Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSr = − exp iθ. Therefore, on the the circle of convergence we have1P (z) =1+expiθ .Unless θ = π this is a finite complex number, and so P (z) converges at all points on thecircle |z| =2exceptatθ = π (i.e. z = −2), where it diverges. Note that P (z) isjustthebinomial expansion of (1 + z/2) −1 , for which it is obvious that z = −2 is a singular point.In general, for power series expansions of complex functions about a given point in thecomplex plane, the circle of convergence extends as far as the nearest singular point. Thisis discussed further in chapter 24. ◭Note that the centre of the circle of convergence does not necessarily lie at theorigin. For example, applying the ratio test to the complex power seriesP (z) =1+ z − 12+(z − 1)24+(z − 1)38+ ··· ,we find that for it to converge we require |(z − 1)/2| < 1. Thus the series convergesfor z lying within a circle of radius 2 centred on the point (1, 0) in the Arganddiagram.4.5.2 Operations with power seriesThe following rules are useful when manipulating power series; they apply topower series in a real or complex variable.(i) If two power series P (x) andQ(x) have regions of convergence that overlapto some extent then the series produced by taking the sum, the difference or theproduct of P (x) andQ(x) converges in the common region.(ii) If two power series P (x) andQ(x) converge for all values of x then oneseries may be substituted into the other to give a third series, which also convergesfor all values of x. For example, consider the power series expansions of sin x ande x given below in subsection 4.6.3,sin x = x − x33! + x55! − x77! + ···e x =1+x + x22! + x33! + x44! + ··· ,both of which converge for all values of x. Substituting the series for sin x intothat for e x we obtaine sin x =1+x + x22! − 3x44! − 8x55! + ··· ,which also converges for all values of x.If, however, either of the power series P (x) andQ(x) has only a limited regionof convergence, or if they both do so, then further care must be taken whensubstituting one series into the other. For example, suppose Q(x) converges forall x, but P (x) onlyconvergesforx within a finite range. We may substitute134