Complex Integration Homework: Compute Integrals w/ Principle Value of z^i

[strangequark]

Homework Statement

Compute the following integrals using the principle value of z^{i}

a.
\int z^{i} dz where \gamma_{1}(t)=e^{it} and \frac{-\pi}{2}\leq t \leq \frac{\pi}{2}

b.
\int z^{i} dz where \gamma_{1}(t)=e^{it} and \frac{\pi}{2}\leq t \leq \frac{3\pi}{2}

Homework Equations

The Attempt at a Solution

There is a "hint" with the problem that says one of the integrals is easier than the other.
I don't see why, for part a, I can't use a branch cut along the negative real axis, so that z^{i} will be analytic along the path.
And for part b, i don't see why I can't simply use a different branch cut, say one along the positive real axis, so that z^{i} will then be analytic along that path.

Then for each, I can just take the antiderivative:

F(z)=\frac{z^{i+1}}{i+1}

and plug in the end points...

If I do, I get:

a. \int z^{i} dz = \frac{i^{i+1}}{i+1}+\frac{(-i)^{i+1}}{i+1} = (\frac{e^{\frac{\pi}{2}}}{2}+\frac{e^{\frac{\pi}{2}}}{2}i)-(-\frac{e^{\frac{\pi}{2}}}{2}-\frac{e^{\frac{\pi}{2}}}{2}i ) = cosh(\frac{\pi}{2})+cosh(\frac{\pi}{2})i

and I will get the same thing for b.

Is there something I am missing? Do I ned to parameterize the integral or is what I am doing correct?

Thanks in advance!

 

[strangequark]

I'm thinking that because f(z)=z^{i} is entire, and that the region in which the curve lies will be simply connected... then the anitderiv exists and since i is just a constant, then the primitive of f(z) will be F(z)=\frac{z^{i+1}}{i+1}...

does anyone have any ideas? I'm really stuck here... thanks