In complex analysis, a residue of a function f is a complex number that is computed about one of the singularities, a, of the function. In mathematical notation, this is concisely written as Res(f,a). Easy to compute, the residue allows the use of the Residue Theorem, which simplifies the calculation of general contour integrals. As long as the function is not overly complicated, computing the residue will be a quick and simple process.
Choose the singularity whose residue you want to identify. A singularity is any point at which the function f becomes undefined. For example, for the function f(z) = 1/z, there is a singularity at z = 0.
Identify the order of the singularity. This is a measure of the function as it approaches the singularity. In the above example of f(z) = 1/z, z=0 is a 1st order singularity. For the function g(z) = (z+7)/(z-6)^2, there is a singularity at z=6 of order 2.
Compute a residue of order 1. If the singularity is of order 1, the residue of a function f, about the singularity a, is simply the limit of (z-a)_f(z) as z goes to a. For the example in step 1, f(z) = 1/z and a=0, (z-a)_f(z) = z/z = 1. So the residue of f(z) about 0 is 1.
Compute a residue of order n. If the singularity is more generally of order n, then the residue is the limit of the (n-1)th derivative of [(z-a)^n f(z)/(n-1)!] as z goes to a. For the second example of step 2, g(z) = (z+7)/(z-6)^2 about a=6, the function inside the square brackets becomes [(z-6)^2 g(z)/(1!)] = [(z+7)]. Taking the first derivative yields a residue of 1.