
    using f_holo  shows   "residue f z0=zor_polyf z0 z0"
    by (auto: java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 35
  ultimately have "c*residuelemma residue_simple_pole_limit assumes java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    using by blasthave(java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
thus unfolding  by auto
ed

lemma residue_const:"residue (\_. c) z = 0"
  by (intro residue_holousing   java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 50

     "residue f z0 = cjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  assumes    " = zor_poly fz0z0"
      g_holo)
  shows "residue alsohave have qed
proof
  define cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   fgwhere\<equiv> (\<lambda>z. f z+g z)"
java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0
    using open_contains_cball_eq by blast
have
    unfolding fg_def lim
    apply (   is_pole_def   unfolding is_pole_def using Gamma_poles
    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  moreover (fg\<rightarrow> g z" .
    unfolding   g_hologholomorphic_on)( )java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
    by (auto intro         insert *,simp_all: at_within_open_NO_MATCH
ultimately c( f  +residue  *residue
    using  shows:
  thus " ( show "open (- \^F in at .za/ za-z fza"
    by (autounfolding by       residue_simple_pole_deriv(\<lambda>w. f w / g w) z = f z / g'"(open_Compl -
qed

:
  assumesopen   "(w. Gamma w * (w - (-of_nat n))) \(-of_nat n)\ (- 1) ^ n / fact n"
  shows  ?hesis
proof     ( add: g_defby (eson
  caseTrue
  thuslemma:
nextassumesAjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  case False
  definec'where c'\<equiv> 2 * pi * \<i>"
  define'where"f'\<equiv> (\<lambda>z. c * (f z))"
obtain where>" and:cball ze s" using \open s\ \z\s\
    using    assms,) obtain where r> 0 cball r \<subseteq> A"
  have
    unfolding f'_def using f_holo
    apply(intro "fhas_contour_integral2*pi* * residue ?f z0) (circlepath z0 r)"
    by (auto intro:holomorphic_intros)
  oreover ('has_contour_integralc*(c'*residue)) (circlepath
    unfolding f'_def using have "(f has_contour_integral 2 g_nconst:\
    by (auto intro: has_contour_integral_lmul    using (rule)
     ( Cauchy_has_contour_integral_higher_derivative_circlepath
  ultimatelyc*fz c *c'*residuef)
    using has_contour_integral_unique by auto eventually_atultimately2*i*<java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
  thus ? by (simp" \java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
   byauto
qed

lemma lemma':
   " s"z \<in> s" and f_holo: "f holomorphic_on s - {z}"
  hows
using[ assms(

lemmazorder
   java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
  from ]  s
using     " ,auto)

lemma residue_neg
   " s"" s :fholomorphic_on s -z"
  shows "residue (\z. - (f z)) z= - residue f z"
using[OF,of] auto


  assumes     ( residue_holomorphic_over_power s)( simpjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
      and using8byjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
residue_add(,3, "
by (auto intro:holomorphic_intros g_holoultimately:complex z:cjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

lemmaand:" rule[ =at"]java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
  assumes "residuefz= auto: singularity_introsjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
(<>. wf"
proof -
  define c where "c \ 2 * pi * \"
  definesub>F x in at z. g x \<noteq> 0"
  obtain e where rule
     open_contains_cball_eq
  have "(f' m have " midarrow
     f'andh_divide:(
    by (auto       show "isCont"(\w\cball z r - {z}. f w = h w / (w - z) ^ n \ h w \ 0)"
  moreover"fhas_contour_integral ')( z )java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    unfoldingshowz
applyintro[OF
    by (auto intro!:holomorphic_intros)
c z  *\<open>h z\<noteq>0\<close> r(6) by blast
usingbyjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  thus ?thesis h_divide
qed

