Off-hand, I would say that oscillates around the unit circle in the complex plane. So that's the oscillation being described. Does that answer your question?
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:
Can someone explain how that integral compares to an oscillation of ?The Fourier Transform's value at a particular value of is a measure of the extent to which oscillates with a frequency 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