Plotting a function rescaled by a parameter along x axis

Question

I would like to know, how to plot a graph, for example of this function:

$f(t)=-\frac{4 \sin \left(\frac{\gamma t}{2}\right)}{\gamma ^2 t}-\frac{4 \sin ^2\left(\frac{\gamma t}{4}\right)}{\gamma }-\frac{2 \cos \left(\frac{\gamma t}{2}\right)}{\gamma }+\frac{\pi t}{2}$ where $t$ is a variable and $\gamma$ is some constant parameter. Is there a possibility to plot it with x axis being in units of notjust $t$, but in units of $\gamma t$?

I was thinking it could be done something like this, but it does not work:

Plot[f(t), {\[Gamma]*t, 0, 40}]

Answer 1

This is your function:

f[g_,t_]:=-4Sin[g t/2]/g^2/t-4Sin[g t/4]^2/g-2Cos[g t/2]/g+Pi t/2

With transformed variable $z=g*t$ you get a new function:

h[g_,z_]=f[g,t]/.t->z/g

$$\frac{\pi z}{2 g}-\frac{4 \sin ^2\left(\frac{z}{4}\right)}{g}-\frac{4 \sin \left(\frac{z}{2}\right)}{g z}-\frac{2 \cos \left(\frac{z}{2}\right)}{g}$$

which you plot (with, for example, $g=1$) as

Plot[h[1, z], {z, 0, 10}]