# Residue Theorem for Laplace Transform

## Main Question or Discussion Point

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything.

For example, if I have this two equations:

$$X(s).(s-1) = -Y(s)+5$$

$$Y(s).(s-4) = 2.X(s)+7$$

I know how to solve them using Simple Fractions, but I need to know how to solve that using Residue Theorem.

Oh, I forgot to mention that I'm looking for the Inverse Transform of Y(s) and X(s)
Thanks!

EDIT:

I know that, for example, for y(t) I'm going to have this:

$$y(t) = Res[Y(s).e^{st}, 2] + Res[Y(s).e^{st}, 3]$$

but I need to know why and a general case (a Theorem, for example)

SteamKing
Staff Emeritus
Homework Helper
I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything.

For example, if I have this two equations:

$$X(s).(s-1) = -Y(s)+5$$

$$Y(s).(s-4) = 2.X(s)+7$$

I know how to solve them using Simple Fractions, but I need to know how to solve that using Residue Theorem.

Oh, I forgot to mention that I'm looking for the Inverse Transform of Y(s) and X(s)
Thanks!

EDIT:

I know that, for example, for y(t) I'm going to have this:

$$y(t) = Res[Y(s).e^{st}, 2] + Res[Y(s).e^{st}, 3]$$

but I need to know why and a general case (a Theorem, for example)
Most LT can be calculated using integral calculus. If you don't want to use a table of LT to calculate the inverse, then residues come in handy, since you need to evaluate a complex integral.

The attached article shows how to use residues to compute an inverse LT:

http://www.staff.city.ac.uk/~george1/laplace_residue.pdf

There are other articles which can be found if you Google "Inverse laplace transform by residue theorem" :)

Perfect! But I have 2 problems:
- I never used a contour like this: D. It was always with the line in the x axis
- Then, I don't understand where $$\int_{C_1}{ } F(s).e^{s.t}ds$$ came from. If you look at the Inverse Transform of Laplace, you can see a $$\frac{1}{2j\pi}$$ and the limits of the integral are awful.

Thanks!

SteamKing
Staff Emeritus