MHB Rieman inversion formula in Laplace transform

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The discussion focuses on solving an exercise involving the inverse Laplace transform using the Riemann inversion formula. The user seeks to find the result of a specific integral related to the function F(s) = 1/(s + 4)^2, evaluated at t-5. Participants suggest using residue calculus or properties of the Laplace transform to compute the solution. The conversation emphasizes the need for clarity on the appropriate methods for finding the inverse transform. The thread highlights the importance of understanding both the Riemann inversion formula and Laplace transform techniques.
lucad93
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Hello everybody! I'm sorry if it's not the right section to post in. I'm trying to solve this exercise:
$$\frac{1}{2i\pi}*\int_{8-i\infty}^{8+i\infty}\frac{e^{s(t-5)}}{(s+4)^2}ds$$
The request is to find the result in function of $$t$$
I know i must use the Riemann inversion formula, and so the request would be to anti-transform this $$\frac{e^{-5s}}{(s+4)^{2}}$$. First question: Am I right? I haven't found a transform that fit with this, how can I do?
ThankYou! :)
 
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Hi lucad93,

Welcome! (Smile) What you're trying to find here is the inverse Laplace transform of the function

$$F(s) = \frac{1}{(s + 4)^2}$$

at $t-5$. Do you have to compute this using residue calculus, or using certain properties of the Laplace transform (e.g., $\mathcal{L}(t^a)(s) = \Gamma(a+1)/s^{a+1}, a > -1$, etc.)?
 

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