A function is meromorphic if it is holomorphic except in a finite number of simple poles, which are points where diverges, but where the product is nonzero and still holomorphic close to . In other words, can be approximated close to :
Where the residue of a simple pole is defined as follows, and represents the rate at which diverges close to :
Cauchy’s residue theorem for meromorphic functions is a generalization of Cauchy’s integral theorem for holomorphic functions, and states that the integral on a contour purely depends on the simple poles enclosed by :
From the definition of a meromorphic function, we know that we can decompose like so, where is holomorphic and are all its poles:
We integrate this over a contour which contains all poles, and apply both Cauchy’s integral theorem and Cauchy’s integral formula to get:
This theorem might not seem very useful, but in fact, by cleverly choosing the contour , it lets us evaluate many integrals along the real axis, most notably Fourier transforms. It can also be used to derive the Kramers-Kronig relations.