Evaluatiom of Integrals Using Contour Integration

In our lectures on integral solutions to differential equations using Laplace kernels ,we encountered integrals of the type-∫ = + C n t xt t f x y 1) exp() () (where t=γ+iτ and C is a closed contour within the complex plane. To evaluate this type of integrals under conditions where the curve partially lies at infinity, one makes use of complex variable methods and in particular the Cauchy Integral Theorem. We present here this evaluation method using several specific examples. Our a starting point is the Cauchy Integral Theorem-∫ = C C within everywhere analytic is z F when dz z F) (0) (and Cauchy's Integral Formula-∫ − = + C n n n z z dz z G i n dz z G d 1 0 0) () (2 !) (π ,where G(z) is analytic everywhere in C but the integrand has an (n+1) order pole at z=z 0 provided n is a positive integer. Consider first the integral-∫ − − ∫ = + + = C C z z z z a dz c bz az dz c b a H ] ][ [) 2 () , , (2 1 2 which has simple poles at-2 2 2 1 b ac i a b z and b ac i a b z − − − = − + − = where we are assuming a, b, and c are real and that ac>b 2. We consider integration around the closed line integral defined as the infinite radius semicircle and the line z=x along the real z axis from minus to plus infinity as shown