- #1

LagrangeEuler

- 717

- 20

- Homework Statement
- 1. Find inverse Laplace transform

[tex]\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}][/tex]

- Relevant Equations
- Inverse Laplace transform can be calculated as sum of residues of ##F(s)e^{st}##.

[tex]\mathcal{L}^{-1}[F(s)]=\sum^n_{k=1}Res[F(s)e^{st},s=\alpha_k][/tex]

[tex]\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2][/tex]

From that I am getting

[tex]f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}[/tex]. And this is not correct. Result should be

[tex]f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})[/tex]

where ##\theta## is Heaviside function. Where is the mistake?

From that I am getting

[tex]f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}[/tex]. And this is not correct. Result should be

[tex]f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})[/tex]

where ##\theta## is Heaviside function. Where is the mistake?