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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2024 • Question 4

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2024 • Question 4

Show that \[ \int _0^\infty \frac {\cos 3x}{(x^2+1)^2}\,dx=\frac {\pi }{e^3}. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Raw LaTeX

Show that
\[
\int_0^\infty \frac{\cos 3x}{(x^2+1)^2}\,dx=\frac{\pi}{e^3}.
\]
Clearly indicate the contour of integration and the treatment of the integral along each
segment.

Topic Tags

Cauchy Integral Formula and Integral Theorems Contour Integration and Residue Calculus Harmonic Functions and Poisson Integral Formula Real Integral Evaluation by Complex Methods