diff --git a/src/HOL/Complex_Analysis/Residue_Theorem.thy b/src/HOL/Complex_Analysis/Residue_Theorem.thy
--- a/src/HOL/Complex_Analysis/Residue_Theorem.thy
+++ b/src/HOL/Complex_Analysis/Residue_Theorem.thy
@@ -1,862 +1,986 @@
 section \<open>The Residue Theorem, the Argument Principle and Rouch\'{e}'s Theorem\<close>
 theory Residue_Theorem
-  imports Complex_Residues
+  imports Complex_Residues "HOL-Library.Landau_Symbols"
 begin
 
 subsection \<open>Cauchy's residue theorem\<close>
 
 lemma get_integrable_path:
   assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
   obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
     "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
 proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
   case 1
   obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
     using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
       valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
   moreover have "f contour_integrable_on g"
     using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
       \<open>f holomorphic_on s - {}\<close>
     by auto
   ultimately show ?case using "1"(1)[of g] by auto
 next
   case idt:(2 p pts)
   obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
     using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
       \<open>a \<in> s - insert p pts\<close>
     by auto
   define a' where "a' \<equiv> a+e/2"
   have "a'\<in>s-{p} -pts"  using e[rule_format,of "a+e/2"] \<open>e>0\<close>
     by (auto simp add:dist_complex_def a'_def)
   then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
     "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
     using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
     by (metis Diff_insert2 open_delete)
   define g where "g \<equiv> linepath a a' +++ g'"
   have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
   moreover have "pathstart g = a" and  "pathfinish g = b" unfolding g_def by auto
   moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
     proof (rule subset_path_image_join)
       have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
         by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
       then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
         by auto
     next
       show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
     qed
   moreover have "f contour_integrable_on g"
     proof -
       have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
         by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
       then have "continuous_on (closed_segment a a') f"
         using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
         apply (elim continuous_on_subset)
         by auto
       then have "f contour_integrable_on linepath a a'"
         using contour_integrable_continuous_linepath by auto
       then show ?thesis unfolding g_def
         apply (rule contour_integrable_joinI)
         by (auto simp add: \<open>e>0\<close>)
     qed
   ultimately show ?case using idt.prems(1)[of g] by auto
 qed
 
 lemma Cauchy_theorem_aux:
   assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
           "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
           "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
           "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
   shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
     using assms
 proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
   case 1
   then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
 next
   case (2 p pts)
   note fin[simp] = \<open>finite (insert p pts)\<close>
     and connected = \<open>connected (s - insert p pts)\<close>
     and valid[simp] = \<open>valid_path g\<close>
     and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
     and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
     and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
     and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
     and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
   have "h p>0" and "p\<in>s"
     and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
     using h \<open>insert p pts \<subseteq> s\<close> by auto
   obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
       "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
     proof -
       have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
         by (simp add: \<open>p \<in> s\<close> dist_norm)
       then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
         by fastforce
       moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
       ultimately show ?thesis
         using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
         by blast
     qed
   obtain n::int where "n=winding_number g p"
     using integer_winding_number[OF _ g_loop,of p] valid path_img
     by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
   define p_circ where "p_circ \<equiv> circlepath p (h p)"
   define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
   define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
   define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
   have n_circ:"valid_path (n_circ k)"
       "winding_number (n_circ k) p = k"
       "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
       "path_image (n_circ k) =  (if k=0 then {p + h p} else sphere p (h p))"
       "p \<notin> path_image (n_circ k)"
       "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
       "f contour_integrable_on (n_circ k)"
       "contour_integral (n_circ k) f = k *  contour_integral p_circ f"
       for k
     proof (induct k)
       case 0
       show "valid_path (n_circ 0)"
         and "path_image (n_circ 0) =  (if 0=0 then {p + h p} else sphere p (h p))"
         and "winding_number (n_circ 0) p = of_nat 0"
         and "pathstart (n_circ 0) = p + h p"
         and "pathfinish (n_circ 0) = p + h p"
         and "p \<notin> path_image (n_circ 0)"
         unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
