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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1997 • Question 9

Question

Math 6210 • Real Analysis Fall 1997 • Question 9

Let \[ f(z)=\sum _{n=1}^{\infty }\frac {1}{n^z} \] where for \(a>0\), \(a^z\) is defined to be \(e^{z\ln a}\), and \(R=\{z\mid z\in \mathbb C\text { and }\Re z>1\}\). Prove that \(f\) is holomorphic on \(R\).

Raw LaTeX

Let
\[
f(z)=\sum_{n=1}^{\infty}\frac{1}{n^z}
\]
where for \(a>0\), \(a^z\) is defined to be \(e^{z\ln a}\), and
\(R=\{z\mid z\in\mathbb C\text{ and }\Re z>1\}\). Prove that \(f\) is holomorphic
on \(R\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem