One sometimes needs to do a partial fraction expansion when one is integrating or when one is doing Laplace transforms. Most people know the standard methods: either gather terms and compare coefficients, or use selected (real) values for .

But sometimes, (NOT all of the time), one can speed things up by using complex numbers.

Here is an example: expand .

Solution: set this up as .

Now clear denominators to obtain .

Setting yields which means

(Yes, I know that we used a number not in the domain of the original fraction…but why can we get away with that? :-))

Now set we obtain . By comparing real and imaginary parts, we obtain and then .

Here is a second, more complicated case. Expand .

Clear denominators again to obtain . Trying yields .

Now use a primitive complex 3’rd root of unity: ; this causes the first term to vanish. The second term becomes immediately:

which simplifies to: .

Comparing real and imaginary parts again: and .

Caveat: one has to be very comfortable with complex arithmetic to use this method, but some engineers and physicists are.

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