Can I obtain the inverse Laplace transform using complex analysis?

[LagrangeEuler]

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

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

 

[pasmith]

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 $t < 5$.

 

[LagrangeEuler]

Thank you. Is it a way to show this somehow? Or to use some version of complex analysis to get this?

 

[LagrangeEuler]

So my mine question, in this case, is can I somehow obtain this result using complex analysis?