Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESyRz 0C 1C 2xFigure 24.12 The region of convergence R for a Laurent series of f(z) abouta point z = z 0 where f(z) has a singularity.classify the nature of that point. If f(z) is actually analytic at z = z 0 ,thenin(24.55) all a n for n0;(ii) it is not possible to find such a lowest value of −p.In case (i), f(z) is of the form (24.54) and is described as having a pole of orderp at z = z 0 ; the value of a −1 (not a −p ) is called the residue of f(z) at the polez = z 0 , and will play an important part in later applications.For case (ii), in which the negatively decreasing powers of z − z 0 do notterminate, f(z) is said to have an essential singularity. These definitions should becompared with those given in section 24.6.◮Find the Laurent series off(z) =1z(z − 2) 3about the singularities z =0and z =2(separately). Hence verify that z =0is a pole oforder 1 and z =2is a pole of order 3, and find the residue of f(z) at each pole.To obtain the Laurent series about z = 0, we make the factor in parentheses in the856