How to Evaluate the Function f(t) Using the Laplace Inverse Transform?

[eljose79]

Let be the function

f(t)=L(-1)1/(exp(-s)-1) where L(-1) means the laplace inverse transform..my doubt is to know what is the value of

f(0),f(1),f(2)...f(n) being n an integer.

 

[quantumdude]

I'll take a stab at this.

First, we need to develop the Laplace transform of a periodic function:

f(t)=f(t+nT) for all integers n.

Start with the definition of L{f(t)}:

L{f(t)}=∫₀^∞e^-stf(t)dt

Let's break this up over intervals of width T:

L{f(t)}=∫₀^Te^-stf(t)dt+∫_T^2Te^-stf(t)dt+∫_2T^3Te^-stf(t)dt+...

Now perform substitutions on each integral such that the limits of each integral are [0,T]:

L{f(t)}=∫₀^Te^-stf(t)dt+∫₀^Te^-s(t+T)f(t+T)dt+∫₀^Te^-s(t+2T)f(t+2T)dt
+∫₀^Te^-s(t+3T)f(t+3T)dt+...

Noting that f(t)=f(t+T)=f(t+2T)=f(t+3T)=..., and factoring ∫₀^Te^-stf(t)dt out of each factor yields:

L{f(t)}=(1+e^-sT+e^-2sT+e^-3sT+...)∫₀^te^-stf(t)dt

The first factor on the right is a geometric series whose sum is:

(1-e^-sT)^-1

So, I have finally:

L{f(t)}=(1-e^-sT)^-1∫₀^Tf(t)dt

Now, we get your function if we let f(t)=d(t+n+1/2). That is, a periodic delta function whose period is 1. I used the half integer n+1/2 so that there is only one delta function in each interval. I could just as easily have chosen n+1/3, n+1/4, or whatever. So, it seems that the function is not unique.

Does that help?

edit: fixed a variety of bracket errors

 

[tacman]

To evaluate the function f(t) = L(-1)1/(exp(-s)-1), we first need to understand what the Laplace inverse transform means. This is a mathematical operation that allows us to find the original function from its Laplace transform. In other words, we are trying to find the function f(t) that, when transformed using the Laplace transform, gives us the function 1/(exp(-s)-1).

Now, to find the value of f(0), we substitute t=0 into the function. This gives us f(0) = L(-1)1/(exp(0)-1) = L(-1)1/0, which is undefined. This is because the denominator becomes 0, which is not allowed in mathematics. Similarly, when we substitute t=1 or t=2, we get undefined values.

However, for integer values of n, we can evaluate the function f(n) by using the definition of the Laplace inverse transform. This involves taking the inverse Laplace transform of the function 1/(exp(-s)-1) and then substituting s=n into the resulting expression. This will give us the value of f(n) for any integer n.

In summary, the value of f(0) and f(n) for non-integer values of n is undefined, but we can find the value of f(n) for integer values of n by using the definition of the Laplace inverse transform.