How do I know "what Fourier transform" to use?

[LCSphysicist]

Homework Statement:: .
Relevant Equations:: .

I am having a hard time thinking about Fourier transform, because there are so many conventions that i think i got more confused each time i think about it.

See an example, "Find the Fourier transform of $$V(t) = Ve^{iwt} \text{ if } nT \leq t \leq n(T + \tau) \text{ for } n = 0,1,...,N-1$$$$V(t) = 0 \text{ otherwise }$$

I don't know what Fourier transform to apply!

There is the convention $F(w) = \int V(t) e^{iwt} dt$, but there is also $F(w) = \int V(t) e^{-iwt} dt$.

Of course the second one would be more properly to this problem, but shouldn't both types of FT gives the same answer? Shouldn't they be equivalent?

Now, to let the things get even worst, is to talk about FT from Position to momentum. Everytime i tried to remember the expression, one new arose.

\begin{align*}
F(k) &= (2\pi)^{n/2} \int e^{-ikr} F(r) d^{n}(r) \\
f(k) &= \int d^3 x e^{-kx} f(x)
\end{align*}

I am not sure of this, but i think that all these expression are equivalent, and OK. THe problem is when the problem ask for the FT, as the one above. How the heck i know what convention i should use?

[Moderator's note: moved from homework to Calculus due to its general nature.]

 

[mathman]

Basic definition: https://en.wikipedia.org/wiki/Fourier_transform

$T(t)=\int_R f(x)e^{-2i\pi xt}dx$.

 

[DaveE]

Basic definition: https://en.wikipedia.org/wiki/Fourier_transform

$T(t)=\int_R f(x)e^{-2i\pi xt}dx$.

Which is the one everyone actually uses, I think. There is some ambiguity in how people deal with the $\frac{1}{2\pi}$ term in the inverse transform. Some put $\frac{1}{\sqrt{2\pi}}$ in front of both transform and inverse.

In any case, there is a burden on people to tell you which way they like to do these things. If you have to guess, I'd always guess the version above. If it's your own work, choose what works for you and tell everyone what you did (life will be easier if you choose the same thing they like too).

 

[jasonRF]

Basic definition: https://en.wikipedia.org/wiki/Fourier_transform

$T(t)=\int_R f(x)e^{-2i\pi xt}dx$.

I use Fourier transforms constantly, but never that convention. The only time I would use it would be when helping answer a question here on Physics forums where the OP used that convention.

The lack of a standard is kind of a pain. My advice is to use the convention most used in whatever field you are working in.

jason