Bit of a complicated integral, need some help

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Bit of a complicated integral, need some help

For personal use I am trying to solve the following integral:

\int_0^1 \frac {\mathrm{d}t} {\prod\limits_{i=0}^n [(x_B t - x_i)^2 + y_i^2]}

After some deliberation, I decided that the easiest way to go about this is to split up the fraction using the method of partial fractions. This yields the following (assuming I got my syntax right):

1 = \sum\limits_{i=0}^{n} [ A_i \prod\limits_{j \in n - \{i\}} [ (x_b t - x_j)^2 + y_i^2 ]]

And then if we substitute \frac { x_i - y_i } { x_B } \sqrt{-1} for t for each i, we'll get a system of linear equations, which then need to be solved for all of the Ai...

... and that's where it goes a bit beyond my grasp. Is there any way I can proceed, or have I gotten in too deep?
"Maybe there are stupid happy people out there... And life isn't fair, and you won't become happier by being jealous of what you can't have... You can never achieve that degree of ignorance... you cannot unknow what you know." -E. Yudkowsky

Aedl Foxe

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Re: Bit of a complicated integral, need some help

First, I think your system of equations is not quite right; because the denominators are quadratic in t the numerators are generally linear in t, so you want (Ai+Bit) there.

But there's a quicker way (than setting up the linear system of equations) to find the partial-fraction expansion, in the special case where all of the denominators are linear polynomials,
\frac{P(t)}{Q(t)} = \sum_m \frac{A_m}{t-t_m}\;.

Multiply the equation through by a factor (t-tk) and evaluate it at t=tk. All of the partial fractions on the right side give zero, except for the kth one. So the right-hand side is just Ak, and you immediately have a formula for the Ak.

In your case you can factor each of the quadratic factors into a complex-conjugate pair to give the desired linear factors,
\frac{A_m+B_mt}{(x_Bt+x_m)^2+y_m^2} = \frac{C_m}{x_Bt+x_m+iy_m} + \frac{D_m}{x_Bt+x_m-iy_m}\;.

Do the above to get Cm and Dm (you should find that Dm=Cm* the complex conjugate), then combine these complex-conjugate partial-fraction pairs to get Am and Bm (which should both be real). Then you can integrate this to give a combination of a logarithm and an arctangent.

eta oin shrdlu

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