3 The Calculus of Residues

Abstract

If f (z) has a pole of order m at z = z 0 , it can be written as Eq. (6.27), or f (z) = φ(z) = a −1 (z − z 0) + a −2 (z − z 0) 2 +. .. + a −m (z − z 0) m , (7.1) where φ(z) is analytic in the neighborhood of z = z 0. Now we have seen that if C encircles z 0 once in a positive sense, C dz 1 (z − z 0) n = 2πiδ n,1 , (7.2) where the Kronecker δ-symbol is defined by δ m,n = 0, m = n, 1, m = n.. (7.3) Proof: By Cauchy's theorem we may take C to be a circle centered on z 0. On the circle, write z = z 0 + re iθ. Then the integral in Eq. (7.2) is i r n−1 2π 0 dθ e i(1−n)θ , (7.4) which evidently integrates to zero if n = 1, but is 2πi if n = 1. QED. Thus if we integrate the function (7.1) on a contour C which encloses z 0 , while φ(z) is analytic on and within C, we find C f (z) dz = 2πia −1. (7.5) Because the coefficient of the (z − z 0) −1 power in the Laurent expansion of f plays a special role, we give it a name, the residue of f (z) at the pole. If C contains a number of poles of f , replace the contour C by contours α, β, γ, .