Contour Integral

Many integrals can be solved with the help of a number of techniques.

substituion
integration by parts

But some are just nasty

∫0∞1x4+1dx\int_{0}^{\infty} \frac{1}{x^4 + 1} dx ∫0∞x4+11dx

It can be even more diffcult by taking it from the real numbers to the set of complex numbers.

But some nice properties happen in the complex plane

If a function is analytic (differentiable everywhere), a simple closed line integral will evaluate to zero.
Why? Because if a complex function is differentiable, then we have ux=vy,uy=−vxu_x = v_y, u_y=-v_xux=vy,uy=−vx, which means ∮Cf(z)dz\oint_C f(z) dz∮Cf(z)dz is a exact differential function, and it equals to F(z1)−F(z0)F(z_1) - F(z_0)F(z1)−F(z0). If z0=z1z_0 = z_1z0=z1, then ∮f(z)dz=0\oint f(z)dz=0∮f(z)dz=0.

If there are distinct points where it’s not analytic, we can still work with that too. Those points are called poles.

Evaluating the integrals

∮Cf(z)dz=2πi∑residues\oint_C f(z)dz = 2\pi i \sum \text{residues} ∮Cf(z)dz=2πi∑residues

Computing the residues at the poles

2 Methods
- Grunt-work derivatives
- series expansion

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