My calculus book has nothing on complex numbers so here i am.
$$ e^z = \sum_0^\infty \frac{z^n}{n!} $$
Given the above function as an example , can someone take me through the steps as to how i would represent it as an integral? and perhaps offer an extra example where i don't know the function , just the sum?
Thank you very much for your time and help.
I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^{(n)}(z)$ can be expressed by the help of Cauchy Integral Formula for $n^{th}$ derivative as, $$a_nn!=f^{(n)}(a)=\frac{n!}{2\pi i}\oint_\gamma \frac{f(z)}{(z-a)^{n+1}}dz$$ $\forall n=0,1,2...$. Notations are defined in the links.