LagrangeEuler
- 711
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- 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]
\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]
From that I am getting
f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be
f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})
where ##\theta## is Heaviside function. Where is the mistake?
From that I am getting
f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be
f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})
where ##\theta## is Heaviside function. Where is the mistake?