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Please answer the question with detailed steps and explanations.
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Please answer the question with detailed steps and explanations.
Please answer the question with detailed steps andexplanations.
4. (10 points) Let &> 0 and let % € C. Suppose f: C\ {z} → C is analytic and f has a simple pole at % or a removable discontinuity at zo. Prove that if there exists MER such that |ƒ (2)| < 12-¹+for all z = C with |z-% > M, then there exists a € C such that f(z) = for all z € C\ {z}. (Hint: use the definition of a simple pole or a removable singularity and then relate f(z) to some integral around a big circle and taking a limit.)
4. (10 points) Let &> 0 and let % € C. Suppose f: C\ {z} → C is analytic and f has a simple pole at % or a removable discontinuity at zo. Prove that if there exists MER such that |ƒ (2)| < 12-¹+for all z = C with |z-% > M, then there exists a € C such that f(z) = for all z € C\ {z}. (Hint: use the definition of a simple pole or a removable singularity and then relate f(z) to some integral around a big circle and taking a limit.)