Evaluating $I(A, n) = \int_{0}^{2\pi} \dfrac{1}{A +\cos^n(\theta)} d\theta$

Question

We want to evaluate the complex-valued integral $$I(A, n) = \int_{0}^{2\pi} \dfrac{1}{A +\cos^n(\theta)} \mathrm d\theta$$ for $n \in \mathbb{N}_+$ and $A \in \mathbb{C}$.

For which $A \in \mathbb{C}$ the integral exists?

For $A = 1$ the integral exists? I'm not sure for which other $A \in \mathbb{C}$ the integral exists

Answer 1

There are two cases: $n$ could be even or odd.

If $n$ is even, then $\{ \cos^n \theta | \theta \in [0; 2 \pi]\} = [0;1]$ hence the integral surely exists for all $A \in \Bbb C \setminus [-1;0]$.

If $A \in [-1;0]$, then only real numbers are involved (no need of complex numbers). The integrand function has a pole at some $\theta$ (such that $A + \cos^n \theta=0$), and so the integral is divergent.

Similarly, if $n$ is odd, then $\{ \cos^n \theta | \theta \in [0; 2 \pi]\} = [-1;1]$ hence the integral surely exists for all $A \in \Bbb C \setminus [-1;1]$.

Again, if $A \in [-1;1]$ you don't need complex numbers, and the integral is again divergent.