Inverse laplace transform with complex inversion theorm

[oddiseas]

Homework Statement

Find the inverse laplace transform of In(1+1/s)

Homework Equations

The Attempt at a Solution

Using the complex inversion theorm, and the sum of the residues.

The only residue is at s=0. and it is a simple pole of degree one.

Therefore lim(s approaches 0)= s*In(1+1/s)e^st and since e^st at s=0 becomes 1, i get:

lim(s approaches 0)=s*In(1+1/s)

Now i have no idea what to do next. Usually i can find f(t) easily at this point but not with this function.

(note: i have already found the answer using a power series and the derivative property,(1-e^-t)/t) i am trying to figure out how to proceed next with this method.)

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[mighty2000]

Hello, as a scientist, I would like to first clarify that the inverse Laplace transform of In(1+1/s) is not a common function that can be easily expressed in terms of elementary functions. Therefore, it is not surprising that you are having difficulty finding the solution using the complex inversion theorem and residues.

However, there are alternative methods that can be used to find the inverse Laplace transform of this function. One approach is to use the Laplace transform table to find a similar function and then use properties of the Laplace transform to manipulate it into the desired form.

For example, we can use the property of the Laplace transform that states that the transform of the derivative of a function is equal to s times the transform of the function minus the initial value of the function. In this case, we can find the inverse Laplace transform of 1/s, which is simply 1, and then use the property to find the inverse Laplace transform of In(1+1/s).

Another approach is to use the Bromwich integral, which is another method for finding inverse Laplace transforms. This method involves integrating a complex function along a contour in the complex plane and using Cauchy's residue theorem to evaluate the integral. However, this method may be more complicated and time-consuming compared to using the Laplace transform table.

In conclusion, while it is possible to find the inverse Laplace transform of In(1+1/s) using the complex inversion theorem and residues, it may be more efficient to use alternative methods such as the Laplace transform table or the Bromwich integral. I hope this helps in your understanding of finding inverse Laplace transforms.