Fourier series for $e^x$

Question

I'm trying to teach myself partial differential equations from Strauss' book. I have run into a very bizarre problem - I cannot figure out what is the Fourier series of $e^x$! And not even Google has helped.

The book's general formula is with .

The book's answer for the $e^x$ fourier series is

But I derived this by hand (and with Mathematica) independently:

So what gives? Did Mathematica and I both fail to do a simple straightforward computation? Or did the book just pull something out of a hat?

In addition, I cannot figure out how to transfer from a complex Fourier series to a real one. I assume one cannot just take the real part?

Answer 1

$$\sinh(l-i n \pi)=\tfrac12(e^{l-i n \pi}-e^{-l+i n \pi})=(-1)^n\sinh(l)$$

since $e^{i\pi}=-1$, and

$$\frac1{l-i n \pi}=\frac{l+i n \pi}{(l-i n \pi)(l+i n \pi)}=\frac{l+i n \pi}{l^2+n^2 \pi^2}$$

so all your quoted expressions are actually equal.