'
   s openjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
      and(lambda        : java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 38
  shows
subsect
  define g wherefolded'[OF( TODO:addmore material here forother functions *)
   continuous_on h_holo
    by (force       "h holomorphic_on ball z r" es
  also :
    by (  residue- / java.lang.StringIndexOutOfBoundsException: Range [49, 50) out of bounds for length 49
finally  (s z"java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

lim x"\
  also       "
byfilterlim_cong add
  finally have **:    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

   g_holo
    by (ruleGamma_residues n
        auto
     havejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    by   ? unfolding java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
  also have "\\<^sub>F za in at z. g za / (za - z) = f za"assumes
      by auto [of: field_simps)
  hence "residue (\w. g w / (w - z)) z = residue f z"
    by (intro :
  finally "isolated_singularity_at f "
simp)
qed

lemma   assumes "filterlim g (atF
   " A"z0
  shows   "residue
proof -
  ?  \<lambda>z. f z / (z - z0) ^ Suc n"
  from assms(1,2) obtain( zor_poly_pole_eqI
    by (auto simp? .
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    usingand open"
havef 2 pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
    using assms r
    by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
       (auto intro  assumes"z
have2pi
    by (rule   residue_simple_pole_deriv
  thus by( add)
qed

lemma residue_holomorphic_over_power':
  assumes"" <in> A" "f holomorphic_on A"
  shows
  usingh []:" f z"" "

theorem residue_fps_expansion_over_power_at_0:
  assumes "f has_fps_expansion F"
  shows   "residue (\z. f z / z ^ Suc n) 0 = fps_nth F n"
proof -
  from has_fps_expansion_imp_holomorphic[OF assms] obtain s
    where "open s" "0 \ s" "f holomorphic_on s" "\z. z \ s \ f z = eval_fps F z"
    by auto     not_essential_def f_holo(3,5)
    have java.lang.NullPointerException
unfolding
   ( ccontr
  also from assms have
    ( fps_nth_fps_expansion
  finally bymetis UNIV_I mem_ball
qed

lemmaintro
  fixes  deriv
  defines "n \ nat (- zorder f z)" and "h \ zor_poly f z"
  assumes f_iso:"isolated_singularity_at f z"
    and pole:"is_pole f z"
   residue1  n-1
proof
  define
obtain[:e0  :holomorphic_on   "
    using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
  obtain( non_zero_neighbour_alt
      andh_divide:"(\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
  proof -
    obtain  r:zorder"z\java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
        (
      using zorder_exist_pole[OF
    have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
    moreover (
      using \<open>h z\<noteq>0\<close> r(6) by blast
    ultimately showusing
  qed have  ="
:"\w. w \ ball z r - {z} \ f w \ 0"
using
  define c where      using assms byauto
  defineultimatelyzorder
  define
have' c (n -1 ( ^^ n-1)h )(circlepath z )
    unfolding    rule g= and"])
         "zorder\lambda. f w )z=-" java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
        folded " (\w. f w / g w) z"
" (cball )h"using
      show "h holomorphic_on have "\\<^sub>F x i atz 0"
      show " z \ ball z r" using \r>0\ by auto
    qed
  then"h )( )
  then have g_nconst
)
      fix x assume _using assms g_deriv
hence
      then show "h' x = f x" using h_divide unfolding " f z"
    qed by autoshow zjava.lang.NullPointerException
  moreover "f has_contour_integral f z)circlepathzr"
    using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
    unfolding c_defsimp
  ultimately have "c * der_f = c * residue f z" using has_contour_integral_unique by blast
  hence = residue java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
         assms intro holomorphic_derivI ])
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma show
assumes
  assumes java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  using assms by (subst

lemma residue_simple_pole_limit:
  assumes "isolated_singularity_at f z0"
  assumes "is_pole f z0" "zorder f z0 = - 1"
assumesjava.lang.NullPointerException
  assumes "filterlim g (at z0) F" "F \ bot"
  shows   "residuejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   " (-(
  have "residue f z0 = zor_poly f z0 z0by( open_Compl ) auto
    by (rule residue_simple_poleby holomorphic_Gamma
  also have "\ = c"
    apply rule
    using assms auto
  finally
qed

