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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2004 • Question 6

Question

Math 6420 • PDE Fall 2004 • Question 6

Show that \[ \int _0^\infty \frac {\cosh ax}{\cosh \pi x}\,dx=\frac {1}{2}\sec \left (\frac {a}{2}\right ), \qquad |a|<\pi \] Use a rectangular contour with corners at \(\pm R\) and \(\pm R+i\).

Raw LaTeX

Show that
\[
  \int_0^\infty \frac{\cosh ax}{\cosh \pi x}\,dx=\frac{1}{2}\sec\left(\frac{a}{2}\right),
  \qquad |a|<\pi
\]
Use a rectangular contour with corners at \(\pm R\) and \(\pm R+i\).

Topic Tags

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