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?