[help] A laplace transform definition problem

In summary, a Laplace transform is a mathematical tool used in engineering and physics to transform a function from the time domain to the frequency domain. The Laplace transform definition problem refers to the challenge of finding the Laplace transform of a given function, which involves applying the integral definition and solving for the unknown variable(s). The integral definition of the Laplace transform is ∫<sub>0</sub><sup>∞</sup> e<sup>-st</sup> f(t) dt, and the steps for solving a Laplace transform definition problem include writing out the integral, simplifying the function, finding the limits of integration, using integration techniques, and applying properties of Laplace transforms. Some common mistakes when solving these problems
  • #1
goodness52200
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http://xs104.xs.to/xs104/06322/ssss.gif [Broken]

Hello all ^^

What is the difference between the two definitions

thanks a lot
 
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  • #2
In a heuristic contest, the first one takes account of distributions/measures (for istance "Dirac delta function" and its derivatives) with support in the origin. The second one doesn't.
 
  • #3


The difference between the two definitions is in the variables used. The first definition uses the variable "s" while the second definition uses "p". These variables represent the complex frequency in the Laplace transform. The variable "s" is commonly used in engineering and physics applications, while "p" is more commonly used in mathematics. Both definitions are equivalent and can be used interchangeably.
 

What is a Laplace transform?

A Laplace transform is a mathematical tool used to transform a function from the time domain to the frequency domain. It is often used in engineering and physics to solve differential equations and analyze systems.

What is the Laplace transform definition problem?

The Laplace transform definition problem refers to the challenge of finding the Laplace transform of a given function. It involves applying the integral definition of the Laplace transform and solving for the unknown variable(s).

What is the integral definition of the Laplace transform?

The integral definition of the Laplace transform is ∫0 e-st f(t) dt, where s is a complex variable and f(t) is the function being transformed.

What are the steps for solving a Laplace transform definition problem?

The steps for solving a Laplace transform definition problem are as follows:

  1. Write out the integral definition of the Laplace transform.
  2. Simplify the function f(t) if possible.
  3. Find the limits of integration and substitute them into the integral.
  4. Use integration techniques to solve the integral and obtain the Laplace transform.
  5. If necessary, use properties of Laplace transforms to simplify the solution.

What are some common mistakes made when solving Laplace transform definition problems?

Some common mistakes made when solving Laplace transform definition problems include:

  • Forgetting to change the limits of integration when simplifying the function f(t).
  • Forgetting to apply the properties of Laplace transforms, such as linearity or time-shifting.
  • Misinterpreting the inverse Laplace transform as the Laplace transform.
  • Forgetting to include the negative sign when using the time-shifting property.
  • Not checking the final solution for errors.

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