Rieman inversion formula in Laplace transform

[lucad93]

Hello everybody! I'm sorry if it's not the right section to post in. I'm trying to solve this exercise:
\(\displaystyle \frac{1}{2i\pi}*\int_{8-i\infty}^{8+i\infty}\frac{e^{s(t-5)}}{(s+4)^2}ds\)
The request is to find the result in function of \(\displaystyle t\)
I know i must use the Riemann inversion formula, and so the request would be to anti-transform this \(\displaystyle \frac{e^{-5s}}{(s+4)^{2}}\). First question: Am I right? I haven't found a transform that fit with this, how can I do?
ThankYou! :)

 

[Euge]

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.)?

 

[math771]

Hi there! It looks like you are on the right track. The Riemann inversion formula is used to find the inverse Laplace transform of a given function. In this case, you are trying to find the inverse Laplace transform of \frac{e^{-5s}}{(s+4)^{2}}. One way to approach this is to use partial fraction decomposition to break down the function into a sum of simpler fractions. Then, you can use known Laplace transforms to find the inverse of each individual fraction. Once you have the inverse Laplace transforms, you can combine them to get the final result in terms of t. I hope this helps! Good luck with your exercise.