# Cauchys Integral Formula

• Aug 20th 2009, 11:40 AM
rak
Cauchys Integral Formula
Hi, I am fine with solving simpler questions involving Cauchys Integral Formula, but for a couple here I've got stuck. I think my problem is actually with algebra more than anything else, so if anyone could help me i'd be grateful!

The question asks to use CIF to evaluate the integral (contour) of:

(exp (i pi z)) / (z^2 +4)(z^2 -4)

With the contour gamma(t)= 9i + 8exp(i t), 0<t<4pi.

And similarly for:

(cos (z)) / (z^4 +4)

I know I need to rearrange the integral for it to equal f(z) / z - w, but I simple can't figure out how! (Headbang)

Sorry for the crude notation, and thanks.
• Aug 21st 2009, 03:03 AM
mr fantastic
Quote:

Originally Posted by rak
[snip]
The question asks to use CIF to evaluate the integral (contour) of:

(exp (i pi z)) / (z^2 +4)(z^2 -4)

With the contour gamma(t)= 9i + 8exp(i t), 0<t<4pi.
[snip]

The singularities of $\frac{e^{i \pi z}}{(z^2 + 4)(z^2 - 4)}$ are $z = \pm 2i$ and $z = \pm 2$. The given contour only encloses $z = 2i$.

Therefore you need to calculate $\oint_{\gamma} \frac{f(z)}{z - 2i} \, dz$ where $f(z) = \frac{e^{i \pi z}}{(z + 2i)(z^2 - 4)}$.

Quote:

Originally Posted by rak
The question asks to use CIF to evaluate the integral (contour) of:

(cos (z)) / (z^4 +4)

With the contour gamma(t)= 9i + 8exp(i t), 0<t<4pi.

Does the given contour enclose any of the zeroes of $z^4 + 4$ ....?