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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2002 • Question 10

Question

Math 6420 • PDE Fall 2002 • Question 10

Show that the conformal transformation \[ w=\frac {z-a}{z+a},\qquad a=\sqrt {c^2-\rho ^2} \] where \(0<\rho <c\), maps the domain bounded by the circle \(|z-c|=\rho \) and the imaginary axis onto an annular region centered about the origin with outer radius \(|w|=1\) and inner radius \(|w|=\delta \). Find the radius \(\delta \).

Raw LaTeX

Show that the conformal transformation
\[
  w=\frac{z-a}{z+a},\qquad a=\sqrt{c^2-\rho^2}
\]
where \(0<\rho<c\), maps the domain bounded by the circle \(|z-c|=\rho\) and the imaginary
axis onto an annular region centered about the origin with outer radius \(|w|=1\) and inner
radius \(|w|=\delta\). Find the radius \(\delta\).

Topic Tags

Conformal Mapping and Bilinear Transformations Distributions and Weak Solutions Fundamental Solutions Linear Programming and Optimization