Mathematical Methods for Physics and Engineering - Matematica.NET

24.11 TAYLOR AND LAURENT SERIESof order p at z = z 0 but is analytic at every other point inside and on C. Thenthe function g(z) =(z − z 0 ) p f(z) is analytic at z = z 0 , and so may be expandedas a Taylor series about z = z 0 :∞∑g(z) = b n (z − z 0 ) n . (24.53)n=0Thus, for all z inside C, f(z) will have a power series representation of the formf(z) =a −p(z − z 0 ) p + ···+ a −1+ a 0 + a 1 (z − z 0 )+a 2 (z − z 0 ) 2 + ··· ,z − z 0(24.54)with a −p ≠ 0. Such a series, which is an extension of the Taylor expansion, iscalled a Laurent series. By comparing the coefficients in (24.53) and (24.54), wesee that a n = b n+p . Now, the coefficients b n in the Taylor expansion of g(z) areseen from (24.52) to be given byb n = g(n) (z 0 )= 1 ∮g(z)dz,n! 2πi (z − z 0 )n+1and so for the coefficients a n in (24.54) we havea n = 12πi∮g(z)1dz =(z − z 0 )n+1+p2πi∮f(z)dz,(z − z 0 )n+1an expression that is valid for both positive and negative n.The terms in the Laurent series with n ≥ 0 are collectively called the analyticpart, whilst the remainder of the series, consisting of terms in inverse powers ofz − z 0 , is called the principal part. Depending on the nature of the point z = z 0 ,the principal part may contain an infinite number of terms, so thatf(z) =+∞∑n=−∞a n (z − z 0 ) n . (24.55)In this case we would expect the principal part to converge only for |(z − z 0 ) −1 |less than some constant, i.e. outside some circle centred on z 0 . However, theanalytic part will converge inside some (different) circle also centred on z 0 .Ifthelatter circle has the greater radius then the Laurent series will converge in theregion R between the two circles (see figure 24.12); otherwise it does not convergeat all.In fact, it may be shown that any function f(z) that is analytic in a regionR between two such circles C 1 and C 2 centred on z = z 0 can be expressed asa Laurent series about z 0 that converges in R. We note that, depending on thenature of the point z = z 0 , the inner circle may be a point (when the principalpart contains only a finite number of terms) and the outer circle may have aninfinite radius.We may use the Laurent series of a function f(z) about any point z = z 0 to855