# Conditions of transformations

1. May 4, 2014

### Jhenrique

The Laplace of 1 is:

$$\int_{0}^{\infty} 1 \exp(-st) dt = \left[ \frac{\exp(-st)}{-s} \right]_{0}^{\infty} = \frac{\exp(-s \infty) - \exp(-s 0)}{-s} = \frac{0 - 1}{-s} = \frac{1}{s}$$

It's result known, however, for this be true is assumed that s>0, because 0 = exp(-∞) = exp(-s∞). But we have a problem, s is a complex number (σ + iω), so you assume that s>0 thus you are saying that ω=0, but in laplace transform ω≠0. The most correct possible would be $\exp(-s∞) = \exp(-(σ + iω)∞) = \exp(-σ∞ - iω∞) = \exp(-\text{sgn}(σ)∞ - i \text{sgn}(ω)∞) = \frac{\exp(-\text{sgn}(σ)∞)}{\exp(i \text{sgn}(ω)∞)}$, but it's become inpraticable... So what is correct form of approach this calculation of a simple way?

2. May 4, 2014

### Staff: Mentor

No, the assumption is that Re(s) > 0. See http://en.wikipedia.org/wiki/Laplace_transform, Table of selected Laplace transforms. Note the column whose heading is "Region of convergence."

3. May 4, 2014

### Jhenrique

Humm, but s is a complex number, so what is the conditition for Im(s)?

4. May 4, 2014

### Staff: Mentor

As far as I know, there aren't any conditions on Im(s).

5. May 4, 2014

### Staff: Mentor

What you wrote above is pretty much meaningless. You cannot use ∞ in arithmetic expressions.

6. May 4, 2014

Why not?

7. May 4, 2014

### Staff: Mentor

Maybe they're not needed. I dug up my book on Advanced Engineering Mathematics, Third Edition (Erwin Kreyszig) and looked at the section where he talks about Laplace tranforms. He doesn't even mention that s is complex. To see a treatment on the Laplace transform that deals with complex values of s, I looked at my copy of Churchill's Complex Analysis, which doesn't have any topics on this tranform, and then my Jerrold Marsden "Basic Complex Analysis," which does include this topic.

The Laplace transform is usually defined as this integral (which I believe you know):
$$F(s) = \int_0^{\infty}e^{-st}f(t)dt$$

The defining theorem for this transform says that there is some real number σ for which the above integral converges if Re(s) > σ, and diverges if Re(s) < σ.

If you want to know more than that, you'll need to study complex analysis.