Let $C$ be the positively oriented unit circle, and consider the function $z^{-3/4}$.
(a). Show that if $f(z)$ is the principal branch of $z^{-3/4}$, i.e., $$f(z)=exp[\frac{-3}{4}Logz],$$ where $|z|>0$, $-\pi<Argz<\pi$, then $$\int_{C}f(z)=4\sqrt{2}i$$
For this, I tried integrating $\int_{-\pi/2}^{\pi/2} (e^{i\theta})^{-3/4}(ie^{i\theta})d\theta$, which got me $4i\sqrt{2-\sqrt{2}}$. Then I tried to integrate the same thing between the bounds of $\pi/2$ and $3\pi/2$, and got another big mess, which didn't add up to what it was supposed to.
Further, for part (b), the instructions say to integrate the same function but $g(z)=exp[\frac{-3}{4}logz]$, and $0<argz<2\pi$. Then the integral should equal $-4+4i$.
How will this one be different than the one before it?