I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following:
$f(x) = \sum_{i=M}^N{c_i x^i}$
I attempt to integrate each term of the power series seperately. For instance, looking at a few terms, we can integrate this to obtain $g(x)$:
$f(x) = \cdots + c_0 x^0 + c_1 x^1 + c^2 x^2 + \cdots$
$g(x) = \cdots + c_0 x^1 + \frac{1}{2}c_1 x^2 + \frac{1}{3}c_2 x^3 + \cdots$
In effect, we take $c_i x_i \mapsto \frac{1}{i+1}c_i x^{i+1}$ integrating terms individually.
Its possible to do this using generating function methods, which is fairly routine.
The Problem
Two terms are problematic. The term $c_{-1} x^{-1}$ should integrate to $\log(x)$. The constant of integration is not known, so this is the second problem term.
The Question
I'm looking for a way to perhaps solve this problem. For instance, finding a way to approximate the two problem terms would be a good start. In the end, I want to get an approximation to the integral within an error bounds. I really want to consider this particular method, though. I'd be interested in seeing related literature.
Some Ideas
We may be able to use approximations to similar integrals to solve for approximations to the two problem terms. I'm interested in trying to get an approximation (to the original integral) within a certain amount of error. If this is possible, I'd consider the problem solved.
An Example: Particular integrals that I'm using this on.
Here's a description of the function(s) I want to integrate. I take:
$\displaystyle x^b \frac{\prod_{i=0}^m{(\pm x^{c_i} \pm 1)}}{\prod_{j=0}^n{(\pm x^{d_j} \pm 1)}}$
Note that both the quotient and the denominator consist of products of the form $\pm x^r \pm 1$. Here, $r \in \mathbb{N}$ and $-(2^s) \leq r \leq 2^s$ for some natural $s$. Also, $b$ is similarly restricted/defined as $r$.
I take this formula and make the substitution $x \mapsto e^{i t}$ and integrate from $t=-\pi$ to $t=\pi$.