Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$ I'm trying to find an example of an analytic function $f$ satisfying the IVP 
$$
f'(z) = z\,f(z), \quad
f(0) = 1, 
$$
and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to proceed. 
Any hints or suggestions would be greatly appreciated.

Can you solve this ODE?: $y'=xy, y(0)=1\,.$
