Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2016 • Question 10

Question

Math 6210 • Real Analysis Summer 2016 • Question 10

Let \(\wp (z)\) be the Weierstrass \(\wp \) function. Show that any even elliptic function \(f(z)\) with periods \(\omega _1,\omega _2\) can be expressed in the form \[ (\text {Constant})\prod _{k=1}^{n} \frac {\wp (z)-\wp (a_k)}{\wp (z)-\wp (b_k)} \] for some Constant, \(a_k,b_k\in \mathbb C\), provided that \(0\) is neither a zero nor a pole. What is the corresponding form if \(f(z)\) either vanishes or has poles at \(z=0\)?

Raw LaTeX

Let \(\wp(z)\) be the Weierstrass \(\wp\) function. Show that any
even elliptic function \(f(z)\) with periods \(\omega_1,\omega_2\) can be
expressed in the form
\[
(\text{Constant})\prod_{k=1}^{n}
\frac{\wp(z)-\wp(a_k)}{\wp(z)-\wp(b_k)}
\]
for some Constant, \(a_k,b_k\in\mathbb C\), provided that \(0\) is neither a zero
nor a pole. What is the corresponding form if \(f(z)\) either vanishes or has
poles at \(z=0\)?

Topic Tags

Isolated Singularities and Residues Picard-Type Theorems and Casorati-Weierstrass Zeros of Holomorphic Functions and the Identity Theorem