I am supposed to find the imaginary part of the analytic complex function, which has real part:
$$e^{y^2-x^2}\cos(2xy)$$
and $z=x+iy$. I know that the answer is $e^{-z^2}+ic$. This can be verified using Cuuchy-Riemann equations:
$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\\ \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x} $$
There is a much shorter way to arrice to the answer than just applying CR, I believe. I remember we considered derivatives with respect to $z$ and $z^*$, but I don't remember the whole method, but it was a huge shortcut compared to CRs in the form above.
How can I arrive to the imaginary part of the analytical complex function with real part $$e^{y^2-x^2}\cos(2xy)$$, without much algebraic juggling?