I am reading about the Fourier series.
I understood the Fourier Series in terms of breaking down a function into a constant and a weighted series of sines and cosines.
I then read on in my text and it looked at the complex representation of a function. I am having difficulty understanding the indexes of the complex coefficients.
Could someone just check my understanding of the following from my text?
The highlighted line is what I am uncertain about (C^-n = C^*n)$
I know what a complex conjugate is, e.g. The complex number a + ib has the complex conjugate a -ib.
Is the meaning of the -n subscript just referring to the symmetry of the function? So if I have four elements in a Fourier series: -n2, -n, n, and n2, the complex value of term -n2 has to equal the complex conjugate of n2?
Thank you!
What that textbook is claiming is that, if$$f(x)=\sum_{n=-\infty}^\infty c_ne^{inx}$$and $f$ is a real function (that is, for each $x$, $f(x)\in\Bbb R$), then$$(\forall n\in\Bbb Z):c_{-n}=\overline{c_n}.$$That's so because, for each $x$\begin{align}\sum_{n=-\infty}^\infty c_ne^{inx}&=f(x)\\&=\overline{f(x)}\text{ (since $f(x)\in\Bbb R$)}\\&=\overline{\sum_{n=-\infty}^\infty c_ne^{inx}}\\&=\sum_{n=-\infty}^\infty\overline{c_ne^{inx}}\\&=\sum_{n=-\infty}^\infty\overline{c_n}e^{-inx}\\&=\sum_{n=-\infty}^\infty\overline{c_{-n}}e^{inx}.\end{align}