Inverse Laplace Transform Step by Step

In summary, the inverse Laplace transform can be done using a contour integral. The function to be inverted must be known in real form and the real inversion can be done using a limiter.
  • #1
steve2k
2
0
Hi - I really need someone to show me step by step how to do an Inverse Laplace transform using a contour integral. The one I would like to understand is the frequency function 1/sqrt(s)

Thank you if you can help me out.

Steve
 
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  • #2
steve2k said:
Hi - I really need someone to show me step by step how to do an Inverse Laplace transform using a contour integral. The one I would like to understand is the frequency function 1/sqrt(s)

Thank you if you can help me out.

Steve
This is the contour integral that gives the inverse.
[tex] f(t)= \frac{1}{2{\pi}i} \int_{a-i\infty}^{a+i\infty} g(s)e^{st} ds [/tex]
or for the specific function.
[tex] f(t)=\frac{1}{2{\pi}i}\int_{a-i\infty}^{a+i\infty} \frac{e^{st}}{\sqrt{s}} ds [/tex]
We need to take "a" far enough to the right that we avoid problems.
Here we may take a=0, as even though the function has problems at zero, they are not major. You can consider a small right half circle and see the integral is small.
[tex] f(t)=\frac{1}{2{\pi}i}\int_{-i\infty}^{i\infty} \frac{e^{st}}{\sqrt{s}} ds [/tex]
we can clean the integral up with a substitution i u=s t
[tex] f(t)=\frac{1}{2{\pi}\sqrt{it}}\int_{-\infty}^{\infty} \frac{e^{iu}}{\sqrt{u}} du [/tex]
This integral can be written in terms of "know" real integrals.
[tex]\int_0^\infty \frac{sin(x)}{\sqrt{x}} dx=\int_0^\infty \frac{cos(x)}{\sqrt{x}} dx=\sqrt{\frac{\pi}{2}}[/tex]
thus the answer
[tex]f(t)=\frac{2+2i}{2{\pi}\sqrt{it}}\sqrt{\frac{\pi}{2}}[/tex]

[tex]f(t)=\frac{1}{\sqrt{{\pi}t}}[/tex]
You can also do a real inversion.
[tex]f(t)=\lim_{k\rightarrow\infty}\frac{(-1)^k}{k!}g^{(k)}(\frac{k}{t})(\frac{k}{t})^{k+1}[/tex]
 
Last edited:
  • #3
Thank you.

Thanks for the reply I really appreciate it!

Steve
 

1. What is an inverse Laplace transform?

An inverse Laplace transform is a mathematical operation that is used to convert a function from the Laplace domain to the time domain. It is the reverse of the Laplace transform and is denoted by the symbol L-1.

2. How do you perform an inverse Laplace transform step by step?

To perform an inverse Laplace transform step by step, you need to follow a series of mathematical operations. First, you need to make sure that the function is in the proper form, which is F(s)= L-1[F(s)]. Then, you can use the table of Laplace transforms to find the corresponding function in the time domain. Finally, you can use partial fraction decomposition and the properties of Laplace transforms to simplify the function and obtain the inverse Laplace transform.

3. What are the properties of inverse Laplace transforms?

The properties of inverse Laplace transforms include linearity, time scaling, shifting, differentiation, integration, convolution, and initial value theorem. These properties allow for easier and quicker calculations of inverse Laplace transforms.

4. When is an inverse Laplace transform used?

An inverse Laplace transform is used in various fields of science and engineering, such as control systems, signal processing, and circuit analysis. It is also used to solve differential equations and to study the behavior of systems over time.

5. What are some common mistakes when performing an inverse Laplace transform?

Some common mistakes when performing an inverse Laplace transform include not using the correct form of the function, not using the correct table of Laplace transforms, and not properly simplifying the function using the properties of Laplace transforms. It is also important to be careful with the algebraic manipulations and to check for any errors in the final result.

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