Hi I'm trying to do a partial fraction decomposition but I'm not sure if what I'm doing is legit math and whether it's a good approach to my problem. I have the following fraction which appears inside an integral:
$$ \frac{e^{c\delta}}{\prod_{i=1}^N e^\delta + e^{\theta_i}} $$
where $c$ and $N$ are integers and $c \leq N$. If I substitute $x=e^\delta$ figured I could rewrite this as:
$$ \frac{x^c}{\prod_{i=1}^N x + e^{\theta_i}} $$ I figured that both the numerator and the denominator are polynomials in $x$ and therefore I could write:
$$ \frac{x^c}{\prod_{i=1}^N x + e^{\theta_i}} = \sum_{i=1}^N \frac{A_i}{x+e^{\theta_i}} $$
multiplying both sides by $\prod_{i=1}^N x + e^{\theta_i}$ gives:
$$ x^c = \sum_{i=1}^N A_i \prod_{j\neq i}^N x+e^{\theta_j} $$ and I figured that I can solve for $A_i$ if I say $x = -e^{\theta_i}$ which gives:
$$ (-e^{\theta_i})^c = A_i \prod_{j\neq i}^N e^{\theta_j} - e^{\theta_i} \\ A_i = \frac{(-e^{\theta_i})^c}{\prod_{j\neq i}^N e^{\theta_j} - e^{\theta_i}} $$ but is this derivation proper? Saying $x = -e^{\theta_i}$ implies $\delta = \theta_l + i \pi$ which suddenly introduces complex numbers (while $\delta$ is only defined on the reals). In addition, by introducing $\frac{1}{\prod_{j\neq i}^N e^{\theta_j} - e^{\theta_i}}$ do I introduce a lot of difficulties when I (later) plan to integrate over $\theta_i$? If this is so, my approach might not be suited for this problem.