Page 427 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 427

418                     FUNCTIONS OF A COMPLEX VARIABLE                   [CHAP. 16


                                       z
                                     þ  p 1
                              Consider     dz.Since z ¼ 0isa branch point, choose C as the contour of Fig. 16-14 where AB
                                      C 1 þ z
                          and GH are actually coincident with x-axis but are shown separated for visual purposes.
                              The integrand has the pole z ¼ 1lying within C.
                              Residue at z ¼ 1 ¼ e  i  is
                                           z  p 1   i p 1  ðp 1Þ i
                                   lim ðz þ 1Þ  ¼ðe Þ  ¼ e
                                  z! 1    1 þ z
                                      þ   p 1
                                         z
                              Then           dz ¼ 2 ie ðp 1Þ i
                                       C 1 þ z
                          or, omitting the integrand,
                                   ð    ð    ð   ð
                                                   ¼ 2 ie ðp 1Þ i
                                     þ    þ   þ
                                   AB  BDEFG  GH  HJA
                              We thus have
                               x         ðRe Þ  iRe d    ðxe  Þ  dx
                             ð  R  p 1  ð 2   i  p 1  i   ð r  2 i p 1
                             r 1 þ x  dx þ  0  1 þ Re i   þ  R 1 þ xe 2 i
                                        i
                              ð 0  i  p 1 ire d                                    Fig. 16-14
                                 ðre Þ           ðp 1Þ i
                             þ          i   ¼ 2 ie
                               2   1 þ re
                          where we have to use z ¼ xe 2 i  for the integral along GH, since the argument of z is increased by 2  in going
                          round the circle BDEFG.
                              Taking the limit as r ! 0 and R !1 and noting that the second and fourth integrals approach zero,
                          we find
                                                 ð  p 1    ð 0  2 iðp 1Þ p 1
                                                   x         e     x
                                                 1
                                                                      dx ¼ 2  e ðp 1Þ i
                                                 0 1 þ x    1   1 þ x
                                                       dx þ
                                                             ð  p 1
                                                             1  x
                          or                       ð1   e 2 iðp 1Þ  Þ  dx ¼ 2 ie ðp 1Þ i
                                                             0 1 þ x
                          so that
                                               ð   p 1        ðp 1Þ i
                                                1  x      2 ie         2 i
                                                0 1 þ x  dx ¼  1   e 2 iðp 1Þ  ¼  e p i    e  p i  ¼  sin p





                                                Supplementary Problems

                     FUNCTIONS, LIMITS, CONTINUITY
                     16.39. Describe the locus represented by (a) jz þ 2   3ij¼ 5;  ðbÞjz þ 2j¼ 2jz   1j;  ðcÞjz þ 5j jz   5j¼ 6.
                          Construct a figure in each case.
                                                    2
                                            2
                          Ans. ðaÞ Circle ðx þ 2Þ þð y   3Þ ¼ 25, center ð 2; 3Þ,radius 5.
                                            2
                                               2
                               (b)Circle ðx   2Þ þ y ¼ 4, center ð2; 0Þ,radius 2.
                                                        2
                                                  2
                               (c)Branch of hyperbola x =9   y =16 ¼ 1, where x A 3.
                     16.40. Determine the region in the z plane represented by each of the following:

                          (a) jz   2 þ ij A 4;  ðbÞjzj @ 3; 0 @ arg z @  ;  ðcÞjz   3jþjz þ 3j < 10.
                                                             4
                          Construct a figure in each case.
                                                                      2
                                                              2
                          Ans.(a) Boundary and exterior of circle ðx   2Þ þð y þ 1Þ ¼ 16.
   422   423   424   425   426   427   428   429   430   431   432