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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2002 • Question 7

Question

Math 6420 • PDE Fall 2002 • Question 7

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(|\operatorname {Re}\alpha |<1\).

Raw LaTeX

Use contour integration to show that
\[
  \int_0^\infty \frac{x^\alpha}{(1+x)^2}\,dx=\frac{\pi\alpha}{\sin\pi\alpha}
\]
where \(|\operatorname{Re}\alpha|<1\).

Topic Tags

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