Laplace transform and fourier transform

In summary, F{f(t)} is the Fourier transform of f(t) and L{f(t)} is the Laplace transform of f(t). The definition of F{f(t)} involves an integral with limits from negative infinity to positive infinity, while the definition of L{f(t)} involves an integral with limits from negative infinity to positive infinity. When s is substituted with jw in the integral for L{f(t)}, it becomes equivalent to the integral for F{f(t)}, therefore F{f(t)} = L{f(t)} where s = jw in L{f(t)}. This is correct as long as the imaginary axis is within the region of convergence.
  • #1
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Homework Statement


F{f(t)} is the Fourier transform of f(t) and L{f(t)} is the Laplace transform of f(t)

why F{f(t)} = L{f(t)} where s = jw in L{f(t)}


The Attempt at a Solution


I suppose the definition of F{f(t)} is

∫[f(t)e^-jwt]dt

where the lower integral limit is -∞ and higher intergral limit is +∞.

And I suppose the definition of L{f(t)} is

∫[f(t)e^-st]dt

where the lower integral limit is -∞ and higher integral limit is +∞.(that is bilateral Laplace transform)

and i think it is obviously to say F{f(t)} = L{f(t)} where s = jw in L{f(t)} just by substitute s = jw in ∫[f(t)e^-st]dt.

My solution is so simple that I can't believe it's a problem assigned by my professor!
Some guy please tell me if I am correct or not, and where it is.

Any reference or advise will be appreciated.

thanks in advance.
 
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  • #2
Yes you are correct, as long as the imaginary axis is inside the region of convergence.
 
  • #3
susskind_leon said:
Yes you are correct, as long as the imaginary axis is inside the region of convergence.

3x~ I am more confident~
 

1. What is the difference between Laplace transform and Fourier transform?

The Laplace transform and Fourier transform are two mathematical techniques used to analyze signals or functions in the time domain. The main difference between the two is that the Laplace transform is used for studying signals that are defined for all time, while the Fourier transform is used for signals that are defined for a finite period of time.

2. How are Laplace and Fourier transforms related?

The Laplace transform is actually an extension of the Fourier transform, as it allows for the analysis of signals that are not periodic. The Fourier transform is a special case of the Laplace transform when the imaginary part of the Laplace variable is equal to zero.

3. What is the purpose of using Laplace and Fourier transforms?

Both techniques are used to convert a function or signal from the time domain to the frequency domain. This allows for a better understanding and analysis of the signal, as well as simplifying calculations for certain operations, such as differentiation and integration.

4. What are the applications of Laplace and Fourier transforms?

The Laplace and Fourier transforms have numerous applications in various fields, including signal processing, control systems, and communication systems. They are also used in solving differential equations, analyzing electrical circuits, and in the study of vibrations and waves.

5. Are there any limitations to using Laplace and Fourier transforms?

One limitation of the Laplace and Fourier transforms is that they assume the signal or function is continuous. They may not be suitable for discrete signals or functions, such as digital signals. Additionally, the calculations involved in these transforms can be complex and time-consuming, making them challenging to use in certain applications.

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