Editing Gamma function (section)

=== Pi function ===
An alternative notation introduced by [[Carl Friedrich Gauss|Gauss]] is the <math>\Pi</math>-function, a shifted version of the gamma function:
<math display="block">\Pi(z) = \Gamma(z+1) = z \Gamma(z) = \int_0^\infty e^{-t} t^z\, dt,</math>
so that <math>\Pi(n) = n!</math> for every non-negative integer <math>n</math>.

Using the pi function, the reflection formula is:
<math display="block">\Pi(z) \Pi(-z) = \frac{\pi z}{\sin( \pi z)} = \frac{1}{\operatorname{sinc}(z)}</math>
using the normalized [[sinc function]]; while the multiplication theorem becomes:
<math display="block">\Pi\left(\frac{z}{m}\right) \, \Pi\left(\frac{z-1}{m}\right) \cdots \Pi\left(\frac{z-m+1}{m}\right) = (2 \pi)^{\frac{m-1}{2}} m^{-z-\frac12} \Pi(z)\ .</math>

The shifted [[reciprocal gamma function]] is sometimes denoted <math display="inline">\pi(z) = \frac{1}{\Pi(z)}\ ,</math>
an [[entire function]].

The [[volume of an n-ball|volume of an {{math|''n''}}-ellipsoid]] with radii {{math|''r''{{sub|1}}, …, ''r''{{sub|''n''}}}} can be expressed as
<math display="block">V_n(r_1,\dotsc,r_n)=\frac{\pi^{\frac{n}{2}}}{\Pi\left(\frac{n}{2}\right)} \prod_{k=1}^n r_k.</math>