Explicit Functions on $\mathbb{C}$ The following question on last years Complex Analysis exam paper, and Im a little stuck on it..
$(i)f(z)=e^{z^2}$ find the explicit formulas for $u(x,y)$ and $v(x,y)$ such that:
$f(x+iy)=u(x,y)+iv(x,y)$
(ii) Find all functions $v: \mathbb{R}^2\rightarrow\mathbb{R}^2$ such that
$f(x+iy)=u(x,y)+iv(x,y)$ 
where $u(x,y)=x^3-3xy^2$ for $(x,y) \in \mathbb{R}^2$ is differentiable on $\mathbb{C}$
.
My Working
(i) $z^2=(x+iy)(x+iy) = x^2-y^2+2ixy$
$e^{z^2}=e^{x^2-y^2+2ixy}$
But im not really sure where to go from here to fin a value for u(x,y) and v(x,y)
(ii) I dont have a clue what to do with this, any help in the right direction would be great.. Im thinking maybe it has something to do with Cauchy Riemann Equations as its differentiable on $\mathbb{C}$

From Euler formula $e^{iz}=\cos z+i\sin z$,
$e^{z^2}=e^{x^2-y^2+2ixy}=e^{x^2-y^2}\cos(2xy)+ie^{x^2-y^2}\sin(2xy)$
$f$ $\mathbb{C}$-differentiable implies $u,v$ $\mathbb{R}$-differentiable plus Cauchy-Riemann equations
$$u_x=v_y$$
$$u_y=-v_x$$
Then 
$$v_y(x,y)=3x^2-3y^2\longrightarrow v=3x^2y-y^3+H(x)$$
$$v_x(x,y)=6xy\longrightarrow v=3x^2y+K(y)$$
By comparison, you get
$$v(x,y)=3x^2y-y^3$$
which is the imaginary part of $f(z)=z^3$
