I am reading a textbook solution on the $1D$ wave PDE with a source $q(x,t)$,
$$\left(\frac 1{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)\varphi(x,t)=q(x,t)$$
We solve this equation by a finding a Green's function such that
$$\left(\frac 1{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)G(x,t;\xi,\tau)=\delta(x-\xi)\delta(t-\tau)$$
If the only waves in the systems are produced by the source, we should demand the Green's function to be causal, in that $G(x,t;\xi,\tau)=0$ if $t<\tau$.
To construct the causal Green's function, we integrate the equation over an infinitesmal time interval from $\tau-\varepsilon$ to $\tau+\varepsilon$ and so find the Cauchy data.
$$G(x,\tau+\varepsilon;\xi,\tau)=0$$
$$\frac d{dt}G(x,\tau+\varepsilon;\xi,\tau)=c^2\delta(x-\xi)$$
The above approach is taken from the textbook mathematics for physics page 221.
This construction is more intuitive in the sense that it does not use contour integral or Fourier transform. However, the solution does not explain how Cauchy data is derived. I am thinking of the following. I am not sure whether I am correct.
$$\int_{\tau-\varepsilon}^{\tau+\varepsilon}\frac 1{c^2}\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)G(x,t;\xi,\tau)dt=\int_{\tau-\varepsilon}^{\tau+\varepsilon}\delta(x-\xi)\delta(t-\tau)dt$$
$$\frac 1{c^2}\frac {\partial}{\partial t}G(x,t;\xi,\tau)\bigg\rvert_{\tau-\varepsilon}^{\tau+\varepsilon}+\frac{d^2}{dx^2}\int_{\tau-\varepsilon}^{\tau+\varepsilon}G(x,t;\xi,\tau)dt=\delta(x-\xi)$$ On the LHS the second term is $0$ as $\varepsilon\to0$ since $G$ is continuous. The first term evaluates to $0$ at $t=\tau-\varepsilon$ by virtue of causality. So we have the second Cauchy data. Is my argument correct? For the first Cauchy data, I believe $$\int_{\tau-\varepsilon}^{\tau+\varepsilon}dt\int^t \frac 1{c^2}\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)G(x,t;\xi,\tau)dt=\int_{\tau-\varepsilon}^{\tau+\varepsilon}dt\int^t\delta(x-\xi)\delta(t-\tau)dt$$ Am I on the right track?