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Give an explicit example of each of the following: (a) Two power series f and g in the complex variable z, both centered at 0, such that Of and g both have radius of convergence 1, but O the power series f + g has radius of convergence strictly greater than 1. (b) A function f: C\ {i, -i} → C which O has simple zeros at 0 and at 1, O has double poles at i and at -i, O is holomorphic on C\ {i, -i}.

Give an explicit example of each of the following: (a) Two power series f and g in the complex variable z, both centered at 0, such that Of and g both have radius of convergence 1, but O the power series f + g has radius of convergence strictly greater than 1. (b) A function f: C\ {i, -i} → C which O has simple zeros at 0 and at 1, O has double poles at i and at -i, O is holomorphic on C\ {i, -i}.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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6. Give an explicit example of each of the following: (a) Two power series f and g in the complex variable z, both centered at 0, such that Of and g both have radius of convergence 1, but O the power series f + g has radius of convergence strictly greater than 1. (b) A function f: C\ {i, -i} → C which O has simple zeros at 0 and at 1, O has double poles at i and at -i, O is holomorphic on C\ {i, -i}.
Transcribed Image Text:6. Give an explicit example of each of the following: (a) Two power series f and g in the complex variable z, both centered at 0, such that Of and g both have radius of convergence 1, but O the power series f + g has radius of convergence strictly greater than 1. (b) A function f: C\ {i, -i} → C which O has simple zeros at 0 and at 1, O has double poles at i and at -i, O is holomorphic on C\ {i, -i}.
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