Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to hard to understand integral's transform.

While an interpretation of Fourier transform is relatively easy to furnish in terms of signal decomposition and harmonics, it seems the "meaning" of Laplace transfrom is more subtle (in spite of the similarities between the two).

I have then been through the derivations.

In Fourier series a function of period 2π is expressed as a series of sines and cosines

$$ f(x) = \sum_{k=0} ^{\infty} a_n cos (nx) + b_n sin (nx)$$

The coefficient a_n and b_n are then found using the orthogonality of functions such as sin(nx), cos(nx).

Considering an aperiodic function one considers the limit of the period to infinity and demonstrates the series converges to an integral.

All this makes a physical meaning very clear: the functions is decomposed onto a basis of harmonic functions.

Why is not the same approach followed for Laplace transforms?

One could start by

$$ f(x) = \sum_{k=0} ^{\infty} a_n e^{b_n x}$$

and try to follow the same approach.

One really would need to have some information on a_n in order to write in the limit the proper integral, for I am not so sure all I am saying make any sense.. If it did, the Laplace transform could be seen as a decomposition on an expnentail basis (by the way, I remember reading somehwere that the exponentials form a complete basis for continous functions, is this true?)

Am I missing something here? Can one follow this approach, and conclude the Laplace transform is the integral representation of a series of exponentials?

Many thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Physical Interpretation of Laplace's Transform

Loading...

Similar Threads for Physical Interpretation Laplace's |
---|

I Meaning of the word "Harmonic" in different contexts |

What is tensor calculus? |

**Physics Forums | Science Articles, Homework Help, Discussion**