src/HOL/Analysis/Great_Picard.thy
changeset 65064 a4abec71279a
parent 65040 5975839e8d25
child 65274 db2de50de28e
equal deleted inserted replaced
65062:dc746d43f40e 65064:a4abec71279a
  1144     with \<open>0 < r\<close> \<open>0 < m\<close> w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
  1144     with \<open>0 < r\<close> \<open>0 < m\<close> w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
  1145       by (auto simp: geq divide_simps hnz)
  1145       by (auto simp: geq divide_simps hnz)
  1146   qed
  1146   qed
  1147   moreover
  1147   moreover
  1148   have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z) =
  1148   have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z) =
  1149         2 * of_real pi * ii * m + 0"
  1149         2 * of_real pi * \<i> * m + 0"
  1150   proof (rule contour_integral_unique [OF has_contour_integral_add])
  1150   proof (rule contour_integral_unique [OF has_contour_integral_add])
  1151     show "((\<lambda>x. m / (x - z0)) has_contour_integral 2 * of_real pi * \<i> * m) (circlepath z0 (r/2))"
  1151     show "((\<lambda>x. m / (x - z0)) has_contour_integral 2 * of_real pi * \<i> * m) (circlepath z0 (r/2))"
  1152       by (force simp: \<open>0 < r\<close> intro: Cauchy_integral_circlepath_simple)
  1152       by (force simp: \<open>0 < r\<close> intro: Cauchy_integral_circlepath_simple)
  1153     show "((\<lambda>x. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
  1153     show "((\<lambda>x. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
  1154       apply (rule Cauchy_theorem_disc_simple [of _ z0 r])
  1154       apply (rule Cauchy_theorem_disc_simple [of _ z0 r])