UMD Analysis Qualifying Exam/Jan10 Complex

Problem 2 The function has a convergent Taylor expansion . Find . Solution . 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, .