I have a Gaussian with unitless output, but which is a function of time. The formula for this function is
$$ f(t)=\exp{(-\frac{t^2}{2c^2})} $$ Where both $t$ and $c$ are in units of time. I want to force the condition that the FWHM is equal to a specific time, $\tau$, so I enforce the condition $c=\frac{\tau}{\sqrt{2\ln{2}}}$ and now I proceed to normalize the function by adding some (presumably unit less) parameter A $$\int_{-\infty}^\infty \! A\exp{(-\frac{t^2}{2c^2})} \, \mathrm{d}t=Ac\sqrt{2\pi}=1$$ $$A=\frac{1}{c\sqrt{2\pi}}$$
However now I run into a problem, which is that since c has units of time, A has units of 1/time, and the output of the original function is no longer unitless. One way to rectify this is to instead integrate not with respect to time but with respect to some unitless parameter (such as $t/c$). Is it possible to normalize the original function (with unitless output) without changing the units of the output?
Thanks
If $f(t)$ is unitless, and $t$ has units of time, then $dt$ also has units of time, as does $$\int_{-\infty}^{\infty} f(t)\,dt.$$ Probabilities are themselves unitless, so if you have a probability density over a variable with some unit $U$, the probability density has to have units of $U^{-1}$, since it is probability per $U$.
So $f(t) = e^{-t^2/c^2}$ fails to be a probability density not just because its integral is not $1$, but also because it has the wrong units for a probability density over time. When you multiply $f(t)$ by $A = \frac 1{c\sqrt{2\pi}}$, you are correcting both problems and turning $f(t)$ into an actual probability density.