I Convergence of this Laplace transformation

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The discussion centers on the Laplace transformation of the function f(t) = e^(-t) * sin(t), leading to the expression X(s) = 1 / (1 + (1 + s)^2). A key point raised is that while substituting s = -1 gives X(-1) = 1, the integral for this value does not converge, as it requires Re(1 + s) > 0. Participants clarify that the Laplace transform can be analytically continued beyond its initial convergence region, allowing for a broader application of the results. The distinction between the defined function F(s) and its analytic continuation G(s) is emphasized, noting that they have different domains and implications for inversion. Understanding these nuances is crucial for correctly applying Laplace transforms in complex analysis.
lukka98
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I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is:
X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ).
Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s).

If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s = -1, so from the formula X(s) = 1) and this seem correctly according to a graph that I see online.

But if I put Re(s) = -1 and Im(s) = 0 in the integral, and I calculate it directly, I should get the same result...(?) but In that case is only the integral of sin(t) from 0 to infty that is not 1 absolutely.

What is wrong?

thanks
 
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The integral \int_0^\infty \sin t\,e^{-(1+s)t}\,dt converges only when \operatorname{Re}(1 + s) > 0. s = -1 is not in this region.

You can, of course, analytically continue the result you get in \operatorname{Re}(1 + s) > 0 into the rest of the s-plane (except the poles at s = -1 \pm i).

LaTeX is enabled; but it often doesn't render when you make your first post in a forum.
 
pasmith said:
The integral \int_0^\infty \sin t\,e^{-(1+s)t}\,dt converges only when \operatorname{Re}(1 + s) > 0. s = -1 is not in this region.

You can, of course, analytically continue the result you get in \operatorname{Re}(1 + s) > 0 into the rest of the s-plane (except the poles at s = -1 \pm i).

LaTeX is enabled; but it often doesn't render when you make your first post in a forum.
Ok, maybe my question was misunderstood, but ##X(s) = \frac{1}{1 + (1+s)^2}##, so for s = -1, I have X(s) = 1;
but ##\int_{0}^{\infty} sin(t) dt## ,that is for s = -1, is indefinite.
And also the graphics solution, looks like X(s) = 1 for s = -1.
 
lukka98 said:
Ok, maybe my question was misunderstood, but ##X(s) = \frac{1}{1 + (1+s)^2}##, so for s = -1, I have X(s) = 1;
but ##\int_{0}^{\infty} sin(t) dt## ,that is for s = -1, is indefinite.
And also the graphics solution, looks like X(s) = 1 for s = -1.

Yes.

There are many functions, such as the Gamma function, which are initially defined by an integral representation valid only in a subset of \mathbb{C} but are then extended to a larger subset of \mathbb{C} by analytic continuation. Laplace transforms often also fall into this category.

One can define a function F : D \to \mathbb{C}: s \mapsto \int_0^\infty f(t)e^{-st}\,dt where D \subset \mathbb{C} is the region where the integral converges. If D satisfies certain conditions then there exists a unique G, defined on as much of \mathbb{C} as possible, such that G(s) = F(s) for every s \in D.

Now F and G are distinct functions because they have different domains, and we commonly define the Laplace transform of f as F when really we mean G: Inverting the transform requires integration over a contour which is almost certainly not contained in D, and if we have a formula for F(s) then it almost certainly makes sense for values of s \notin D. But we must remember that if s \notin D then <br /> G(s) \neq \int_0^\infty f(t)e^{-st}\,dt.
 
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