ABSTRACT
Let O be an open set in C, and let f: O ~ a → C be analytic, where a ∈ O. Recall from Sec. 10.5 that if f has a pole at a, then by the definition of a pole there exists a polynomial Q in 1/(z - a) without a constant term, called the principal part of f at a, and such that f - Q has a removable singularity at a. The coefficient of 1/(z - a) in Q is called the residue of f at a. Henceforth, we use the notation Res (f; a) for the residue of f at a. In Exercise 10.5.1, it is shown that if (1/2Πi)f(z) is integrated over a suitably chosen circle with center at a, the result is Res (f; a). This fact is the motivation for the word "residue": The "residue" is "what is left" after integration.
