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.