Partial Fractions (Laplace Transform, complex roots)

[sandy.bridge]

Hello all,
Say one wants to find the inverse Laplace Transform of a function, and the method for attaining the solution is executed via partial fractions. Do the real numbers go with the complex numbers when determining the constants of partials? Perhaps this is wordy. I'll provide a theoretical example:

Say we have:
$$\frac{A}{s+1-\sqrt{3}j}+\frac{B}{s+1+\sqrt{3}j}$$ where A+B=s+2.

Do we say:
$$A(1+\sqrt{3}i)+B(1-\sqrt{3}i)=2$$
or are the complex numbers treated separately?

 

[Char. Limit]

If you're trying to find the partial fraction decomposition of 1/(s^2 + 2s + 4), which is what i think you're doing here, you should get

$$\frac{A}{s + 1 - \sqrt{3} i} + \frac{B}{s + 1 + \sqrt{3} i} = \frac{1}{(s + 1 - \sqrt{3} i) (s + 1 + \sqrt{3} i)}$$

Multiply both sides by (s+1-sqrt(3)i)(s+1+sqrt(3)i) to get the right side as just 1. Then you'll have the real part of the left side equal to 1, and the imaginary part equal to 0. That should give you two equations for A and B.

 

[HallsofIvy]

A+ iB= C+ iD if and only if A= C and B= D.