\tilde{f}(z, 0)=\int_{-\infty}^{\infty} \tilde{A}(k) e^{i k z} d k \Rightarrow \tilde{f}(z, 0)^{*}=\int_{-\infty}^{\infty} \tilde{A}(k)^{*} e^{-i k z} d k . \text { Let } l \equiv-k ; \text { then } \tilde{f}(z, 0)^{*}=
\int_{\infty}^{-\infty} \tilde{A}(-l)^{*} e^{i l z}(-d l)=\int_{-\infty}^{\infty} \tilde{A}(-l)^{*} e^{i l z} d l=\int_{-\infty}^{\infty} \tilde{A}(-k)^{*} e^{i k z} d k \text { (renaming the dummy variable } l \rightarrow k ).
f(z, 0)=\operatorname{Re}[\tilde{f}(z, 0)]=\frac{1}{2}\left[\tilde{f}(z, 0)+\tilde{f}(z, 0)^{*}\right]=\int_{-\infty}^{\infty} \frac{1}{2}\left[\tilde{A}(k)+\tilde{A}(-k)^{*}\right] e^{i k z} d k . Therefore
\frac{1}{2}\left[\tilde{A}(k)+\tilde{A}(-k)^{*}\right]=\frac{1}{2 \pi} \int_{-\infty}^{\infty} f(z, 0) e^{-i k z} d z.
\text { Meanwhile, } \dot{\tilde{f}}(z, t)=\int_{-\infty}^{\infty} \tilde{A}(k)(-i \omega) e^{i(k z-\omega t)} d k \Rightarrow \dot{\tilde{f}}(z, 0)=\int_{-\infty}^{\infty}[-i \omega \tilde{A}(k)] e^{i k z} d k.
(Note that \omega=|k| v, here, so it does not come outside the integral.)
\dot{\tilde{f}}(z, 0)^{*}=\int_{-\infty}^{\infty}\left[i \omega \tilde{A}(k)^{*}\right] e^{-i k z} d k=\int_{-\infty}^{\infty}\left[i|k| v \tilde{A}(k)^{*}\right] e^{-i k z} d k=\int_{\infty}^{-\infty}\left[i|l| v \tilde{A}(-l)^{*}\right] e^{i l z}(-d l)
=\int_{-\infty}^{\infty}\left[i|k| v \tilde{A}(-k)^{*}\right] e^{i k z} d k=\int_{-\infty}^{\infty}\left[i \omega \tilde{A}(-k)^{*}\right] e^{i k z} d k .
\dot{f}(z, 0)=\operatorname{Re}[\dot{\tilde{f}}(z, 0)]=\frac{1}{2}\left[\dot{\tilde{f}}(z, 0)+\dot{\tilde{f}}(z, 0)^{*}\right]=\int_{-\infty}^{\infty} \frac{1}{2}\left[-i \omega \tilde{A}(k)+i \omega \tilde{A}(-k)^{*}\right] e^{i k z} d k .
\frac{-i \omega}{2}\left[\tilde{A}(k)-\tilde{A}(-k)^{*}\right]=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \dot{f}(z, 0) e^{-i k z} d z, \text { or } \frac{1}{2}\left[\tilde{A}(k)-\tilde{A}(-k)^{*}\right]=\frac{1}{2 \pi} \int_{-\infty}^{\infty}\left[\frac{i}{\omega} \dot{f}(z, 0)\right] e^{-i k z} d z.
Adding these two results, we get \tilde{A}(k)=\frac{1}{2 \pi} \int_{-\infty}^{\infty}\left[f(z, 0)+\frac{i}{\omega} \dot{f}(z, 0)\right] e^{-i k z} d z . qed