         by (auto simp add: dist_norm)
       show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
         unfolding n_circ_def p_circ_pt_def
         apply (auto intro!:winding_number_trivial)
         by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
       show "f contour_integrable_on (n_circ 0)"
         unfolding n_circ_def p_circ_pt_def
         by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
       show "contour_integral (n_circ 0) f = of_nat 0  *  contour_integral p_circ f"
         unfolding n_circ_def p_circ_pt_def by auto
     next
       case (Suc k)
       have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
       have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
         using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
       have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
         proof -
           have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
           then show ?thesis using h_p pcirc(1) by auto
         qed
       have pcirc_integrable:"f contour_integrable_on p_circ"
         by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
           contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
           holomorphic_on_subset[OF holo])
       show "valid_path (n_circ (Suc k))"
         using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
       show "path_image (n_circ (Suc k))
           = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
         proof -
           have "path_image p_circ = sphere p (h p)"
             unfolding p_circ_def using \<open>0 < h p\<close> by auto
           then show ?thesis unfolding n_Suc  using Suc.hyps(5)  \<open>h p>0\<close>
             by (auto simp add:  path_image_join[OF pcirc(3)]  dist_norm)
         qed
       then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
       show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
         proof -
           have "winding_number p_circ p = 1"
             by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
           moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
           then have "winding_number (p_circ +++ n_circ k) p
               = winding_number p_circ p + winding_number (n_circ k) p"
             using  valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
             apply (intro winding_number_join)
             by auto
           ultimately show ?thesis using Suc(2) unfolding n_circ_def
             by auto
         qed
       show "pathstart (n_circ (Suc k)) = p + h p"
         by (simp add: n_circ_def p_circ_def)
       show "pathfinish (n_circ (Suc k)) = p + h p"
         using Suc(4) unfolding n_circ_def by auto
       show "winding_number (n_circ (Suc k)) p'=0 \<and>  p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
         proof -
           have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
           moreover have "p' \<notin> path_image (n_circ k)"
             using Suc.hyps(7) that by blast
           moreover have "winding_number p_circ p' = 0"
             proof -
               have "path_image p_circ \<subseteq> cball p (h p)"
                 using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
               moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
               ultimately show ?thesis unfolding p_circ_def
                 apply (intro winding_number_zero_outside)
                 by auto
             qed
           ultimately show ?thesis
             unfolding n_Suc
             apply (subst winding_number_join)
             by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
         qed
       show "f contour_integrable_on (n_circ (Suc k))"
         unfolding n_Suc
         by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
       show "contour_integral (n_circ (Suc k)) f = (Suc k) *  contour_integral p_circ f"
         unfolding n_Suc
         by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
           Suc(9) algebra_simps)
     qed
   have cp[simp]:"pathstart cp = p + h p"  "pathfinish cp = p + h p"
          "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
          "winding_number cp p = - n"
          "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
          "f contour_integrable_on cp"
          "contour_integral cp f = - n * contour_integral p_circ f"
     proof -
       show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
         using n_circ unfolding cp_def by auto
     next
       have "sphere p (h p) \<subseteq>  s - insert p pts"
         using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
       moreover  have "p + complex_of_real (h p) \<in> s - insert p pts"
         using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
       ultimately show "path_image cp \<subseteq>  s - insert p pts" unfolding cp_def
         using n_circ(5)  by auto
     next
       show "winding_number cp p = - n"
         unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
         by (auto simp: valid_path_imp_path)
     next
       show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
         unfolding cp_def
         apply (auto)
         apply (subst winding_number_reversepath)
         by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
     next
       show "f contour_integrable_on cp" unfolding cp_def
         using contour_integrable_reversepath_eq n_circ(1,8) by auto
     next
       show "contour_integral cp f = - n * contour_integral p_circ f"
         unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
         by auto
     qed
   define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
   have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
     proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
       show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
       show "open (s - {p})" using \<open>open s\<close> by auto
       show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close>  by blast
       show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
