I'm given the function
$f(x) = Acos(kx)$
From the tabulated Fourier transform of $cos$, the transform is of the form
$\hat{f}(k) = \pi (\delta (k-k_o) + \delta (k+k_o))$
but there are a couple questions I have. First and foremost, the tabulated Fourier transform pairs correspond to the definitions
$f(t) = \frac{1}{2\pi}\int^\infty_{-\infty}F(w)e^{jwt}dw, \quad F(w)=\int^\infty_{-\infty}f(t)e^{-jwt}dt$
but the book I'm following defines the transform pairs with an alternative definition, where the coefficient is interchanged as
$f(t) = \int^\infty_{-\infty}F(w)e^{jwt}dw, \quad F(w)=\frac{1}{2\pi}\int^\infty_{-\infty}f(t)e^{-jwt}dt$
which makes me wonder if tabulating the solution from the source is incorrect due to the different coefficient conventions. Do I have to do something to the tabulated solutions such that they correspond to my conventions?
Second question is what is $k_o$ in the tabulated solution? Is that simply the value defined in my problem? In other words, in the solution it has a numerical value, whereas "k" remains the variable.
The last issue is the constant A. My intuition says to simply multiplied that to the tabulated transform solution, but my mathematical intuition often finds a way to fail me (why I'm here).
To help facilitate my understanding, I've assumed some random numerical values in my initial function
$f(x) = 3cos(0.7x)$
and from the tabulated Fourier transform, along with my assumptions, I believe the solution is
$\hat{f}(k) = 3\pi (\delta (k-0.7) + \delta (k+0.7))$
Wolfram gives me something similar,
3 sqrt(π/2) (δ(0.7 - ω) + δ(ω + 0.7))
I think they use a different convention in the definition of the transform pairs? This further emphasises my first question on understanding how to treat the tabulated transform solutions when different conventions are presented. I also noticed they have a sign difference with 0.7 - ω whereas I have k - 0.7. I'm not sure where the sign change comes from...maybe related to the convention?