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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2010 • Question 9

Question

Math 6420 • PDE Fall 2010 • Question 9

The operator \(Lu=-u''\) on \(L^2(-\infty ,\infty )\) has a Green’s function defined by \(-G''-\lambda G=\delta (x-\xi )\) given by \[ G(x,\xi ;\lambda )=-\frac {e^{i\sqrt {\lambda }|x-\xi |}}{2i\sqrt {\lambda }}. \] Assume the representation \[ -\frac {1}{2\pi i}\int _{C_\infty }G(x,\xi ;\lambda )\,d\lambda =\delta (x-\xi ) \] and show how it leads to the Fourier Transform Theorem that \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-i\mu x}F(\mu )\,d\mu , \qquad F(\mu )=\int _{-\infty }^{\infty }e^{i\mu x}f(x)\,dx. \]

Raw LaTeX

The operator \(Lu=-u''\) on \(L^2(-\infty,\infty)\) has a Green's function defined by
\(-G''-\lambda G=\delta(x-\xi)\) given by
\[
  G(x,\xi;\lambda)=-\frac{e^{i\sqrt{\lambda}|x-\xi|}}{2i\sqrt{\lambda}}.
\]
Assume the representation
\[
  -\frac{1}{2\pi i}\int_{C_\infty}G(x,\xi;\lambda)\,d\lambda=\delta(x-\xi)
\]
and show how it leads to the Fourier Transform Theorem that
\[
  f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\mu x}F(\mu)\,d\mu,
  \qquad
  F(\mu)=\int_{-\infty}^{\infty}e^{i\mu x}f(x)\,dx.
\]

Topic Tags

Distributions and Weak Solutions Fourier Transform and Sampling Theory Fundamental Solutions Linear Programming and Optimization