Find complex Fourier coefficients of $f(-x), f^*(x)$

Question

For $f(-x)$ i have tried to replace the $k$ with $k'=-k$ but still i can't find any relationship between the coefficients. What could be a better way to approach this problem?

Answer 1

Hint: change of variable $t=-x$.

Answer 2

You have

$$f(-x)=\sum_{k=-\infty}^{\infty}c_ke^{-ikx}=\sum_{k=-\infty}^{\infty}c_{-k}e^{ikx}\tag{1}$$

So the Fourier coefficients of $f(-x)$ are $c_{-k}$.

For $\bar{f}(x)$ you get

$$\bar{f}(x)=\sum_{k=-\infty}^{\infty}\bar{c}_ke^{-ikx}=\sum_{k=-\infty}^{\infty}\bar{c}_{-k}e^{ikx}\tag{2}$$

From (2) the Fourier coefficients of $\bar{f}(x)$ are $\bar{c}_{-k}$.