This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ rather than in terms of wavenumber $k$ and position $x$, so I will use the substitutions $k \rightarrow \nu$ and $x \rightarrow t$ rewrite this as: $$F(\nu) = \int_{-\infty}^{\infty} f(t) e^{-i2\pi \nu t}dt \; \; \; \text{(eq. 1).}$$
Is this substitution valid, or did I miss a factor?
Now, angular frequency $\omega$ and linear frequency $\nu$ are related by $\omega = 2 \pi \nu$ so I can rewrite in terms of the angular frequency $\omega$: $$F({\omega\over2\pi}) = F(\nu) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t}dt \; \; \; \text{(eq. 2).}$$ However, my physics professor's distributed notes give this definition of the Fourier transform $F(\omega)$ of $f(t)$: $$F(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} f(t) e^{-i\omega t}dt \; \; \; \text{(eq. 3)}$$
How can I convert equation (2) to equation (3), to obtain the $\frac{1}{2\pi}$ factor?