To say what contour integration means, we first have to define path integration. We can parameterise a parth G from A to B in the complex plane as, say, z(t) from t=a to t=b. Then, we define the path integral of a complex function f(z) along G as

ò_G f(z) dz=ò_a^b f(z(t))×z ' (t) dt

[I assume you know how to do that integral - it is simply

ò_a^b f(z(t))×z ' (t) dt=ò_a^b Re(f(z(t))×z ' (t))dt+i ò_a^b Im(f(z(t))×z ' (t))dt]

We can define contour integration in a similar manner. A contour is simply a closed curve in the complex plane, and we can evaluate the contour integral as sum of 2 path integrals.

Some 'nasty' integrals can be evaluated easily by choosing appropriate contours and applying Cauchy's Residue Theorem, for example

ò_-¥^¥ 1/(1+x⁴) dx

This integral can be evaluated by elementary means (by partial fractions) or by considering the contour integral along the semicircle with base {(r,0): -r £ r £ r}.

See Michael's link for a demonstration of using this technique.

f(z)=¼+a₂ (z-z)^-2+a₁(z-z)^-1+a₀+a₁(z-z)+¼

generally speaking, this series converges for z in some annulus centred at z. The residue of f(z) at z=z is a_-1, written Res(f,z).

A number z is a pole of order k (integer) if (z-z)^k f(z) is analytic at z but (z-z)^k-1f(z) is not. If there is no such k then z is called an essential singularity.