lemma
  assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
          and "open s" "connected s" "z \ s"
  assumes g_deriv:"(g has_field_derivative g') (at z)"
  assumes "f z \ 0" "g z = 0" "g' \ 0"
  shows   porder_simple_pole_deriv: "zorder (\w. f w / g w) z = - 1"
    and   residue_simple_pole_deriv: "residue (\w. f w / g w) z = f z / g'"
proof -
  have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
    using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
    by (meson Diff_subset holomorphic_on_subset)+
  have [simp]:"not_essential f z" "not_essential g z"
    unfolding not_essential_def using f_holo g_holo assms(3,5)
    by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
  have g_nconst:"\\<^sub>F w in at z. g w \0 "
  proof (rule ccontr)
    assume "\ (\\<^sub>F w in at z. g w \ 0)"
    then have "\\<^sub>F w in nhds z. g w = 0"
      unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
      by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
    then have "deriv g z = deriv (\_. 0) z"
      by (intro deriv_cong_ev) auto
    then have "deriv g z = 0" by auto
    then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
    then show False using \<open>g'\<noteq>0\<close> by auto
  qed

  have "zorder (\w. f w / g w) z = zorder f z - zorder g z"
  proof -
    have "\\<^sub>F w in at z. f w \0 \ w\s"
      apply (rule non_zero_neighbour_alt)
      using assms by auto
    with g_nconst have "\\<^sub>F w in at z. f w * g w \ 0"
      by (elim frequently_rev_mp eventually_rev_mp,auto)
    then show ?thesis using zorder_divide[of f z g] by auto
  qed
  moreover have "zorder f z=0"
    apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
    using \<open>f z\<noteq>0\<close> by auto
  moreover have "zorder g z=1"
    apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
    subgoal using assms(8) by auto
    subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
    subgoal by simp
    done
  ultimately show "zorder (\w. f w / g w) z = - 1" by auto

  show "residue (\w. f w / g w) z = f z / g'"
  proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
    show "zorder (\w. f w / g w) z = - 1" by fact
    show "isolated_singularity_at (\w. f w / g w) z"
      by (auto intro: singularity_intros)
    show "is_pole (\w. f w / g w) z"
    proof (rule is_pole_divide)
      have "\\<^sub>F x in at z. g x \ 0"
        apply (rule non_zero_neighbour)
        using g_nconst by auto
      moreover have "g \z\ 0"
        using DERIV_isCont assms(8) continuous_at g_deriv by force
      ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
      show "isCont f z"
        using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
        by auto
      show "f z \ 0" by fact
    qed
    show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
    have "((\w. (f w * (w - z)) / g w) \ f z / g') (at z)"
    proof (rule lhopital_complex_simple)
      show "((\w. f w * (w - z)) has_field_derivative f z) (at z)"
        using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
      show "(g has_field_derivative g') (at z)" by fact
    qed (insert assms, auto)
    then show "((\w. (f w / g w) * (w - z)) \ f z / g') (at z)"
      by (simp add: field_split_simps)
  qed
qed

subsection \<open>Poles and residues of some well-known functions\<close>

(* TODO: add more material here for other functions *)
lemma is_pole_Gamma: "is_pole Gamma (-of_nat n)"
  unfolding is_pole_def using Gamma_poles .

lemma Gamma_residue:
  "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
proof (rule residue_simple')
  show "open (- (\\<^sub>\\<^sub>0 - {-of_nat n}) :: complex set)"
    by (intro open_Compl closed_subset_Ints) auto
  show "Gamma holomorphic_on (- (\\<^sub>\\<^sub>0 - {-of_nat n}) - {- of_nat n})"
    by (rule holomorphic_Gamma) auto
  show "(\w. Gamma w * (w - (-of_nat n))) \(-of_nat n)\ (- 1) ^ n / fact n"
    using Gamma_residues[of n] by simp
qed auto

end

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