       show "valid_path g'"
         unfolding g'_def cp_def using n_circ valid pg g_loop
         by (auto intro!:valid_path_join )
       show "pathfinish g' = pathstart g'"
         unfolding g'_def cp_def using pg(2) by simp
       show "path_image g' \<subseteq> s - {p} - pts"
         proof -
           define s' where "s' \<equiv> s - {p} - pts"
           have s':"s' = s-insert p pts " unfolding s'_def by auto
           then show ?thesis using path_img pg(4) cp(4)
             unfolding g'_def
             apply (fold s'_def s')
             apply (intro subset_path_image_join)
             by auto
         qed
       note path_join_imp[simp]
       show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
         proof clarify
           fix z assume z:"z\<notin>s - {p}"
           have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
               + winding_number (pg +++ cp +++ (reversepath pg)) z"
             proof (rule winding_number_join)
               show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
               show "z \<notin> path_image g" using z path_img by auto
               show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
                 by (simp add: valid_path_imp_path)
             next
               have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
                 using pg(4) cp(4) by (auto simp:subset_path_image_join)
               then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
             next
               show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
             qed
           also have "... = winding_number g z + (winding_number pg z
               + winding_number (cp +++ (reversepath pg)) z)"
             proof (subst add_left_cancel,rule winding_number_join)
               show "path pg" and "path (cp +++ reversepath pg)"
                and "pathfinish pg = pathstart (cp +++ reversepath pg)"
                 by (auto simp add: valid_path_imp_path)
               show "z \<notin> path_image pg" using pg(4) z by blast
               show "z \<notin> path_image (cp +++ reversepath pg)" using z
                 by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
                   not_in_path_image_join path_image_reversepath singletonD)
             qed
           also have "... = winding_number g z + (winding_number pg z
               + (winding_number cp z + winding_number (reversepath pg) z))"
             apply (auto intro!:winding_number_join simp: valid_path_imp_path)
             apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
             by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
           also have "... = winding_number g z + winding_number cp z"
             apply (subst winding_number_reversepath)
             apply (auto simp: valid_path_imp_path)
             by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
           finally have "winding_number g' z = winding_number g z + winding_number cp z"
             unfolding g'_def .
           moreover have "winding_number g z + winding_number cp z = 0"
             using winding z \<open>n=winding_number g p\<close> by auto
           ultimately show "winding_number g' z = 0" unfolding g'_def by auto
         qed
       show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
         using h by fastforce
     qed
   moreover have "contour_integral g' f = contour_integral g f
       - winding_number g p * contour_integral p_circ f"
     proof -
       have "contour_integral g' f =  contour_integral g f
         + contour_integral (pg +++ cp +++ reversepath pg) f"
         unfolding g'_def
         apply (subst contour_integral_join)
         by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
           intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
           contour_integrable_reversepath)
       also have "... = contour_integral g f + contour_integral pg f
           + contour_integral (cp +++ reversepath pg) f"
         apply (subst contour_integral_join)
         by (auto simp add:contour_integrable_reversepath)
       also have "... = contour_integral g f + contour_integral pg f
           + contour_integral cp f + contour_integral (reversepath pg) f"
         apply (subst contour_integral_join)
         by (auto simp add:contour_integrable_reversepath)
       also have "... = contour_integral g f + contour_integral cp f"
         using contour_integral_reversepath
         by (auto simp add:contour_integrable_reversepath)
       also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
         using \<open>n=winding_number g p\<close> by auto
       finally show ?thesis .
     qed
   moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
     proof -
       have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
         using "2.prems"(8) that
         apply blast
         apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
         by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
       have "winding_number g' p' = winding_number g p'
           + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
         apply (subst winding_number_join)
         apply (simp_all add: valid_path_imp_path)
         apply (intro not_in_path_image_join)
         by auto
       also have "... = winding_number g p' + winding_number pg p'
           + winding_number (cp +++ reversepath pg) p'"
         apply (subst winding_number_join)
         apply (simp_all add: valid_path_imp_path)
         apply (intro not_in_path_image_join)
         by auto
       also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
           + winding_number (reversepath pg) p'"
         apply (subst winding_number_join)
         by (simp_all add: valid_path_imp_path)
       also have "... = winding_number g p' + winding_number cp p'"
         apply (subst winding_number_reversepath)
         by (simp_all add: valid_path_imp_path)
       also have "... = winding_number g p'" using that by auto
       finally show ?thesis .
