| definition | $f:$ ?? |
| definition | $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$ |
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The limit is clear using
$\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^{a\,n+b}=\lim_{n\to\infty}\left(1+\dfrac{a\,x}{n}\right)^n={\mathrm e}^{a\,x}$.
Here the normalization:
Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-6,6}]
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Wikipedia: Student's t-distribution