I have to solve an equation but I don't know if it's analytic or not, but I suppose it is.
It would be easy if it was $e^z$ without the exponent because I would only need to transform it to $e^x(\cos y+i\sin y)$ but this one has an exponent of $2$.
If I solve the polynomial it would be $e^{(x^2-y^2+2ixy)}$ and if I use the function exponential of complex number like the previous example, it would be $e^{(x^2-y^2)}(\cos 2xy+i\sin 2xy)$ and when I derive it, they won't be the same ($U_x$ is not $V_y$, $U_y$ is not $-V_x$).
As I say, I don't know if it's analytic or not.
Thanks for the answer.
Upon careful scrutiny it is revealed that Cauchy-Riemann does in fact hold for the function $e^{z^2}$; with
$z = x + iy, \tag 0$
we have
$z^2 = x^2 - y^2 + 2ixy, \tag{0.1}$
as is in fact both easily calculated and well-known. Then
$e^{z^2} = e^{x^2 - y^2 + 2ixy} = e^{x^2 - y^2}(\cos 2xy + i \sin 2xy), \tag 1$
and thus
$U = e^{x^2 - y^2}\cos 2xy, \tag 2$
$V = e^{x^2 - y^2}\sin 2xy; \tag 3$
we differentiate:
$U_x = 2xe^{x^2 - y^2}\cos 2xy - 2ye^{x^2 - y^2}\sin 2xy, \tag 4$
$U_y = -2ye^{x^2 - y^2}\cos 2xy - 2x e^{x^2 - y^2}\sin 2xy, \tag 5$
$V_x = 2x e^{x^2 - y^2}\sin 2xy + 2y e^{x^2 - y^2}\cos 2xy; \tag 6$
$V_y = -2y e^{x^2 - y^2}\sin 2xy + 2x e^{x^2 - y^2}\cos 2xy. \tag 7$
Inspection of (4)-(7) shows that
$U_x = V_y, \; U_y = -V_x, \tag 8$
as we would expect since $e^{z^2}$ is the composition of the two holomorphic functions $z^2$ and $\exp(\cdot)$, hence itself holomorphic.