     qed
   ultimately show ?case unfolding p_circ_def
     apply (subst (asm) sum.cong[OF refl,
         of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
     by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
 qed
 
 lemma Cauchy_theorem_singularities:
   assumes "open s" "connected s" "finite pts" and
           holo:"f holomorphic_on s-pts" and
           "valid_path g" and
           loop:"pathfinish g = pathstart g" and
           "path_image g \<subseteq> s-pts" and
           homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0" and
           avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
   shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
     (is "?L=?R")
 proof -
   define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
   define pts1 where "pts1 \<equiv> pts \<inter> s"
   define pts2 where "pts2 \<equiv> pts - pts1"
   have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
     unfolding pts1_def pts2_def by auto
   have "contour_integral g f =  (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
     proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
       have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
       then show "connected (s - pts1)"
         using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
     next
       show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
       show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
       show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
       show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
         by (simp add: avoid pts1_def)
     qed
   moreover have "sum circ pts2=0"
     proof -
       have "winding_number g p=0" when "p\<in>pts2" for p
         using  \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
       thus ?thesis unfolding circ_def
         apply (intro sum.neutral)
         by auto
     qed
   moreover have "?R=sum circ pts1 + sum circ pts2"
     unfolding circ_def
     using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
     by blast
   ultimately show ?thesis
     apply (fold circ_def)
     by auto
 qed
 
 theorem Residue_theorem:
   fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
     and g::"real \<Rightarrow> complex"
   assumes "open s" "connected s" "finite pts" and
           holo:"f holomorphic_on s-pts" and
           "valid_path g" and
           loop:"pathfinish g = pathstart g" and
           "path_image g \<subseteq> s-pts" and
           homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
   shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
 proof -
   define c where "c \<equiv>  2 * pi * \<i>"
   obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
     using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
   have "contour_integral g f
       = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
     using Cauchy_theorem_singularities[OF assms avoid] .
   also have "... = (\<Sum>p\<in>pts.  c * winding_number g p * residue f p)"
     proof (intro sum.cong)
       show "pts = pts" by simp
     next
       fix x assume "x \<in> pts"
       show "winding_number g x * contour_integral (circlepath x (h x)) f
           = c * winding_number g x * residue f x"
         proof (cases "x\<in>s")
           case False
           then have "winding_number g x=0" using homo by auto
           thus ?thesis by auto
         next
           case True
           have "contour_integral (circlepath x (h x)) f = c* residue f x"
             using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
             apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
             by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
           then show ?thesis by auto
         qed
     qed
   also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
     by (simp add: sum_distrib_left algebra_simps)
   finally show ?thesis unfolding c_def .
 qed
 
 subsection \<open>The argument principle\<close>
 
 theorem argument_principle:
   fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
   defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
   assumes "open s" and
           "connected s" and
           f_holo:"f holomorphic_on s-poles" and
           h_holo:"h holomorphic_on s" and
           "valid_path g" and
           loop:"pathfinish g = pathstart g" and
           path_img:"path_image g \<subseteq> s - pz" and
           homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
           finite:"finite pz" and
           poles:"\<forall>p\<in>poles. is_pole f p"
   shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
           (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
     (is "?L=?R")
 proof -
   define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
   define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
   define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
   define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
 
   have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
   proof -
     obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
       using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
     have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
     proof -
       define po where "po \<equiv> zorder f p"
       define pp where "pp \<equiv> zor_poly f p"
       define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
       define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
       obtain r where "pp p\<noteq>0" "r>0" and
           "r<e1" and
           pp_holo:"pp holomorphic_on cball p r" and
           pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
       proof -
         have "isolated_singularity_at f p"
         proof -
           have "f holomorphic_on ball p e1 - {p}"
             apply (intro holomorphic_on_subset[OF f_holo])
             using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
           then show ?thesis unfolding isolated_singularity_at_def
             using \<open>e1>0\<close> analytic_on_open open_delete by blast
         qed
         moreover have "not_essential f p"
         proof (cases "is_pole f p")
           case True
           then show ?thesis unfolding not_essential_def by auto
         next
           case False
           then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
           moreover have "open (s-poles)"
             using \<open>open s\<close>
             apply (elim open_Diff)
             apply (rule finite_imp_closed)
             using finite unfolding pz_def by simp
           ultimately have "isCont f p"
             using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
             by auto
           then show ?thesis unfolding isCont_def not_essential_def by auto
         qed
         moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
         proof (rule ccontr)
           assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
           then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
           then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
             unfolding eventually_at by (auto simp add:dist_commute)
           then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
           moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
           ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
           then have "infinite pz"
