# Complex power series to calculate Fourier series

terhorst
I never took complex analysis in undergrad and always regretted it, so I'm working through the book Visual Complex Analysis on my own. Really enjoying it so far.

## Homework Statement

Actually you can view the problem http://books.google.de/books?id=ogz5FjmiqlQC&printsec=frontcover&hl=en#PPA117,M1", it's #21 at the top of the page.

Show that the Fourier series for $$[\cos(\sin \theta)]e^{\cos \theta}$$ is $$\sum^{\infty}_{n=0} \frac{\cos n \theta}{n!}$$.

## Homework Equations

The author suggests substituting $$z=re^{i \theta}$$ into the power series for $$e^z$$ and then isolating the real and imaginary parts.

## The Attempt at a Solution

I don't really understand why the substitution. So then you end up with $$\sum \frac{r^n [\cos(n \theta) + i \sin (n \theta)]}{n !}$$. But I don't see how this gets me any closer to equating $$[\cos(\sin \theta)]e^{\cos \theta}$$ and $$\sum \frac{r^n \cos(n \theta)}{n!}$$.

A nudge in the right direction would be appreciated. Thanks for looking.

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Homework Helper
Show that the Fourier series for $$[\cos(\sin \theta)]e^{\cos \theta}$$ is $$\sum^{\infty}_{n=0} \frac{\cos n \theta}{n!}$$.

The author suggests substituting $$z=re^{i \theta}$$ into the power series for $$e^z$$ and then isolating the real and imaginary parts.

Hi terhorst!

(have a theta: θ )

Hints:

i] use r = 1

ii] what is cos(eisinθ) ?