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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2022 • Question 3

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2022 • Question 3

  1. Use principal value integrals to show that \[ \int _0^\infty \frac {\cos kx-\cos mx}{x^2}\,dx=-\frac {\pi }{2}(k-m),\qquad k,m>0. \]
  2. Deduce from (a) that \[ \int _0^\infty \frac {\sin ^2 x}{x^2}\,dx=\frac {\pi }{2}. \]

Raw LaTeX

\begin{enumerate}
\item
Use principal value integrals to show that
\[
\int_0^\infty \frac{\cos kx-\cos mx}{x^2}\,dx=-\frac{\pi}{2}(k-m),\qquad k,m>0.
\]

\item
Deduce from (a) that
\[
\int_0^\infty \frac{\sin^2 x}{x^2}\,dx=\frac{\pi}{2}.
\]
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Harmonic Functions and Poisson Integral Formula Principal Value Integrals Real Integral Evaluation by Complex Methods