Mathematical Methods for Physics and Engineering - Matematica.NET

24.14 EXERCISESWe have seen that ∫ and ∫ vanish, and if we denote z by x along the line AB then itΓ γhas the value z = x exp 2πi along the line DC (note that exp 2πi must not be set equal to1 until after the substitution for z has been made in ∫ ). Substituting these expressions,DCThusand∫ ∞0∫dx0(x + a) 3 x + 1/2∞dx[x exp 2πi + a] 3 x 1/2 exp( 1 2(1 − 1 ) ∫ ∞dx 3π=exp πi 0 (x + a) 3 x1/2 4a 5/2I = 1 2 ×3π4a 5/2 . ◭2πi)=3π4a 5/2 .Several other examples of integrals of multivalued functions around a varietyof contours are included in the exercises that follow.24.14 Exercises24.1 Find an analytic function of z = x + iy whose imaginary part is(y cos y + x sin y)expx.24.2 Find a function f(z), analytic in a suitable part of the Argand diagram, for whichsin 2xRe f =cosh 2y − cos 2x .Wherearethesingularitiesoff(z)?24.3 Find the radii of convergence of the following Taylor series:∞∑ z n(a)ln n , (b) ∑ ∞n!z nn , n n=2n=1∞∑∞∑( ) n 2n + p(c) z n n ln n , (d)z n , with p real.nn=1n=124.4 Find the Taylor series expansion about the origin of the function f(z) defined by∞∑ ( pz)f(z) = (−1) r+1 sin ,rr=1where p is a constant. Hence verify that f(z) is a convergent series for all z.24.5 Determine the types of singularities (if any) possessed by the following functionsat z =0andz = ∞:(a) (z − 2) −1 , (b) (1 + z 3 )/z 2 , (c) sinh(1/z),(d) e z /z 3 , (e) z 1/2 /(1 + z 2 ) 1/2 .24.6 Identify the zeros, poles and essential singularities of the following functions:(a) tan z, (b) [(z − 2)/z 2 ] sin[1/(1 − z)], (c) exp(1/z),(d) tan(1/z), (e) z 2/3 .867