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548 chapter 20AIm k'BCIm k'– k kIm k'– kkRe k' Re k' Re k'k – k + eFigure 20.4 Three different paths in the complex k ′ plane used to evaluate line integrals whenthere are singularities. Here the singularities are at k and −k, and the integration variable is k ′ .20.4.1 Singular Integrals (Math)A singular integralG =∫ bag(k) dk, (20.23)is one in which the integrand g(k) is singular at a point k 0 within the interval [a, b]and yet the integral G is still finite. (If the integral itself were infinite, we could notcompute it.) Unfortunately, computers are notoriously bad at dealing with infinitenumbers, and if an integration point gets too near the singularity, overwhelmingsubtractive cancellation or overflow may occur. Consequently, we apply someresults from complex analysis before evaluating singular integrals numerically. 3In Figure 20.4 we show three ways in which the singularity of an integrand canbe avoided. The paths in Figures 20.4A and 20.4B move the singularity slightly offthe real k axis by giving the singularity a small imaginary part ±iɛ. The Cauchyprincipal-value prescription P (Figure 20.4C) is seen to follow a path that “pinches”both sides of the singularity at k 0 but does not to pass through it:∫ [+∞∫ k0−ɛ ∫ ]+∞P f(k) dk = lim f(k) dk + f(k) dk . (20.24)ɛ→0−∞−∞k 0+ɛThe preceding three prescriptions are related by the identity∫ +∞−∞∫f(k) dk+∞k − k 0 ± iɛ = P−∞f(k) dk ′k − k 0∓ iπf(k 0 ), (20.25)which follows from Cauchy’s residue theorem.3 [S&T 93] describe a different approach using Maple and Mathematica.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 548