I have following term, which I would like to decompose into partial fractions:
$ \frac{1}{x^2(1+x^2)} \overset{!}{=} \frac{Ax+B}{1+x^2} + \frac{C}{x} + \frac{D}{x^2}$
Multiplying everything out gives:
$ 1 = (Ax+B)x^2+Cx(1+x^2)+D(1+x^2)$
By plugging in arbitrary values for x such as 0, 1, 2, or 3, it leads to a system of equations that yields the correct coefficients:
$A=C=0, \qquad B=-1, \qquad D = 1$
However, I could have reached this solution much faster by making use of complex numbers, such as plugging in 0, i, -i, and 1. But in my problem, x is defined to be a real number.
My question: Is it always legitimate to substitute complex numbers when finding partial fraction decompositions, or are there some cases where you run into problems?