Trigonometric integral

A trigonometric integral is an integral in the form:

$${\int }_0^{2\pi }F(\cos \theta ,\sin \theta )d\theta$$

We can evaluate these integrals with residues by writing them as contour integrals on the unit circle centred at the origin. Just like how we might parameterise to the $\theta$ domain, we essentially go backwards. Our core “identities” are:¹

$$\cos \theta =\frac{e^{-\theta }+e^{-i\theta }}{2}=\frac{z+\frac{1}{z}}{2}$$ $$\sin \theta =\frac{e^{i\theta }-e^{-i\theta }}{2i}=\frac{z-\frac{1}{z}}{2i}$$ $$dz=i{e}^{i\theta }d\theta$$ $$d\theta =\frac{dz}{iz}$$

By converting a function $f$ from the $\theta$ domain to the $z$ domain, we can then compute using Cauchy’s residue theorem. It can get pretty tricky to simplify the algebra after this conversion.

Anything leftover from the conversion with a pole within the unit circle is used in the residue computation.

Footnotes

1. Side note: looks super similar to multiple angle identities. So cool. ↩