Consider the two simple closed contours shown in Fig. 111

Chapter 0, Problem 7.33

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Consider the two simple closed contours shown in Fig. 111 and obtained by dividing into two pieces the annul us formed by the circles C,, and CR in Fig. 110 (Sec. 91 ). The legs L and -L of those contours arc directed Ii ne segments along any ray arg::. = l.~1. where ;r < Oo < 3;r /2. Also. r,, and )',-, arc the indicated portions of C,,. while r R and YR make up CR. \' \' I I I I I I I I R R .\ .\ FIGURE Ill (ii) Show how it follows from Cauchy's residue theorem that when the branch --" /1(::.) = -~ ::.+I ( ;r 3;r)of the multiple-valued function:-"/(::.+ I) is integrated around the closed contouron the left in Fig. I 11.!R .r~" c/r + ~ fi(::.) c/::. + r fi(::_) d::. + ~ f1 (::_) d::. = 2;ri __ l~~s1 fi(::.) . . , / . I .Ir i; ./ L .fr ..(h) Apply the Cauchy-Goursat theorem to the branch.... -ii(;r 5;r) 1::.1 > 0. 2 < arg::. < 2 ';(::_) = -~-' - ::. +Iof::.-"/ (::. + I). integrated around the closL'CI contour on the right in Fig. 11 I. to showthat_ !R ,.-ot'-i"2.t.1.7 f, ! f ---dr + .12. . I. . Y>:(c) Point out why. in the last lines in parts (a) and (h). the branches / 1 (::.)and _h(':.) of::. -a/ (::. + I ) can he replaced hy the branch_-a((::.) = -~-. ::. +I (1::.1 > o.o < arg: < 2;r).Then. by adding corresponding sides of those two Ii ncs. derive equation (~). Sec. l) I.which was obtained only formally there.