How do you integrate around each singularity separately in complex analysis?

stripes · Apr 15, 2014

Homework Statement

$Typeset problem: Prove (1/(2πi)) times the contour integral over C of e^(zt)/(z^2+1) dz equals sin t, where t>0 and C is |z|=3. Asks to do it two ways: (i) partial fractions of 1/(z^2+1), (ii) integrating around each singularity separately.$

Homework Equations

The Attempt at a Solution

I did (i) by breaking the integrand into partial fractions and then using the Cauchy Integral Formula for each integral. I got the correct answer.

What does (ii) even mean? WHat does it mean to integrate "around each singularity separately"? The singularities are i and -i but I figure the only way to do this is partial fractions. Any help would be great.

stripes · Apr 16, 2014

Alright, I got it. In case anyone ever comes across this:

All I did was break it down into two smaller contours, each one containing one of the singularities but not the other. Then the integral over the entire C is equal to the sum of two integrals; one over C1 (containing one singularity) and the other over C2 (containing the other singularity). Each integrand is the same, but you're just taking one integral over C1, plus the other integral over C2. Then apply the Cauchy integral formula to each integral and you end up with the exponential form of sin(t).

How do you integrate around each singularity separately in complex analysis?

Homework Statement

Homework Equations

The Attempt at a Solution

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