What do Laplace and Fourier transforms actually show about a function?

[dijkarte]

Thought it's pretty related to this forum. I'm familiar with applications of both Laplace and Fourier in physics and differential equations. However I still struggling trying to figure out the intuitive interpretation of both transforms or at least a mathematical illustration that shows their nature. We apply Laplace to a function but what do we get relative to the function? All the texts I have been reading state formulas and with a minimal proof/deduction if any. And the only interpretation they have is it transforms from one domain to another. Any good detailed reading that explains why and what exactly these transforms do?

Thanks.

 

[dijkarte]

As I'm doing some reading about the derivation of Fourier series in complex form, how the following sums up? Sorry that I lack the common sense of an average math student, but I insist to learn :)

sum(1, inf){ c[n]*exp(inwt) + c[-n]*exp(-inwt) } = sum(-inf, inf){ c[n]*exp(inwt) } ?

No single book shows the derivation.

 

[micromass]

Read this thread: https://www.physicsforums.com/showthread.php?t=461628