Figure A 2: Derivation of the contour deformation principle Principal value integrals. We must occasionally carry out integrations of the form where f(x) has a finite number of singularities xk(k=1,., n)along the real axis. Such one singularity present at point x1, for instance, we detine improper integral. With just singularities in the integrand force us to interpret I as ar →0 provided that both limits exist. When both limits do not exist, we may still be able to obtain a well-defined result by computing f(x)dx+f(r)d (i.e by taking n= E so that the limits are "symmetric"). This quantity is called the Cauchy principal value of I and is denoted f(r) P V f(x)dx+ f(r)dx+ f(x)d. In a large class of problems f(z)(i.e, f(x) with x replaced by the complex variable is analytic everywhere except for the presence of finitely many simple poles. Some of these may lie on the real axis(at points xI and some may not Consider now the integration contour C shown in Figure A.3. We choose R so large and E so small that C encloses all the poles of f that lie in the upper half of the complex 0 2001 by CRC Press LLCFigure A.2: Derivation of the contour deformation principle. Principal value integrals. We must occasionally carry out integrations of the form I = ∞ −∞ f (x) dx where f (x) has a finite number of singularities xk (k = 1,..., n) along the real axis. Such singularities in the integrand force us to interpret I as an improper integral. With just one singularity present at point x1, for instance, we define ∞ −∞ f (x) dx = lim ε→0 x1−ε −∞ f (x) dx + lim η→0 ∞ x1+η f (x) dx provided that both limits exist. When both limits do not exist, we may still be able to obtain a well-defined result by computing lim ε→0  x1−ε −∞ f (x) dx + ∞ x1+ε f (x) dx (i.e., by taking η = ε so that the limits are “symmetric”). This quantity is called the Cauchy principal value of I and is denoted P.V. ∞ −∞ f (x) dx. More generally, we have P.V. ∞ −∞ f (x) dx = lim ε→0  x1−ε −∞ f (x) dx + x2−ε x1+ε f (x) dx + +···+ xn−ε xn−1+ε f (x) dx + ∞ xn+ε f (x) dx for n singularities x1 < ··· < xn. In a large class of problems f (z) (i.e., f (x) with x replaced by the complex variable z) is analytic everywhere except for the presence of finitely many simple poles. Some of these may lie on the real axis (at points x1 < · · · < xn, say), and some may not. Consider now the integration contour C shown in Figure A.3. We choose R so large and ε so small that C encloses all the poles of f that lie in the upper half of the complex