I have a general question in partial differential equations.
Can we say that when an even function is expressed as a Fourier series, the Fourier cosine series is also the Fourier series?
My thinking is that a Fourier series has the form,
$$f(x) = \frac{a_0}{2}+\sum^{\infty}_{n=1} a_n cos(nx) + \sum^\infty_{n=1}b_nsin(nx)$$
where $$a_0 = \frac{1}{\pi}\int ^\infty _{-\infty}f(x)dx$$, $$a_n = \frac{1}{\pi}\int ^\infty _{\infty}f(x)cos(nx)dx$$, $$b_n = \frac{1}{\pi}f(x)sin(nx)dx$$
where $cos$ is a even function and $sin$ is a odd function. Then if $f(x)$ is even, multiplying a even and odd function together gives a odd function which is $0$ which would eliminate the $b_n$ term, leaving just $a_0$ and $a_n$ which is the fourier cosine series. Therefore yes this is true