Laplace transformation t^(3/2)

In summary, the conversation involved a question about finding the Laplace transform of a function with t3/2, which led to a formula being used to find the transform and a clarification on the difference between t1/2 and t3/2. The solution was provided using the gamma function.
  • #1
jrsweet
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Homework Statement


Just a quick question concerning a Laplace transformation...
Find the Laplace transform of the following function:
f(t)=10t3/2-e(-7t)


Homework Equations





The Attempt at a Solution


I wasn't sure what to do with the t3/2 so I just followed the formula for t1/2 and I got this:

(10pi(3/2))/(2s5/2)-(1/(s+7))

Is there anything wrong with this?
 
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  • #2
t1/2 and t3/2 are different functions, so it should be expected that their Laplace transforms would be different.
L{tk - 1} = [itex]\Gamma[/itex](k)/sk, where [itex]\Gamma[/itex] represents the gamma function.

[tex]\Gamma(z)~=~\int_0^{\infty} t^{z - 1}~e^{-t}dt[/tex]
Hope that helps.
 

1. What is Laplace transformation t^(3/2)?

Laplace transformation t^(3/2) is a mathematical operation that converts a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems in the frequency domain.

2. How is Laplace transformation t^(3/2) performed?

To perform Laplace transformation t^(3/2), you first need to know the Laplace transform of the function t^(3/2). In this case, the Laplace transform is 3/(s^(5/2)). Then, using the basic Laplace transform properties, you can apply the Laplace transform to the original function, t^(3/2).

3. What are the applications of Laplace transformation t^(3/2)?

Laplace transformation t^(3/2) is commonly used in engineering and physics to analyze systems in the frequency domain. It is also used to solve differential equations, particularly those that involve fractional powers of time.

4. What are some important properties of Laplace transformation t^(3/2)?

Some important properties of Laplace transformation t^(3/2) include linearity, time-shifting, differentiation, and integration. These properties make it a powerful tool for solving differential equations and analyzing systems in the frequency domain.

5. Are there any limitations to Laplace transformation t^(3/2)?

Yes, there are some limitations to Laplace transformation t^(3/2). It may not be applicable to all functions, particularly those that do not have a finite or well-defined Laplace transform. Additionally, it cannot be used to solve differential equations that involve discontinuous or impulsive functions.

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