Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESy Cy yrθxrθxxC ′(a) (b) (c)Figure 24.1 (a) A closed contour not enclosing the origin; (b) a closedcontour enclosing the origin; (c) a possible branch cut for f(z) =z 1/2 .circuit around any one branch point, and so the function in question remainssingle-valued.For the function f(z) =z 1/2 , we may take as a branch cut any curve startingat the origin z = 0 and extending out to |z| = ∞ in any direction, since allsuch curves would equally well prevent us from making a closed loop aroundthe branch point at the origin. It is usual, however, to take the cut along eitherthe real or the imaginary axis. For example, in figure 24.1(c), we take the cut asthe positive real axis. By agreeing not to cross this cut, we restrict θ to lie in therange 0 ≤ θ