equal
deleted
inserted
replaced
862 by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero) |
862 by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero) |
863 moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0" |
863 moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0" |
864 proof - |
864 proof - |
865 define h where [abs_def]: "h z = (z - \<xi>)^2 * f z" for z |
865 define h where [abs_def]: "h z = (z - \<xi>)^2 * f z" for z |
866 have h0: "(h has_field_derivative 0) (at \<xi>)" |
866 have h0: "(h has_field_derivative 0) (at \<xi>)" |
867 apply (simp add: h_def Derivative.DERIV_within_iff) |
867 apply (simp add: h_def has_field_derivative_iff) |
868 apply (rule Lim_transform_within [OF that, of 1]) |
868 apply (rule Lim_transform_within [OF that, of 1]) |
869 apply (auto simp: divide_simps power2_eq_square) |
869 apply (auto simp: divide_simps power2_eq_square) |
870 done |
870 done |
871 have holh: "h holomorphic_on S" |
871 have holh: "h holomorphic_on S" |
872 proof (simp add: holomorphic_on_def, clarify) |
872 proof (simp add: holomorphic_on_def, clarify) |
877 using field_differentiable_at_within field_differentiable_def h0 by blast |
877 using field_differentiable_at_within field_differentiable_def h0 by blast |
878 next |
878 next |
879 case False |
879 case False |
880 then have "f field_differentiable at z within S" |
880 then have "f field_differentiable at z within S" |
881 using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close> |
881 using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close> |
882 unfolding field_differentiable_def DERIV_within_iff |
882 unfolding field_differentiable_def has_field_derivative_iff |
883 by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at]) |
883 by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at]) |
884 then show ?thesis |
884 then show ?thesis |
885 by (simp add: h_def power2_eq_square derivative_intros) |
885 by (simp add: h_def power2_eq_square derivative_intros) |
886 qed |
886 qed |
887 qed |
887 qed |
1717 using cnjs apply auto |
1717 using cnjs apply auto |
1718 done |
1718 done |
1719 have "cnj \<circ> f \<circ> cnj field_differentiable at x within S \<inter> {z. Im z < 0}" |
1719 have "cnj \<circ> f \<circ> cnj field_differentiable at x within S \<inter> {z. Im z < 0}" |
1720 if "x \<in> S" "Im x < 0" "f field_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x |
1720 if "x \<in> S" "Im x < 0" "f field_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x |
1721 using that |
1721 using that |
1722 apply (simp add: field_differentiable_def Derivative.DERIV_within_iff Lim_within dist_norm, clarify) |
1722 apply (simp add: field_differentiable_def has_field_derivative_iff Lim_within dist_norm, clarify) |
1723 apply (rule_tac x="cnj f'" in exI) |
1723 apply (rule_tac x="cnj f'" in exI) |
1724 apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify) |
1724 apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify) |
1725 apply (drule_tac x="cnj xa" in bspec) |
1725 apply (drule_tac x="cnj xa" in bspec) |
1726 using cnjs apply force |
1726 using cnjs apply force |
1727 apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj) |
1727 apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj) |