Laplace Inversion: Why Contour Must Exceed Singularities

In summary, Laplace inversion is a mathematical technique used to find the original function from its Laplace transform. It is important in various fields such as engineering, physics, and mathematics. A contour in Laplace inversion is a closed curve in the complex plane along which the inversion integral is evaluated. It must exceed singularities and be chosen carefully to avoid branch cuts. The contour is chosen based on the location and type of singularities of the Laplace transform. Laplace inversion can only be applied to functions with a Laplace transform and that satisfy certain conditions, such as being continuous and having a finite number of singularities within the chosen contour.
  • #1
bruno67
32
0
Maybe I am just being stupid, but I don't understand why in the Laplace inversion formula

[tex](\mathcal{L}^{-1} F)(t) = \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{st} F(s) ds[/tex]
the contour of integration must be chosen so that [itex]\sigma[/itex] is greater than the real part of all singularities of [itex]F(s)[/itex]. I would be very grateful if someone could explain this.
 
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  • #2
Is it so that f(t) vanishes for t<0?
 

FAQ: Laplace Inversion: Why Contour Must Exceed Singularities

What is Laplace inversion and why is it important?

Laplace inversion is a mathematical technique used to find the original function from its Laplace transform. It is important because it allows us to solve problems in various fields such as engineering, physics, and mathematics.

What is a contour in Laplace inversion?

A contour is a path in the complex plane along which the Laplace inversion integral is evaluated. It is typically a closed curve that encloses all the singularities of the Laplace transform function.

Why does the contour in Laplace inversion need to exceed singularities?

The contour must exceed singularities because if it passes through or too close to a singularity, the integral may not converge and the inversion may not be accurate. The contour must also be chosen carefully to avoid crossing any branch cuts of the Laplace transform.

How is the contour chosen in Laplace inversion?

The contour is chosen based on the location and type of singularities of the Laplace transform. It should enclose all the singularities and be chosen such that the integral along the contour can be evaluated using known methods.

Can Laplace inversion be applied to any function?

No, Laplace inversion can only be applied to functions that have a Laplace transform. The function must also satisfy certain conditions, such as being continuous and having a finite number of singularities within the chosen contour.

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