Inverse Laplace transform. Bromwitch integral

In summary, the inverse Laplace transform is given by integrating along a straight line in the complex plane from c-i\infty to c+i\infty, where c is a constant chosen such that the integral path is in the right half plane where F(s) is analytic. The use of the imaginary unit i is due to the fact that the traditional Laplace transform involves an exponential function with a complex argument.
  • #1
LagrangeEuler
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Inverse Laplace transform
[tex]\mathcal{L}^{-1}[F(p)]=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}F(s)e^{st}dp=f(t)[/tex]

Question if we integrate along a straight line in complex plane where axis are [tex]Re(p)[/tex], [tex]Im(p)[/tex], why we integrate from [tex]c-i \ínfty[/tex] to [tex]c+\infty[/tex]? So my question is, because [tex]Im(p)[/tex] are also real numbers why we integrate from [tex]c-i\infty[/tex] to [tex]c+i\infty[/tex].
 
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  • #3
I'm guessing that you meant that the integral is with respect to [itex]s[/itex], not [itex]p[/itex]. Anyway, for the traditional Laplace transform,
F(s) = \int^{\infty}_{0} dt e^{-s t} f(t)

[itex]F(s)[/itex] is only analytic for some right half plane (that is [itex]\Re(s)>s_0[/itex]). So for the inverse transform you pick a [itex]c>s_0[/itex] for your path.

Does that help?

jason
 
  • #4
Yes. Mistake. I understand that. I am not very good in complex analysis. My question is if [tex[Im(s)[/tex] are real numbers why I integrate from [tex]c-i\infty [/tex] to [tex]c+i\infty [/tex]? Why I have this [tex]i[/tex]? Thanks for the answer.
 

Related to Inverse Laplace transform. Bromwitch integral

1. What is the Inverse Laplace transform?

The Inverse Laplace transform is a mathematical operation that takes a function in the complex frequency domain and transforms it back into the corresponding function in the time domain. It is used to solve differential equations and is an important tool in engineering and physics.

2. How is the Inverse Laplace transform calculated?

The Inverse Laplace transform can be calculated using the Bromwich integral, which is an integral over a contour in the complex plane. This integral can be evaluated using various techniques, such as the residue theorem or partial fraction decomposition.

3. What is the relationship between the Laplace transform and the Inverse Laplace transform?

The Laplace transform and the Inverse Laplace transform are inverse operations of each other. This means that the Laplace transform takes a function in the time domain and transforms it into the complex frequency domain, while the Inverse Laplace transform takes a function in the complex frequency domain and transforms it back into the time domain.

4. What is the significance of the Bromwich integral in the Inverse Laplace transform?

The Bromwich integral is the mathematical expression that represents the Inverse Laplace transform. It allows us to calculate the Inverse Laplace transform of a function by integrating over a contour in the complex plane. This integral is essential in solving differential equations and is used in many fields of science and engineering.

5. Can the Inverse Laplace transform be calculated numerically?

Yes, the Inverse Laplace transform can be calculated numerically using various techniques such as the Fast Fourier Transform (FFT) or the Stehfest algorithm. These techniques are useful when the Bromwich integral cannot be evaluated analytically, or when dealing with complex functions or noisy data.

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