During my research I came across a coupeld, complex double integral of the following form:
$$ f(k)=\int_{-L}^{+L} \int_{-l}^{+l} \exp \Big\{-j \frac{\pi}{2} \Big[ (y-x)^2-(x-k)^2 \Big] \Big\} \, dx \, dy $$
I am looking for a closed expression in terms of fundamental mathematical expressions/elemantary functions.
My approach so far is the substititon: $$\phi=\phi(y) =(y-x) ,\\ \phi_{u\backslash d}=(\pm a-x) \\ $$
resulting in the following expression:
$$ \int_{-L}^{+L} \int_{\phi_d}^{\phi_u} \exp \Big\{-j \frac{\pi}{2} \Big[\phi^2-(x-k)^2 \Big] \Big\} \, d\phi \, dx $$
which in turn results in:
$$ \int_{-A}^{+A} \exp \Big\{(x-k)^2 \Big\} \cdot \big[C(\phi)-j S(\phi)\big]_{\phi_d}^{\phi_u} \, dx ,\\ \tag1 $$ in which $C(\phi)$ and $S(\phi)$ are the Fresnel Integrals. From here it is getting more complicated trough the coupeld nature of the integrals. A further substitution and therefore a change of the limits $\phi_{u\backslash d}$ with regards to $x$ leads to an integration over the Fresnel and the exponential terms. So far I couldt not find any closed expression.
Any hint or adivce to achieve a closed expression in terms of fundamental mathematical expressions is appreciated.