What is the proof for the Laplace transform of f(t)/t?

In summary, in order to find the Laplace transform of \frac{f(t)}{t}, we can use Lemma 10.3.9 from the provided link. This lemma states that the Laplace transform of a function F(s) is equal to the negative derivative of the inverse Laplace transform of G(s). By taking the antiderivative of both sides and letting the upper bound approach infinity, we can find the Laplace transform of \frac{f(t)}{t} as \int_s^\infty F(s)\, ds.
  • #1
fishingspree2
139
0
Hello, I am trying to find the laplace transform of [tex]\frac{f(t)}{t}[/tex]

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node92.html
Lemma 10.3.9

I don't understand why we are taking the limit as s goes to infinity

Can anyone help me?
thank you
 
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  • #2
Look at it this way. You have this relationship:

[tex]F(s) = -\frac d {ds} G(s) = -G'(s)[/tex]

Take the anitiderivative of both sides from s to b

[tex]\int_s^b F(u)\, du = -(G(b) - G(s)) = G(s)-G(b)[/tex]

This must hold for all b, so let [itex]b\rightarrow \infty[/itex], knowing [itex]G(b)\rightarrow 0[/itex] and switching sides:

[tex]G(s) = \int_s^\infty F(u)\, du = \int_s^\infty F(s)\, ds [/tex]

This builds in the constant of integration in the formula.
 

What is a Laplace transform proof?

A Laplace transform proof is a mathematical technique used to solve differential equations. It involves transforming the original equation from the time domain to the frequency domain using the Laplace transform, which simplifies the equation and makes it easier to solve.

Why do we use Laplace transform proofs?

Laplace transform proofs are used because they allow us to solve complex differential equations that may be difficult or impossible to solve using traditional methods. The Laplace transform also has many useful properties, such as linearity and the ability to handle initial conditions, making it a powerful tool in solving mathematical problems.

What are the steps involved in a Laplace transform proof?

The first step in a Laplace transform proof is to take the Laplace transform of the original equation. This involves replacing the time variable with the Laplace variable s and integrating over time. The next step is to manipulate the transformed equation to solve for the desired variable. Finally, the inverse Laplace transform is taken to convert the solution back to the time domain.

What are the applications of Laplace transform proofs?

Laplace transform proofs have many applications in engineering, physics, and other scientific fields. They are commonly used to analyze systems with time-dependent inputs, such as electrical circuits, mechanical systems, and chemical reactions. They are also used in control theory and signal processing.

What are the limitations of Laplace transform proofs?

One limitation of Laplace transform proofs is that they can only be applied to linear differential equations. They are also not always applicable to systems with discontinuous or non-smooth inputs. Additionally, the Laplace transform may not converge for certain functions, making it impossible to solve using this method.

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