3-D generalization of the Gaussian point spread function

Take a look here to see the definition and integration. In short, if you want a Gaussian of the form: $$N\exp\left(-\frac{x^2+y^2+z^2 + ...}{2\sigma^2}\right) \,,$$ then the constant $N$ depends on the number of variables $n$: $$N = \frac1{\sigma^{n}(2\pi)^{n/2}} \,.$$ So in your case, $n=3$, and the normalization constant is: $$\frac{1}{\sigma^3 (2\pi)^{3/2}} \,.$$ Note that for the 2D case, this is $1/ 2\pi \sigma^2$, i.e. you are missing a $\sigma$. This is intuitive, since $\sigma$ has the dimension of the length of whatever you are trying to measure. Integration over $n$ dimensions adds $n$ powers of that length, and since after integration we get a unit-less quantity (probability), the power of $\sigma$ in the Gaussian must always be $-n$.