The introduction to the Laplace transform chapter in my DE textbook says (I'm paraphrasing some of this):

In general, integral transforms address the question: How much is a given function "like" a particular standard function?

For example, if represents a radio signal and we want to compare it to with frequency , just integrate:

.

If goes up when goes up, and goes down when goes down, then the integral will be large, because is very much "like" . So far, this makes perfect sense.

Then, it says you can compare to , where is Complex, using:

Which can be split into

and .

The last (bold) of those integrals is called the Fourier Transform (which is what I really need to learn). Then the book says:

The Fourier Transform's value at a particular value of is a measure of the extent to which oscillates with a frequency of .Can someone explain how that integral compares to an oscillation of ?

I am aware of Euler's equation:

If I apply it to just the Fourier integral, I get:

.

Unless the is always zero, I don't see how this can make that comparison.

Thanks for your explanations.

Jeff