Confusion over real functions as a complex Fourier series

Question

I am a bit confused with the following, I mean I do understand what these notes say, but does that mean that for real functions that the sum is only from 0 to infinity rather than negative infinity to positive? Are terms doubling up? And do complex conjugates need to be taken?

Answer 1

He is just breaking 3.22 into two sums in the following way $$ S(x)=\frac{a_0}{2}+ \sum_{n=1}^\infty \frac{a_n}{2}\left(e^{\frac{in\pi x}{l}}+e^{\frac{-in\pi x}{l}} \right)+\frac{-ib_n}{2}\left(e^{\frac{in\pi x}{l}}-e^{\frac{-in\pi x}{l}} \right) $$ $$ S(x)=\frac{a_0}{2}+ \sum_{n=1}^\infty \left(\frac{a_n}{2}-\frac{ib_n}{2} \right)e^{\frac{in\pi x}{l}}+\sum_{n=1}^\infty \left(\frac{a_n}{2}+\frac{ib_n}{2} \right)e^{-\frac{in\pi x}{l}}. $$ Now, on the right most sum let $n \to -n$ and you have $$ S(x)=\frac{a_0}{2}+ \sum_{n=1}^\infty \left(\frac{a_n}{2}-\frac{ib_n}{2} \right)e^{\frac{in\pi x}{l}}+\sum_{n=-1}^{-\infty} \left(\frac{a_{-n}}{2}+\frac{ib_{-n}}{2} \right)e^{\frac{in\pi x}{l}}. $$ Here I think there is a typo in your book and the $c_n$ should be defined as follows $$ \begin{cases} c_0=\frac{a_0}{2}, \\ c_n=\frac{a_n}{2}-\frac{ib_n}{2}, \ n>0,\\ c_n=\frac{a_{-n}}{2}-\frac{ib_{-n}}{2}, n<0. \end{cases} $$ Notice that as he defined you would have to know the values $a_n, b_n$ for negative $n$ and in fact he defined $c_n$ in two different ways for positive values of $n$. Which makes no sense. But using the definition I give above notice that $$ S(x)=c_0+ \sum_{n=1}^\infty c_n e^{\frac{in\pi x}{l}}+\sum_{n=-1}^{-\infty} c_n e^{\frac{in\pi x}{l}}=\sum_{n=-\infty}^\infty c_n e^{\frac{in\pi x}{l}}. $$