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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2022 • Question 2

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2022 • Question 2

Use a sector contour with radius \(R\) centered at the origin with angle \(0\le \theta \le 2\pi /5\) to find, for \(a>0\), \[ \int _0^\infty \frac {dx}{x^5+a^5}=\frac {\pi }{5a^4\sin (\pi /5)}. \]

Raw LaTeX

Use a sector contour with radius \(R\) centered at the origin with angle \(0\le \theta \le 2\pi/5\) to find,
for \(a>0\),
\[
\int_0^\infty \frac{dx}{x^5+a^5}=\frac{\pi}{5a^4\sin(\pi/5)}.
\]

Topic Tags

Contour Integration and Residue Calculus Keyhole and Sector Contours Real Integral Evaluation by Complex Methods