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I want to find a partial fraction expansion for the following: ($b$ is a constant)

$$\frac{1}{(b^2+x^2)(b^2+y^2)}$$

As there are two variables, I am unsure what form the decomposition should be of. I am looking for one where the coefficients of the terms in the numerator are independent of $x$ and $y$. My only idea at the moment is this form: ($f$ and $g$ are polynomial functions here)

$$\frac{1}{(b^2+x^2)(b^2+y^2)} = \frac{f(x,y)}{(b^2+x^2)} + \frac{g(x,y)}{(b^2+y^2)}$$

How should I chose the numerators?

Comp_Warrior
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1 Answers1

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A nice technique is to pretend that $b$ is the only variable of your function. Since: $$\operatorname{Res}\left(\frac{1}{(b^2+x^2)(b^2+y^2)},b=ix\right)=\frac{i}{2x(x^2-y^2)},$$ it follows that (not surpringly) $$f(x,y)=\frac{1}{y^2-x^2}=-g(x,y).$$

Jack D'Aurizio
  • 353,855