What Is the V-Hat Object in RC Circuit Filtering?

[NeuroRYoung]

I bought a biophysics book the other day, and immediately began reading it. Sadly, I'm only 12 pages in now, because I seem to have run up against a wall in my understanding. You see, I don't understand a step being taken in the math, and am not sure what I should learn to understand it better.

image of paragraph.

Specifically, what loses me is the V-hat object - no idea what that is. And furthermore, I don't know where the 2pi*i term comes from. I know the bare basics of Fourier Transform, but I think there's something missing. And Koch doesn't define the V-hat object. (I'm life sciences, please forgive my ignorance.)

Any tips would be majorly appreciated!

 

[atyy]

Does Koch define things in Appendix B?

 

[NeuroRYoung]

Ya, I read Appendix B. It doesn't seem to go into the matter. Or at least if it does, I can't see the connection.

Appendix B
pg.1 .... http://imgur.com/mZCodBh
pg.2 .... http://imgur.com/OtEQ7J6
pg.3 .... http://imgur.com/TVA7EV1

Moreover, V-hat never showed up before that paragraph. Nor is it in the symbol table in the beginning of the book. I must be missing something - Koch is a physicist; I'm guessing it's something imported the hat notation from his field?

 

[atyy]

Hmm, I don't know what the V-hat is, but the 2*pi*i object comes from the Fourier transform of the derivative eg. http://math.stackexchange.com/questions/430858/fourier-transform-of-derivative (Nicholas R. Peterson's f is Koch's V, and his $\xi$ is Koch's f).

 

[NeuroRYoung]

@atyy Super! Thanks for the helpful link. Clarifies at least some of what's going on.

 

[atyy]

This may also be helpful. It uses the Laplace transform, instead of the Fourier transform, to solve an RC circuit, but the ideas are the same, since the Laplace and Fourier transforms are similar.
http://www.stanford.edu/~boyd/ee102/laplace.pdf