Suppose a punctured disk *D* = {*z* : 0 < |*z* - *c*| < *R*} in the complex plane is given and *f* is a holomorphic function defined (at least) on *D*. The residue of *f* at *c*, written as Res(*f*, *c*), is then defined as the coefficient *a*_{-1} of (*z*-*c*)^{-1} in the Laurent series expansion of *f* around *c*. This coefficient can often be computed by combining several known Taylor series; it is also possible to use the integral formula given in the Laurent series article:

If the function *f* can be continued to a holomorphic function on the whole disk {*z* : |*z* - *c*| < *R*}, then Res(*f*, *c*) = 0. The converse is not generally true.