# Laplace transformation t^(3/2)

• jrsweet
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.
jrsweet

## Homework Statement

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

## 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?

t1/2 and t3/2 are different functions, so it should be expected that their Laplace transforms would be different.
L{tk - 1} = $\Gamma$(k)/sk, where $\Gamma$ represents the gamma function.

$$\Gamma(z)~=~\int_0^{\infty} t^{z - 1}~e^{-t}dt$$
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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