Let $$G_\beta(w) = e^{\beta w^2}$$
Now I get the process of applying a fourier transform (or inverse) to get a new gaussian:
$$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the derivation the constant $G_\beta(0)$ never arose. I know how to calculate the constant, but where did it come from in the derivation?
edit: Fourier form used: $$\int_{-\infty}^{\infty} f(w) e^{-iwx} dw$$
I guess $\beta>0 $ to have $G_{\beta}\left(\omega\right)=e^{-\beta\omega^{2}} $ integrable. The fourier transform $\widehat{G_{\beta}} $ has the following property :$$-ix\widehat{G_{\beta}}\left(x\right)=\widehat{\frac{d}{d\omega}G_{\beta}}\left(x\right). $$ Computing the derivative and using the above property again yields to an ODE :$$-ix\widehat{G_{\beta}}\left(x\right)=\widehat{-2\beta\omega G_{\beta}}\left(x\right)=-2\beta\left(-i\frac{d}{dx}\right)\widehat{G_{\beta}}\left(x\right)=2i\beta\frac{d}{dx}\widehat{G_{\beta}}\left(x\right). $$ The general solution is then $\widehat{G_{\beta}}\left(x\right)=Ce^{-\frac{x^{2}}{4\beta}}$ and $C$ is found by taking $x=0$ for example.Here, the constant appears in the solution of a Cauchy' problem for an ODE.