# Complex analysis, Cauchy's integral theorem

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• Aug 20th 2011, 11:45 PM
olski1
Complex analysis, Cauchy's integral theorem
Use cauchy's integral formula to evaluate
$\oint \frac{e^{sin(z)}}{z^2(z-\frac{\pi}{4})}$

over the curve,
$c:=|z|=\frac{\pi}{8})$

Okay, so i used the method shown in lectures,

Since the only singularity inside C is for z=0,

we let ,

$f(z)=\frac{e^{sin(z)}}{(z-\frac{\pi}{4})}$

hence, by cauchy's formula, z=0

$(2\pi i).\frac{e^{sin(0)}}{(0-\frac{\pi}{4})}= -8i$

But the answer says -32/pi*(1+pi/4)i

Just starting this topic, so still trying to wrap my head around it.

Thanks in advanced
• Aug 21st 2011, 12:06 AM
FernandoRevilla
Re: Complex analysis, Cauchy's integral theorem
Quote:

Originally Posted by olski1
Since the only singularity inside C is for z=0, we let , $f(z)=\frac{e^{sin(z)}}{(z-\frac{\pi}{4})}$ hence, by cauchy's formula, z=0 $(2\pi i).\frac{e^{sin(0)}}{(0-\frac{\pi}{4})}= -8i$

It should be $2\pi i f'(0)$ , not $2\pi i f(0)$ .