Calculating an integral without using the Cauchy Integral Formula

Question

I'm trying to solve an exercise in Stein complex analysis book, and totally stuck up right now. enter image description here

I can't find a pdf of this book..so I take the photo. The problem requires me to show the equality 'without' using the Cauchy Integral Formula. I at first thought it would be an easy task...but the calculation seems to be quite nontrivial. Some complicated exponential terms come out on the denominator..... Could anyone help me how to solve this problem?

This is the chapter one of Stein's complex analysis book. I am not allowed to use the Cauchy theorem. — Keith, Jan 02 '15 at 14:34
I think I have to solve this problem by just direct calculation of line integral. — Keith, Jan 02 '15 at 14:34

Tim Raczkowski · Answer 1 · 2015-01-02T14:53:48.837

2

Split apart the integrand by partial fractions, and parameterize $\gamma$ as $re^{it}$ with $0<t<2\pi$.

edited Jan 02 '15 at 14:53

answered Jan 02 '15 at 14:36

Tim Raczkowski

20,046

Wouldn't you parametrize $\gamma$ as $\color{red}{r}e^{it}$? – kobe Jan 02 '15 at 14:52
Yes, my mistake. I'll edit it. – Tim Raczkowski Jan 02 '15 at 14:53

Calculating an integral without using the Cauchy Integral Formula

1 Answers1