Partial Fraction Decomposition with Complex non-linear factors

Question

Is it possible to write $1/(x^2+1)^2$ as $A/(x+i)^2+B/(x-i)^2$? If so, I am having trouble with the answer and I really can't see where I'm going wrong, an help?

Answer 1

You can write: \begin{align} \frac{1}{(x^2+1)^2} & = \frac{1}{(x+i)^2 (x-i)^2} = \frac{A}{x+i} + \frac{B}{(x+i)^2} + \frac{C}{x-i} + \frac{D}{(x-i)^2}. \end{align} The algebra is all done the same way as if real numbers were involved, but the arithmetic will require things like replacing $i^2$ with $-1$ and multiplying a numerator and denominator both by the conjugate of the denominator.

PS: I'm getting $A=-C=i/4$ and $B=D=-1/4$.