The factorial

$$n!=1\cdot2\cdots n$$

is naturally defined for nonnegative integers. The Gamma function extends the same multiplicative pattern to a continuous variable.

For $s>0$,

$$\Gamma(s)=\int_0^\infty t^{s-1}e^{-t},dt.$$

Recurrence

Integration by parts gives

$$\Gamma(s+1)=s\Gamma(s).$$

Since

$$\Gamma(1)=1,$$

we obtain

$$\Gamma(n+1)=n!$$

for positive integers $n$.

Half-integers

A central value is

$$\Gamma\left(\frac12\right)=\sqrt\pi.$$

The recurrence then determines all positive half-integer values, explaining why powers of $\pi$ appear in formulas that alternate between odd and even dimensions.

Hypersphere volume

The volume of an $n$-dimensional ball can be written compactly as

$$V_n(R)=\frac{\pi^{n/2}}{\Gamma\left(\frac n2+1\right)}R^n.$$

For integer $n$, this one expression reproduces the familiar lengths, areas and volumes and continues them consistently to every higher dimension.

The Gamma function is therefore more than notation for exotic factorials: it is the natural analytic object generated when factorial-like recurrences interact with integration.