Complex Analysis: Theory and Applications · Problem 7.1.9

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · uniform-convergence

Problem

Consider the function ζ(z)=∑n=1∞n−z\zeta(z) = \displaystyle\sum_{n=1}^{\infty} n^{-z}, where n−z=exp⁡(−zln⁡n)n^{-z} = \exp(-z\ln n). ((a)) Show that ζ(z)\zeta(z) converges uniformly on A={z:Re(z)≥2}A = \{z : \mathrm{Re}(z) \geq 2\}. ((b)) Let DD be a closed disk contained in {z:Re(z)>1}\{z : \mathrm{Re}(z) > 1\}. Show that ζ(z)\zeta(z) converges uniformly on DD.

Related theory

Browse the problem library