MATH 463 Lecture : MATH463_BOYLE-M_FALL2014_0101_MID_SOL
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Exam 2 - fall 2014 - math 463 - boyle. There are problems on both sides of this paper: (a) what is the radius of convergence of p . From the geometric series, we know pn((4 + 3i)z)n converges if |(4 + 3i)z| < 1 and diverges if |(4 + 3i)z| > 1. The border |(4 + 3i)z| = 1 happens at |z| = 1/|4 + 3i| = 1/5. (b) let f (z) = ez/(z2 + 4) . What is the radius of convergence of its taylor series p . This radius of convergence is 2, because it is the minimum distance from (1+i) to a point at which f is not analytic. 2. (a) what is the order m of the pole at z = 0 of the function f (z) = ez2. Answer m = 6 . f (z) is a ratio of analytic functions. The denominator has at z0 = 0 a.
