Suppose we have the complex wave function : \begin{equation*} f(z,t)=Ae^{i\left( kz -\omega t\right)} \end{equation*} So i read in a book that
"Any wave can be expressed as a linear combination of sinusoidal ": \begin{equation*} f(z,t) = \int_{-\infty}^{+\infty} A(k)e^{ikz-i\omega t}dk \end{equation*} But it doesn't provide details or a proof about that . Sorry if this topic is not very relevant to pure mathematics but i want to see a formal proof about that, where can i find it ? I suppose that is related to fourier ( transform).
Thanks in advance.
The key to understanding here is that if $f$ and $g$ are solutions of the wave equation
$$u_{tt}=cu_{xx},$$
then $f(x,t)+g(x,t)$ is also a solution. In physics this is known as superposition principle.
Now, your integral is a Fourier transform of a particular solution. You can see that it only has plane waves in the integrand (the solutions of the form of your first formula), modulated by a $k$-dependent function $A$. This just uses the continuous superposition of plane waves with all $k\in\mathbb R$ taken with different amplitudes given by $A$.
Given nice enough $A$, you can verify that the integral does indeed satisfy the wave equation.
To get back the wave function of
$$f(x,t)=Ce^{i(kx-\omega t)},$$
you can set $A$ in the Fourier integral to
$$A(\kappa)=C\delta(\kappa-k),$$
where $\delta$ is the Dirac delta, a distribution.