Contour integration

The residue theorem, invented by Cauchy, is a method of evaluating definite integrals.

Consider the following contour in the complex plane :

where R is a limit going to infinity...or something

Let's find

$\int \limits _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\mathrm {d} x$

the first half of the contour is a straight line that goes from -R to R hence, this can be represented as, (H is the entire contour, because i said so)

$\oint \limits _{H}{\frac {1}{x^{2}+1}}\mathrm {d} x=\int \limits _{\gamma _{R}}{\frac {1}{x^{2}+1}}\mathrm {d} x+\int \limits _{R}^{R}{\frac {1}{x^{2}+1}}\mathrm {d} x$

the other half of the contour, γR, can be represented as the equation :

$x=Re^{iv}$

where v is a vriable that goes from pi to 0 the other half of the contour is just a straight line going from -R to R, hence, we have

$\oint \limits _{x}=Re^{iv}$