Let's take a time domain function $x(t) = \cos( 2 \pi f_0 t) $. Its Fourier transform can be represented as
$$X(f) = \frac{1}{2} \left[ \delta(f - f_0) + \delta(f + f_0) \right]\tag{1}$$
as well as
$$X(j\omega) = \pi \left[ \delta(\omega - \omega_0) + \delta(\omega + \omega_0) \right]\tag{2}$$ which was initially wrong, corrected for clarity.
Considering $ \omega = 2 \pi f $, I can interchange between the two expressions. I can tell that the $j$ in the argument of the second expression can be accounted for by the $j$ in the RHS of the first one but I can't seem to get $(1)$ from $(2)$.
Starting with $(2)$:
\begin{align} X(j\omega) &= \pi \left[ \delta(\omega - \omega_0) + \delta(\omega + \omega_0) \right]\\ X(j 2 \pi f) &= \pi \left[ \delta( 2 \pi f - 2 \pi f_0) + \delta( 2 \pi f + 2 \pi f_0) \right] \end{align}
I can't see how to get $(1)$ from here. I know it is very simple but I can't get any good results by Googling it up. I'd appreciate if you do not downvote my post.