Let us use 1hc 1hcorcm 10 cvalua1c 1hc imcgral I 4:. 5 I

Chapter 0, Problem 6.7

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Let us use 1hc 1hcorcm 10 cvalua1c 1hc imcgral I 4:. 5 I ---{:. .c:.(:.-1) \llAP. 6 where C is !he cirdc 1:.1 = 2. described in the coumerclockwise direction (Fig. 94). The incegmnd has the l\\'O isolated singularities:. = 0 and :. = 1. both of which arc interior to C. The corresponding residues 81 at:. = 0 and 82 at:. = I arc readily found with the aid of 1he Maclaurin series represcnrncion (Sec. 64) I ) I _ :. = l +:. + :.- + ( 1:.1 < I). We observe first chat when 0 < 1:.1 < I. 4:. - 5 4:. - 5 - I ( 5) ) _ _ = _ --- = 4 - :- (-I - :. - :.- - ): .. ( .. - I ) .. I - .. .. and by idcncifying the coefficient of I/:. in chc produce on the right here. we find that (.J) Also. since 4:. - 5 :.(:. - I) 4(:. - I) - l :. - I I + (:. - I ) = (4 - ~) [I - (:. - l) + (:. - I ) 2 - . J .. l when 0 < I:. - 11 < l. ic follows chat (4) Thus (5) 82 = -1. .. - - - ") . - . 1 4- s - - - d .. - _Jrt ( 8 I + 8 2) - 8 ;r I. r .. (,, I ) \' .,,,,.- - ......