Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 10

Question

Math 6420 • PDE Fall 1996 • Question 10

Suppose \(f(\theta )\) is a \(2\pi \) periodic function. Define its Hilbert Transform as follows: Let \(u(r,\theta )\) be the solution of Laplace’s equation on the unit circle satisfying Neumann data \(u_r(1,\theta )=f(\theta )\). The Hilbert transform of \(f\) is \(H(f)=u_\theta (1,\theta )\).

Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier coefficients.

Raw LaTeX

Suppose \(f(\theta)\) is a \(2\pi\) periodic function. Define its
Hilbert Transform as follows: Let \(u(r,\theta)\) be the solution of Laplace's equation on
the unit circle satisfying Neumann data \(u_r(1,\theta)=f(\theta)\). The Hilbert transform of
\(f\) is \(H(f)=u_\theta(1,\theta)\).

Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier
coefficients.

Topic Tags

Boundary Value Problems Laplace Equation Linear Programming and Optimization Variable-Coefficient Diffusion Problems