Figure A 1: Derivation of the residue theorem. Figure A l depicts a simple closed curve C enclosing n isolated singularities of a function f(z). We assume that f(z) is analytic on and elsewhere within C. Around each singular point zk we have drawn a circle Ck so small that it encloses no singular point other than Zk; taken together, the Ck(k=1,., n) and C form the boundary of a region in which f(z)is everywhere analytic. By the Cauchy-Goursat theorem f(z)dz+ f(z)dz=0 f(z)dz f(z)dz where now the integrations are all performed in a counterclockwise sense. By(A 12) f(z)dz=2mj∑k (A.14) where rI,..., In are the residues of f(z) at the singularities within C Contour deformat Suppose f is analytic in a region D and r is a simple closed curve in D. If r can be continuously deformed to another simple closed curve r passing out of D, then ∫(z)dz=f(x)d (A.15) To see this, consider Figure A2 where we have introduced another set of curves +y; these new curves are assumed parallel and infinitesimally close to each other. Let c be the composite curve consisting of r, +y, - and -y, in that order. Since f is analytic f(z)dz=f(z)dz+ f(z)dz+ f(z)dz+f(z)dz=0 But -r, f(a)dz=-fr f(a)dz and _ y f(z)dz=-+y f(z)dz, hence(A15) The contour deformation principle often permits us to replace an integration contour by one that is more convenient 0 2001 by CRC Press LLCFigure A.1: Derivation of the residue theorem. Figure A.1 depicts a simple closed curve C enclosing n isolated singularities of a function f (z). We assume that f (z) is analytic on and elsewhere within C. Around each singular point zk we have drawn a circle Ck so small that it encloses no singular point other than zk ; taken together, the Ck (k = 1,..., n) and C form the boundary of a region in which f (z) is everywhere analytic. By the Cauchy–Goursat theorem C f (z) dz + n k=1 Ck f (z) dz = 0. Hence 1 2π j C f (z) dz = n k=1 1 2π j Ck f (z) dz, where now the integrations are all performed in a counterclockwise sense. By (A.12) C f (z) dz = 2π j n k=1 rk (A.14) where r1,...,rn are the residues of f (z) at the singularities within C. Contour deformation. Suppose f is analytic in a region D and  is a simple closed curve in D. If  can be continuously deformed to another simple closed curve  without passing out of D, then  f (z) dz =  f (z) dz. (A.15) To see this, consider Figure A.2 where we have introduced another set of curves ±γ ; these new curves are assumed parallel and infinitesimally close to each other. Let C be the composite curve consisting of , +γ , − , and −γ , in that order. Since f is analytic on and within C, we have C f (z) dz =  f (z) dz + +γ f (z) dz + − f (z) dz + −γ f (z) dz = 0. But − f (z) dz = −  f (z) dz and −γ f (z) dz = − +γ f (z) dz, hence (A.15) follows. The contour deformation principle often permits us to replace an integration contour by one that is more convenient.