Fast partial-fraction decomposition

Question

I'm studying Laplace transformations for my differential equations class and typically there's a partial fraction decomposition involved, which can be very long and demanding for calculations by hand, if done the standard way.

I am aware of some of the tricks used to speed-up this procedure (like the usage of limits at infinity, or multiplying with denominators and taking particular values of $s$), however I am not able to apply them at this example:

$$\frac {s} {(s^2+2s+5)(s^2+4)} = \frac {\alpha s + \beta} {s^2+2s+5} + \frac {\gamma s + \delta} {s^2+4}$$

For example, if I attempt to extract a relation for $\gamma$ and $\delta$, by multiplying with $s^2+4$ and taking $s=2i$, I get and equation that involves complex numbers; that makes me feel I have not gained much in terms of number of operations.

Are there any better tricks for this example?

Answer 1

Let's apply the Heaviside cover-up method, nonlinear version, as described by Bill Dubuque.

$$\rm \frac{x}{(x^2\!+2x+5)(x^2\!+4)} \ =\ \frac{ax+b}{x^2\!+2x+5}\, +\ \frac{cx+d}{x^2\!+4}$$

Clearing denominators yields

$$\rm x\, =\, (x^2\!+4)(ax+b)\, +\, (x^2\!+2x+5)(cx+d) $$

Evaluating this mod $\rm\ x^2\! +\! 4\,\: $ i.e.$\:$ iteratively applying the rewrite rule $\rm\ x^2 \to -4\,\ $ yields

$$\rm x\, =\, (c\!+\!2d)\ x\,+\, d\!-\!8c\ \ \Rightarrow\ \ c = 1/17,\,\ d= 8/17$$

Evaluating it mod $\rm\ x^2\! +\! 2x\!+\!5\:,\: $ i.e.$\:$ iteratively applying rewrite rule $\rm\ x^2 \to -2x\!-\!5\: $ yields

$$\rm x\, =\, (3a\!-\!2b)\ x \,+\, 10a\!-\!b\ \ \Rightarrow\ \ a = -1/17,\,\ b = -10/17$$

That seems easy enough to me.

Answer 2

Well, the following is certainly a trick. Whether it's better or not remains to be decided....

First, I want to make the factors in the denominator symmetric with respect to the origin (right now they're of the form $s^2+4$ and $(s+1)^2+4$, hence symmetric with respect to $1/2$). So setting $s=t-1/2$, we have the equivalent problem of doing partial fractions on $$ \frac{16 t-8}{(4 t^2-4t+17)(4 t^2+4t+17)}. $$

Let's start by noting (using the symmetry) that $$ \frac{1}{4 t^2-4t+17} - \frac{1}{4 t^2+4t+17} = \frac{8t}{(4 t^2-4t+17)(4 t^2+4t+17)}. $$ How can we divide everything in sight by $t$? Notice that the inverse of $t$ modulo $4t^2-4t+17$ is $(-4t+4)/17$, while the inverse of $t$ modulo $4t^2+4t+17$ is $(-4t-4)/17$. So we get \begin{align*} \frac{8}{(4 t^2-4t+17)(4 t^2+4t+17)} &= \frac{1/t}{4 t^2-4t+17} - \frac{1/t}{4 t^2+4t+17} \\ &\equiv \frac{-4t+4}{17(4 t^2-4t+17)} + \frac{4t+4}{17(4 t^2+4t+17)} \pmod1. \end{align*} But this (mod 1)-equivalence must actually be an equality(!): by the theory of partial fractions, the left-hand side (whose numberator has larger degree than the denominator) has a representation in the form of the right-hand side.

Between the last two identities, we see that $$ \frac{16 t-8}{(4 t^2-4t+17)(4 t^2+4t+17)} = \frac{42 t+30}{17 (4 t^2-4 t+17)}-\frac{4t+38}{17 (4 t^2+4 t+17)}; $$ now plugging $t=s+1/2$ back in yields $$ \frac s{(s^2+2s+5)(s^2+4)} = \frac{-s-10}{17 (s^2+2 s+5)}+\frac{s+8}{17 (s^2+4)}. $$