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# HW8 - MATH 185 SOLUTIONS OF HW VIII(1(48 1(each 8pts All...

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MATH 185 SOLUTIONS OF HW VIII (1) (48. 1)(each 8pts) All problems use the Cauchy integral formula.(so, after I’ll denote it by C.I.F.) (a) Let f ( z ) = e - z then, by the C.I.F, the integral is equal to 2 πif ( - πi/ 2) = 2 π since f ( z ) is analytic in the interior of C(and on C) and πi/ 2 lies in the interior of C. (b) Let f ( z ) = cos z z 2 +8 then, by the C.I.F, the integral is equal to 2 πif (0) = πi/ 4 since f ( z ) is analytic in the interior of C(and on C) and 0 lies in the interior of C. (c) Let f ( z ) = z/ 2 then, by, the C.I.F, the integral is equal to 2 πif ( - 1 / 2) = - πi/ 2 since f ( z ) is analytic in the interior of C(and on C) and - 1 / 2 lies in the interior of C. (d) Let f ( z ) = cosh z then, by the C.I.F, the integral is equal to 2 πif (3) (0) / 3! = 0 since f ( z ) is analytic in the interior of C(and on C) and 0 lies in the in- terior of C. (e) Let f ( z ) = tan( z/ 2) then, by the C.I.F, the integral is equal to 2 πif ( x 0 ) = πi sec 2 ( x 0 / 2) since f ( z ) is analytic in the interior of C(and on C) and x 0 lies in the interior of C.

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