Mathematical Methods for Physics and Engineering - Matematica.NET

24.6 SINGULARITIES AND ZEROS OF COMPLEX FUNCTIONSyzyyir 1θ 1r 2iixxx−iθ 2−i−i(a) (b) (c)Figure 24.2 (a) Coordinates used in the analysis of the branch points off(z) =(z 2 +1) 1/2 ; (b) one possible arrangement of branch cuts; (c) anotherpossible branch cut, which is finite.(iii) z = −i but not z = i, thenθ 1 → θ 1 , θ 2 → θ 2 +2π and so f(z) →−f(z);(iv) both branch points, then θ 1 → θ 1 +2π, θ 2 → θ 2 +2π and so f(z) → f(z).Thus, as expected, f(z) changes value around loops containing either z = i or z = −i (butnot both). We must therefore choose branch cuts that prevent us from making a completeloop around either branch point; one suitable choice is shown in figure 24.2(b).For this f(z), however, we have noted that after traversing a loop containing both branchpoints the function returns to its original value. Thus we may choose an alternative, finite,branch cut that allows this possibility but still prevents us from making a complete looparound just one of the points. A suitable cut is shown in figure 24.2(c). ◭24.6 Singularities and zeros of complex functionsA singular point of a complex function f(z) is any point in the Argand diagramat which f(z) fails to be analytic. We have already met one sort of singularity,the branch point, and in this section we will consider other types of singularityas well as discuss the zeros of complex functions.If f(z) has a singular point at z = z 0 but is analytic at all points in someneighbourhood containing z 0 but no other singularities, then z = z 0 is called anisolated singularity. (Clearly, branch points are not isolated singularities.)The most important type of isolated singularity is the pole. Iff(z) has the formf(z) =g(z)(z − z 0 ) n , (24.23)where n is a positive integer, g(z) is analytic at all points in some neighbourhoodcontaining z = z 0 and g(z 0 ) ≠0,thenf(z) has a pole of order n at z = z 0 .Analternative (though equivalent) definition is thatlim [(z − z 0 ) n f(z)] = a, (24.24)z→z 0837