About Fourier transform and complex conjugate

Question

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of $\mathscr{F}[h(\tau)]=H(f)$.

Thanks!

Answer 1

It is just a change of variable: $$ \int_{-\infty}^\infty h(-\tau)\,e^{i\tau t}\,d\tau=\int_{-\infty}^\infty h(\tau)\,e^{-i\tau t}\,d\tau=\int_{-\infty}^\infty h(\tau)\,\overline{e^{i\tau t}}\,d\tau=\overline{\int_{-\infty}^\infty h(\tau)\,e^{i\tau t}\,d\tau} $$ because $h$ is real valued.