# Thread: Complex An

1. ## Complex An

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

2. Originally Posted by confusedmathmajor
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.

3. I like EmptySet's solution. Here is another. Let $\displaystyle [0,z]$ represent the line joining $\displaystyle 0$ and $\displaystyle z$.

Define, $\displaystyle F(z) = \int_{[0,z]} e^{w^2} dw$.