             unfolding pz_def infinite_super by auto
           then show False using \<open>finite pz\<close> by auto
         qed
         ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
                   "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
           using zorder_exist[of f p,folded po_def pp_def] by auto
         define r1 where "r1=min r e1 / 2"
         have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
         moreover have "r1>0" "pp holomorphic_on cball p r1"
                   "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
           unfolding r1_def using \<open>e1>0\<close> r by auto
         ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
       qed
 
       define e2 where "e2 \<equiv> r/2"
       have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
       define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
       define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
       have "((\<lambda>w.  prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
       proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
         have "ball p r \<subseteq> s"
           using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
         then have "cball p e2 \<subseteq> s"
           using \<open>r>0\<close> unfolding e2_def by auto
         then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
           using h_holo by (auto intro!: holomorphic_intros)
         then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
           using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
           unfolding prin_def by (auto simp add: mult.assoc)
         have "anal holomorphic_on ball p r" unfolding anal_def
           using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
           by (auto intro!: holomorphic_intros)
         then show "(anal has_contour_integral 0) (circlepath p e2)"
           using e2_def \<open>r>0\<close>
           by (auto elim!: Cauchy_theorem_disc_simple)
       qed
       then have "cont ff' p e2" unfolding cont_def po_def
       proof (elim has_contour_integral_eq)
         fix w assume "w \<in> path_image (circlepath p e2)"
         then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
         define wp where "wp \<equiv> w-p"
         have "wp\<noteq>0" and "pp w \<noteq>0"
           unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
         moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
         proof (rule DERIV_imp_deriv)
           have "(pp has_field_derivative (deriv pp w)) (at w)"
             using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
             by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
           then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
                   + deriv pp w * (w - p) powr of_int po) (at w)"
             unfolding f'_def using \<open>w\<noteq>p\<close>
             by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
         qed
         ultimately show "prin w + anal w = ff' w"
           unfolding ff'_def prin_def anal_def
           apply simp
           apply (unfold f'_def)
           apply (fold wp_def)
           apply (auto simp add:field_simps)
           by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
       qed
       then have "cont ff p e2" unfolding cont_def
       proof (elim has_contour_integral_eq)
         fix w assume "w \<in> path_image (circlepath p e2)"
         then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
         have "deriv f' w =  deriv f w"
         proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
           show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
             by (auto intro!: holomorphic_intros)
         next
           have "ball p e1 - {p} \<subseteq> s - poles"
             using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
             by auto
           then have "ball p r - {p} \<subseteq> s - poles"
             apply (elim dual_order.trans)
             using \<open>r<e1\<close> by auto
           then show "f holomorphic_on ball p r - {p}" using f_holo
             by auto
         next
           show "open (ball p r - {p})" by auto
           show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
         next
           fix x assume "x \<in> ball p r - {p}"
           then show "f' x = f x"
             using pp_po unfolding f'_def by auto
         qed
         moreover have " f' w  =  f w "
           using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
           unfolding f'_def by auto
         ultimately show "ff' w = ff w"
           unfolding ff'_def ff_def by simp
       qed
       moreover have "cball p e2 \<subseteq> ball p e1"
         using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
       ultimately show ?thesis using \<open>e2>0\<close> by auto
     qed
     then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
       by auto
     define e4 where "e4 \<equiv> if p\<in>pz then e2 else  e1"
     have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
     moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
     moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
       by (auto simp add: e2 e4_def)
     ultimately show ?thesis by auto
   qed
   then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
       \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
     by metis
   define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
   define w where "w \<equiv> \<lambda>p. winding_number g p"
   have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
   proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
         path_img homo])
     have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
     then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
       by (auto intro!: holomorphic_intros simp add:pz_def)
   next
     show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
       using get_e using avoid_def by blast
   qed
   also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
   proof (rule sum.cong[of pz pz,simplified])
     fix p assume "p \<in> pz"
     show "w p * ci p = c * w p * h p * (zorder f p)"
     proof (cases "p\<in>s")
       assume "p \<in> s"
       have "ci p = c * h p * (zorder f p)" unfolding ci_def
         apply (rule contour_integral_unique)
         using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
       thus ?thesis by auto
     next
       assume "p\<notin>s"
       then have "w p=0" using homo unfolding w_def by auto
       then show ?thesis by auto
     qed
   qed
   also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
     unfolding sum_distrib_left by (simp add:algebra_simps)
   finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
   then show ?thesis unfolding ff_def c_def w_def by simp
 qed
 
+
+subsection \<open>Coefficient asymptotics for generating functions\<close>
+
+text \<open>
+  For a formal power series that has a meromorphic continuation on some disc in the
+  context plane, we can use the Residue Theorem to extract precise asymptotic information
+  from the residues at the poles. This can be used to derive the asymptotic behaviour
+  of the coefficients (\<open>a\<^sub>n \<sim> \<dots>\<close>). If the function is meromorphic on the entire 
+  complex plane, one can even derive a full asymptotic expansion.
