Complex Variables :: Contour Integration

You will be surprised to know what a variety of definite proper and improper integrals can be computed using the residue theorem and  methods of contour integration in complex plane.

Here is a partial list.
A word of caution about this list:
The integrals are defined and exist for a range of parameters that  appear in the integrand. The parameters may be real or complex. It is an exrecise for the reader  to find suitable ranges for which the integrals exist. It is not expected that a complete answer abour the range will be attempted.

Following is an incomplete list of sample definite  integral problems that can be solved by the method of contour integration in complex plane.

1. (0,2\pi) integrals of rational functions of trigonometric fucntions \(\itn_0^{2\pi) \frac{d\theta }{(7+6\cos\theta)}=\frac{2\pi}{\surd 13} \)
2. Certain integrals involving fractional powers of algeraic fucntions of \(x\).\(\int_0^1 \frac{x^{2n} \,dx}{(x(1-x^2))^{1/3}}\)
3. Integrals of functionsof trigonometric fuunction over infinite range \((0,\infty)\) \(\int_0^\infty \cos ax^2 \cos 2bx dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}} (\cos(b^2/a) + \sin (b^2/a))\)
4. Integrals involving exponentials and trigonomteric fucntions\(\int_0^\infty \exp(-x^2\cos(2\alpha)) \cos(x^2\sin(2\alpha))=\frac {\surd \pi}{2} \cos \alpha\)
5. More complicated integrals and trigonometric functions\( \int_0^\infty \exp(-x^p \cos p\lambda)\cos(x^ p\sin  p\lambda) = (1/p)\,\cos (\lambda)  \Gamma(1/p)\)
6. \( \int_0^\infty x e^{-px} \cos(2x^2 +px) =0\)
7. Rational functions\(\int_0^\infty \frac{dx}{1+x^6}\)
8. Integrals involving \(\log x\) and rational functions
9. Integrals involving fractional powers and \(\log x\).
10. Integrals involving hyperbolic and exponential functions
11. Integrals having singularity in the range of integration,principal value and more
12. What you see in not what you get.  These involve some wonderful tricks  of combininig different pieces of integral along a contour. The most famous and amazing example is the way Gaussian integral \(\int_{-\infty}^\infty e^{-x^2} dx =\surd \pi\)can be proved. Reference Polya, McRobert

You will also find a large  number of problems and solved examples scattered over three different chapters in the book:

A. K. Kapoor, "Complex Variables -- Principles and Problem Sessions" Cambridge University  Press (2011)
Errata and Solutions to Problems in this book are available on this site: Click here to see all