Evaluating a complex path integral using the complex fundamental theorem of calculus.

Question

I am currently stuck on a question that I think would be quite simple with some direction. It is:

Evaluate the integral $\int_{\gamma|[0,1]}f(z)dz$ where $f(z) = z^3-z^2+z$ where $\gamma(0) = i + 1$ and $\gamma(1) = i - 1$.

I am almost certain that I have to use the fundamental theorem of calculus in the complex plane such that:

$\int_{\gamma|[a,b]}f'(z)dz = f(\gamma(b)) -f(\gamma(a))$

but I am not sure how to get the normal path integral of $f(z)$ and not $f'(z)$. Im thinking along the lines of introducing a function $g(z) = f'(z)$ but am not sure how to proceed. Any help is appreciated.

Answer 1

If you define $F(z)=\frac14z^4-\frac13z^3+\frac12z^2$, then, since $F'=f$,$$\int_\gamma f(z)\,\mathrm dz=F\bigl(\gamma(1)\bigr)-F\bigl(\gamma(0)\bigr).$$