+
+  We will first show the relationship between the coefficients and the sum over the residues
+  with an explicit remainder term (the contour integral along the circle used in the
+  Residue theorem).
+\<close>
+theorem
+  fixes f :: "complex \<Rightarrow> complex" and n :: nat and r :: real
+  defines "g \<equiv> (\<lambda>w. f w / w ^ Suc n)" and "\<gamma> \<equiv> circlepath 0 r"
+  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
+  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
+  shows   fps_coeff_conv_residues:
+            "(deriv ^^ n) f 0 / fact n = 
+                contour_integral \<gamma> g / (2 * pi * \<i>) - (\<Sum>z\<in>S. residue g z)" (is ?thesis1)
+    and   fps_coeff_residues_bound:
+            "(\<And>z. norm z = r \<Longrightarrow> z \<notin> k \<Longrightarrow> norm (f z) \<le> C) \<Longrightarrow> C \<ge> 0 \<Longrightarrow> finite k \<Longrightarrow>
+               norm ((deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z)) \<le> C / r ^ n"
+proof -
+  have holo: "g holomorphic_on A - insert 0 S"
+    unfolding g_def using assms by (auto intro!: holomorphic_intros)
+  have "contour_integral \<gamma> g = 2 * pi * \<i> * (\<Sum>z\<in>insert 0 S. winding_number \<gamma> z * residue g z)"
+  proof (rule Residue_theorem)
+    show "g holomorphic_on A - insert 0 S" by fact
+    from assms show "\<forall>z. z \<notin> A \<longrightarrow> winding_number \<gamma> z = 0"
+      unfolding \<gamma>_def by (intro allI impI winding_number_zero_outside[of _ "cball 0 r"]) auto
+  qed (insert assms, auto simp: \<gamma>_def)
+  also have "winding_number \<gamma> z = 1" if "z \<in> insert 0 S" for z
+    unfolding \<gamma>_def using assms that by (intro winding_number_circlepath) auto
+  hence "(\<Sum>z\<in>insert 0 S. winding_number \<gamma> z * residue g z) = (\<Sum>z\<in>insert 0 S. residue g z)"
+    by (intro sum.cong) simp_all
+  also have "\<dots> = residue g 0 + (\<Sum>z\<in>S. residue g z)" 
+    using \<open>0 \<notin> S\<close> and \<open>finite S\<close> by (subst sum.insert) auto
+  also from \<open>r > 0\<close> have "0 \<in> cball 0 r" by simp
+  with assms have "0 \<in> A - S" by blast
+  with assms have "residue g 0 = (deriv ^^ n) f 0 / fact n"
+    unfolding g_def by (subst residue_holomorphic_over_power'[of "A - S"])
+                       (auto simp: finite_imp_closed)
+  finally show ?thesis1
+    by (simp add: field_simps)
+
+  assume C: "\<And>z. norm z = r \<Longrightarrow> z \<notin> k \<Longrightarrow> norm (f z) \<le> C" "C \<ge> 0" and k: "finite k"
+  have "(deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z) = contour_integral \<gamma> g / (2 * pi * \<i>)"
+    using \<open>?thesis1\<close> by (simp add: algebra_simps)
+  also have "norm \<dots> = norm (contour_integral \<gamma> g) / (2 * pi)"
+    by (simp add: norm_divide norm_mult)
+  also have "norm (contour_integral \<gamma> g) \<le> C / r ^ Suc n * (2 * pi * r)"
+  proof (rule has_contour_integral_bound_circlepath_strong)
+    from \<open>open A\<close> and \<open>finite S\<close> have "open (A - insert 0 S)"
+      by (blast intro: finite_imp_closed)
+    with assms show "(g has_contour_integral contour_integral \<gamma> g) (circlepath 0 r)"
+      unfolding \<gamma>_def
+      by (intro has_contour_integral_integral contour_integrable_holomorphic_simple [OF holo]) auto
+  next
+    fix z assume z: "norm (z - 0) = r" "z \<notin> k"
+    hence "norm (g z) = norm (f z) / r ^ Suc n"
+      by (simp add: norm_divide g_def norm_mult norm_power)
+    also have "\<dots> \<le> C / r ^ Suc n" 
+      using k and \<open>r > 0\<close> and z by (intro divide_right_mono C zero_le_power) auto
+    finally show "norm (g z) \<le> C / r ^ Suc n" .
