equal
deleted
inserted
replaced
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]) |