# Complex analysis and Taylor development

• Jan 5th 2009, 10:03 AM
asi123
Complex analysis and Taylor development
Hey guys.
I need to prove this equation using Taylor development.
It's pretty obvious that if I'm only using the first part of the Taylor development (the part I marked in red) I can prove it, but my question is, is that enough? can I do something like that?

Thanks a lot.
• Jan 5th 2009, 12:07 PM
ThePerfectHacker
If $\gamma$ is a contour containing $ia\not = 0$ then:
$\frac{1}{2\pi i} \oint_{\gamma} \frac{e^{iz}}{z-ia} dz = f ' (ia)$ where $f(z) = e^{iz}$ by Cauchy's theorem.
Therefore,
$\frac{1}{2\pi i} \oint_{\gamma} \frac{e^{iz}}{z-ia}dz = i e^{-a} \implies \frac{1}{2a i}\oint_{\gamma} \frac{e^{iz}}{z-ia} dz = \frac{\pi i e^{-a}}{a}$
Thus, your equation seems to be wrong.