+  qed (insert C(2) k \<open>r > 0\<close>, auto)
+  also from \<open>r > 0\<close> have "C / r ^ Suc n * (2 * pi * r) / (2 * pi) = C / r ^ n"
+    by simp
+  finally show "norm ((deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z)) \<le> \<dots>"
+    by - (simp_all add: divide_right_mono)
+qed
+
+text \<open>
+  Since the circle is fixed, we can get an upper bound on the values of the generating
+  function on the circle and therefore show that the integral is $O(r^{-n})$.
+\<close>
+corollary fps_coeff_residues_bigo:
+  fixes f :: "complex \<Rightarrow> complex" and r :: real
+  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
+  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
+  assumes g: "eventually (\<lambda>n. g n = -(\<Sum>z\<in>S. residue (\<lambda>z. f z / z ^ Suc n) z)) sequentially"
+                (is "eventually (\<lambda>n. _ = -?g' n) _")
+  shows   "(\<lambda>n. (deriv ^^ n) f 0 / fact n - g n) \<in> O(\<lambda>n. 1 / r ^ n)" (is "(\<lambda>n. ?c n - _) \<in> O(_)")
+proof -
+  from assms have "compact (f ` sphere 0 r)"
+    by (intro compact_continuous_image holomorphic_on_imp_continuous_on
+              holomorphic_on_subset[OF \<open>f holomorphic_on A - S\<close>]) auto
+  hence "bounded (f ` sphere 0 r)" by (rule compact_imp_bounded)
+  then obtain C where C: "\<And>z. z \<in> sphere 0 r \<Longrightarrow> norm (f z) \<le> C"
+    by (auto simp: bounded_iff sphere_def)
+  have "0 \<le> norm (f (of_real r))" by simp
+  also from C[of "of_real r"] and \<open>r > 0\<close> have "\<dots> \<le> C" by simp
+  finally have C_nonneg: "C \<ge> 0" .
+
+  have "(\<lambda>n. ?c n + ?g' n) \<in> O(\<lambda>n. of_real (1 / r ^ n))"
+  proof (intro bigoI[of _ C] always_eventually allI )
+    fix n :: nat
+    from assms and C and C_nonneg have "norm (?c n + ?g' n) \<le> C / r ^ n"
+      by (intro fps_coeff_residues_bound[where A = A and k = "{}"]) auto
+    also have "\<dots> = C * norm (complex_of_real (1 / r ^ n))" 
+      using \<open>r > 0\<close> by (simp add: norm_divide norm_power)
+    finally show "norm (?c n + ?g' n) \<le> \<dots>" .
+  qed
+  also have "?this \<longleftrightarrow> (\<lambda>n. ?c n - g n) \<in> O(\<lambda>n. of_real (1 / r ^ n))"
+    by (intro landau_o.big.in_cong eventually_mono[OF g]) simp_all
+  finally show ?thesis .
+qed
+
+corollary fps_coeff_residues_bigo':
+  fixes f :: "complex \<Rightarrow> complex" and r :: real
+  assumes exp: "f has_fps_expansion F"
+  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
+  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
+  assumes "eventually (\<lambda>n. g n = -(\<Sum>z\<in>S. residue (\<lambda>z. f z / z ^ Suc n) z)) sequentially"
+             (is "eventually (\<lambda>n. _ = -?g' n) _")
+  shows   "(\<lambda>n. fps_nth F n - g n) \<in> O(\<lambda>n. 1 / r ^ n)" (is "(\<lambda>n. ?c n - _) \<in> O(_)")
+proof -
+  have "fps_nth F = (\<lambda>n. (deriv ^^ n) f 0 / fact n)"
+    using fps_nth_fps_expansion[OF exp] by (intro ext) simp_all
+  with fps_coeff_residues_bigo[OF assms(2-)] show ?thesis by simp
+qed
+
 subsection \<open>Rouche's theorem \<close>
 
 theorem Rouche_theorem:
   fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
   defines "fg\<equiv>(\<lambda>p. f p + g p)"
   defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
   assumes
     "open s" and "connected s" and
     "finite zeros_fg" and
     "finite zeros_f" and
     f_holo:"f holomorphic_on s" and
     g_holo:"g holomorphic_on s" and
     "valid_path \<gamma>" and
     loop:"pathfinish \<gamma> = pathstart \<gamma>" and
     path_img:"path_image \<gamma> \<subseteq> s " and
     path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
     homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
   shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
           = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
 proof -
   have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
   proof -
     have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
     proof -
       have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
       moreover have "f z = - g z"  using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
       then have "cmod (f z) = cmod (g z)" by auto
       ultimately show False by auto
     qed
     then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
   qed
   have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
   proof -
     have False when "z\<in>path_image \<gamma>" and "f z =0" for z
     proof -
       have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
       then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
       then show False by auto
     qed
     then show ?thesis unfolding zeros_f_def using path_img by auto
   qed
   define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
   define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
   define h where "h \<equiv> \<lambda>p. g p / f p + 1"
   obtain spikes
     where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
     using \<open>valid_path \<gamma>\<close>
     by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
   proof -
     have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
     proof -
       have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
       proof -
         have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
           apply (cases "cmod (f p) = 0")
           by (auto simp add: norm_divide)
         then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
       qed
       then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
         by (simp add: image_subset_iff path_image_compose)
       moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
       ultimately show "?thesis"
         using  convex_in_outside[of "ball 1 1" 0] outside_mono by blast
     qed
     have valid_h:"valid_path (h \<circ> \<gamma>)"
     proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
