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)
The use of the residue theorem comes from the definition of the ILT:
$$f(t) = \frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, F(s) e^{s t} $$
where the line $\operatorname{Re}{s}=c$ is to the right of the rightmost pole of $F$. To evaluate this integral when $F$ has simple poles, one closes a contour to the left of the line of integration for $t \gt 0$; the contour then contains all of the poles of $F$. The ILT is then the sum of the residues of the poles of $F$. To evaluate for $t \lt 0$, one closes the contour to the right; since there are no poles there, the ILT for $t \lt 0$ is zero.
With that in mind, let's solve the above system for $X$ and $Y$:
$$\begin{align}(s-1) X + Y &= 5 \\ -2 X+(s-4) Y &= 7 \\ \end{align}$$
Solving, I get
$$X(s) = \frac{5 s-27}{(s-2)(s-3)} $$ $$Y(s) = \frac{7 s+3}{(s-2)(s-3)} $$
So, for $t \gt 0$, $x(t)$ is equal to the sum of the residues of $X(s) e^{s t}$ at $s=2$ and $s=3$, and zero otherwise. This is then
$$x(t) = \left (17 e^{2 t}- 12 e^{3 t} \right ) \theta(t) $$
where $\theta$ is the Heaviside step function, zero for $t \lt 0$, one for $t \gt 0$. Similarly,
$$y(t) = \left (-17 e^{2 t}+24 e^{3 t} \right ) \theta(t) $$