Hey I really need help with this questions, I honestly don't even know where to start...
1)Explain why F(z)=e/\(z/\2) has an anti derivative in the whole plane?
/\ is the exponent carrot thingy
$\displaystyle F(z)=e^{z^2}$
This is a composition of two functions that are analytic on $\displaystyle \forall z \in \mathbb{C}$
let $\displaystyle g(z)=e^z \mbox{ and } h(z)=z^2$
so $\displaystyle F(z)=g(h(z))=e^{z^2}=\sum_{n=0}^{\infty}\frac{z^{2 n}}{n!}$
By the ratio test this converges for all values of z So we can integrate for the antiderivative.