       show "h holomorphic_on s - zeros_f"
         unfolding h_def using f_holo g_holo
         by (auto intro!: holomorphic_intros simp add:zeros_f_def)
     next
       show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
         by auto
     qed
     have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
     proof -
       have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
       then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
         using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
         unfolding c_def by auto
       moreover have "winding_number (h o \<gamma>) 0 = 0"
       proof -
         have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
         moreover have "path (h o \<gamma>)"
           using valid_h  by (simp add: valid_path_imp_path)
         moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
           by (simp add: loop pathfinish_compose pathstart_compose)
         ultimately show ?thesis using winding_number_zero_in_outside by auto
       qed
       ultimately show ?thesis by auto
     qed
     moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
       when "x\<in>{0..1} - spikes" for x
     proof (rule vector_derivative_chain_at_general)
       show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
     next
       define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
       define t where "t \<equiv> \<gamma> x"
       have "f t\<noteq>0" unfolding zeros_f_def t_def
         by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
       moreover have "t\<in>s"
         using contra_subsetD path_image_def path_fg t_def that by fastforce
       ultimately have "(h has_field_derivative der t) (at t)"
         unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
         by (auto intro!: holomorphic_derivI derivative_eq_intros)
       then show "h field_differentiable at (\<gamma> x)"
         unfolding t_def field_differentiable_def by blast
     qed
     then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
                   = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
       unfolding has_contour_integral
       apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
       by auto
     ultimately show ?thesis by auto
   qed
   then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
     using  contour_integral_unique by simp
   moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
       + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
   proof -
     have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
     proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
       show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
         by auto
       then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
         using f_holo
         by (auto intro!: holomorphic_intros simp add:zeros_f_def)
     qed
     moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
       using h_contour
       by (simp add: has_contour_integral_integrable)
     ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
                         contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
       using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
       by auto
     moreover have "deriv fg p / fg p =  deriv f p / f p + deriv h p / h p"
                       when "p\<in> path_image \<gamma>" for p
     proof -
       have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
         by auto
       have "h p\<noteq>0"
       proof (rule ccontr)
         assume "\<not> h p \<noteq> 0"
         then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
         then have "cmod (g p/f p) = 1" by auto
         moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
           apply (cases "cmod (f p) = 0")
           by (auto simp add: norm_divide)
         ultimately show False by auto
       qed
       have der_fg:"deriv fg p =  deriv f p + deriv g p" unfolding fg_def
         using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _  \<open>open s\<close>] path_img that
         by auto
       have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
       proof -
         define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
         have "p\<in>s" using path_img that by auto
         then have "(h has_field_derivative der p) (at p)"
           unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
           by (auto intro!: derivative_eq_intros holomorphic_derivI)
         then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
       qed
       show ?thesis
         apply (simp only:der_fg der_h)
         apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
         by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
     qed
     then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
                   = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
       by (elim contour_integral_eq)
     ultimately show ?thesis by auto
   qed
   moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
     unfolding c_def zeros_fg_def w_def
   proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
         , of _ "{}" "\<lambda>_. 1",simplified])
     show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
     show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
     show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
   qed
   moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
     unfolding c_def zeros_f_def w_def
   proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
         , of _ "{}" "\<lambda>_. 1",simplified])
     show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
     show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
     show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
   qed
   ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
     by auto
   then show ?thesis unfolding c_def using w_def by auto
 qed
 
 end
\ No newline at end of file