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, $\gamma _{R}$, can be represented as the equation : $x=Re^{iv}$ where v is a variable that goes from $\pi$to 0

differentating, we have

$\mathrm {d} x=Rie^{iv}\mathrm {d} v$

plugging it back into our original integrals

$\oint \limits _{H}{\frac {1}{x^{2}+1}}\mathrm {d} x=\int \limits _{\pi }^{0}{\frac {Rie^{iv}\mathrm {d} v}{(Re^{iv})^{2}+1}}\mathrm {d} x+\int \limits _{R}^{R}{\frac {1}{x^{2}+1}}\mathrm {d} x$