Can I obtain the inverse Laplace transform using complex analysis?

LagrangeEuler
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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?
 
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LagrangeEuler said:
Homework Statement:: 1. Find inverse Laplace transform
\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]
Relevant Equations:: Inverse Laplace transform can be calculated as sum of residues of ##F(s)e^{st}##.

This is only valid if \lim_{\operatorname{Re}(s) \to -\infty} F(s)e^{st} = 0. That is not the case for t < 5.
 
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Thank you. Is it a way to show this somehow? Or to use some version of complex analysis to get this?
 
So my mine question, in this case, is can I somehow obtain this result using complex analysis?
 
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