UMD Analysis Qualifying Exam/Jan10 Complex

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Problem 2[edit]

The function has a convergent Taylor expansion . Find .


. By using the definition , we get that if and only if . It is not hard to show that this happens if and only if and . Therefore, the only zeros of all occur on the real axis at integer distances away from 1/2. Therefore, is analytic everywhere except at these points.

Our Taylor series is centered at . By simple geometry, the shortest distance from to or (the closest poles of ) is . This is the radius of convergence of the Taylor series.

From calculus (root test), we know that . Therefore, .