# Inverse Laplace Transform

## Main Question or Discussion Point

My book on signal processing says that:

$$f(t) = \frac{1}{2\pi j} \int_{c-j\infty}^{c+j\infty} F(s) e^{st} ds = \lim_{\Delta s \to 0} \sum_{n = -\infty}^{\infty} \Big[ \frac{F(n\Delta s)\Delta s}{2\pi j} \Big] e^{n\Delta s t}$$

I don't get this. How/Why can you write a integration over a complex variable as the above sum?

edit: I forgot a coefficient $$\frac{1}{2\pi j}$$ on the LHS. Sorry about that. Its fixed now.

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## Answers and Replies

Isn't that definition of a Reimann integral?

I understand how you could be confused! It would have to be explained what \Delta s is in the sum since the integral seems to have s lie on a vertical line c + jy in the complex plane (real y). So do we perhaps have \Delta s =
j\Delta y???

Supposing the above and not knowing anything else about F, using a Riemann sum, you should have limit as \Delta s -> 0 of sum over n of the following terms:
F(c+n\Delta s)\exp((c+n\Delta s)t)\Delta s
which is the limit as \Delta y-> 0 of the sum of terms

F(c+n j \Delta y)\exp((c+ n j \Delta y)t)j\Delta y

The 2\pi j in the denominator seems something like Cauchy's integral thm., but we need to know quite a bit more about F to get that. (Is F an entire fuction? Does it vanish very rapidly away from the y-axis in the complex plane?)

Generally speaking, the independence of c on the RHS makes the formula dubious.

Can you still get the money back for the book?

I understand how you could be confused! It would have to be explained what $$\Delta s$$ is in the sum since the integral seems to have $s$ lie on a vertical line $$c + jy$$ in the complex plane (real y). So do we perhaps have $$\Delta s = j\Delta y$$???

Supposing the above and not knowing anything else about $$F$$, using a Riemann sum, you should have limit as $$\Delta s -> 0$$ of sum over $n$ of the following terms:
$$F(c+n\Delta s)\exp((c+n\Delta s)t)\Delta s$$
which is the limit as $$\Delta y-> 0$$ of the sum of terms

$$F(c+n j \Delta y)\exp((c+ n j \Delta y)t)j\Delta y$$

The $$2\pi j$$ in the denominator seems something like Cauchy's integral thm., but we need to know quite a bit more about $$F$$ to get that. (Is $$F$$ an entire function? Does it vanish very rapidly away from the y-axis in the complex plane?)

Generally speaking, the independence of $c$ on the RHS makes the formula dubious.

Can you still get the money back for the book?
Edited gammamcc's post to look nice...