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&576 Advanced Calculus, Fifth EditionFigure 8.17 Cauchy residue theorem. *I ' -If C is a simple closed path in D, within which f (z) is analytic except forisolated singularities at z,, ..., zk, then by Theorem 16,f f(z)dz = f f(z)dz+...+ f(z.)dz,f C CI CIiewhere C, encloses only the singularity at zl , C2 encloses only z2, . . . , as in Fig. 8.17.We thus obtain the following basic theorem:THEOREM 28 (Cauchy's Residue Theorem) If f (z) is analytic in a domainD and C is a simple closed curve in D within which f (z) is analytic except forisolated singularities at 2,. . . . , zk, thenf f (z)dz = 2irilRes [ f (2). 211 + . . . + Res L f (z), zkl). (8.76)CThis theorem permits rapid evaluation of integrals on closed paths, whenever itis possible to compute the coefficient a_, of the Laurent expansion at each singularityinside the path. Various techniques for obtaining the Laurent expansion are illustratedin the problems preceding this section. However, if we wish only the term in (z -zo)-'of the expansion, various simplifications are possible. We give several rules here:RULE IAt a simple pole 7.0 (that is, a pole of first order),Res [ f (z), 201 = lim (z - 20) f (2).2-*ZIIRULE I1 At a pole zo of order N (N = 2,3, . . .),where g(z) = (z - 20)~f (2).
Chapter 8 Functions of a Complex Variable 577RULE I11 If A(z) and B(z) are analytic in a neighborhood of zo, A(zo) # 0, and -B(z) has a zero at zo of order 1, thenhas a pole of first order at zo, andt:
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Differential Calculusof Functionsof
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Vector Integral CalculusPART I.TWO-
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Chapter 5 Vector Integral Calculus
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Infinite Series6.1 INTRODUCTIONAn i
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Fourier Series andOrthogonal Functi
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