Laplace transform of exponential function

In summary, the individual is attempting to find the Laplace transform of exp(-2r/a)r^2 and has verified through integration by parts that the integral of this expression is equal to a^3/4. However, they are unable to obtain the Laplace transform using this method and are questioning the validity of the result given by Mathematica Online. They are also trying to determine if it is possible to find the Laplace transform for this expression.
  • #1
jaejoon89
195
0
I'm trying to find the laplace transform (if possible) of exp(-2r/a)r^2.

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I did integration by parts to check that the integral of exp(-2r/a)r^2 from 0 to infinity is a^3 / 4. but i cannot get the laplace transform to work out. the answer mathematica online gives would have to have the term in the denominator "as" = 0 for the whole thing to become a^3 / 4. I'm trying to figure out how this would happen, and if you can even do the laplace transform for this in the first place.

http://www.wolframalpha.com/input/?i=LT+e^(-2t/a)+t^2+
 
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  • #2
The whole thing isn't a^3/4. It's a function of s. I agree with Mathematica.
 

FAQ: Laplace transform of exponential function

What is the Laplace transform of an exponential function?

The Laplace transform of an exponential function is a mathematical operation that converts a function of time into a function of complex frequency. It is denoted by the symbol L{f(t)} and is defined as the integral of the function multiplied by the exponential function e^-st, where s is a complex number.

What is the formula for the Laplace transform of an exponential function?

The formula for the Laplace transform of an exponential function is L{e^at} = 1/(s-a), where a is a constant. This means that the transformed function is a fraction with the denominator being the difference between the complex number s and the constant a.

What is the Laplace transform of e^-at?

The Laplace transform of e^-at is 1/(s+a), where a is a constant. This is the inverse of the formula L{e^at} = 1/(s-a). In other words, the transformed function is a fraction with the denominator being the sum of the complex number s and the constant a.

What is the significance of the Laplace transform of an exponential function?

The Laplace transform of an exponential function is an important tool in mathematics and engineering, as it allows us to solve differential equations and analyze complex systems. It is also used to find the frequency response of a system, which is crucial in understanding its behavior and stability.

What are some properties of the Laplace transform of an exponential function?

Some properties of the Laplace transform of an exponential function include linearity, time shifting, and differentiation. Linearity means that the transform of a linear combination of functions is equal to the linear combination of their individual transforms. Time shifting allows us to shift the function in time by adding a constant in the exponential term. Differentiation in the time domain corresponds to multiplication by the Laplace variable s in the frequency domain.

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