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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 7

Question

Math 6210 • Real Analysis Fall 2007 • Question 7

Let \(H=\{z\in \mathbb C:\operatorname {Im}z>0\}\) be the upper half plane and let \(\Delta =\{z\in \mathbb C:|z|<1\}\) be the unit disk.

a. Find a conformal map from \(H\) to \(\Delta \) that takes \(i\) to \(0\).

b. Let \(f:H\to H\) be a holomorphic map with \(f(i)=i\). Show that \(|f'(i)|\le 1\) and that if \(|f'(i)|=1\) then \(f\) is of the form \(f(z)=\frac {az+b}{cz+d}\) with \(a,b,c,d\in \mathbb R\).

Raw LaTeX

Let \(H=\{z\in\mathbb C:\operatorname{Im}z>0\}\) be the upper half
plane and let \(\Delta=\{z\in\mathbb C:|z|<1\}\) be the unit disk.

a. Find a conformal map from \(H\) to \(\Delta\) that takes \(i\) to \(0\).

b. Let \(f:H\to H\) be a holomorphic map with \(f(i)=i\). Show that
\(|f'(i)|\le 1\) and that if \(|f'(i)|=1\) then \(f\) is of the form
\(f(z)=\frac{az+b}{cz+d}\) with \(a,b,c,d\in\mathbb R\).

Topic Tags

Argument Principle and Rouche's Theorem Conformal Moduli and Annuli Holomorphic Functions and Cauchy-Riemann Equations Hyperbolic Geometry in the Disk and Half-Plane