Editing Gamma function (section)

=== Main definition ===
The notation <math>\Gamma (z)</math> is due to [[Adrien-Marie Legendre|Legendre]].<ref name="Davis" /> If the real part of the complex number&nbsp;{{mvar|z}} is strictly positive (<math>\Re (z) > 0</math>), then the [[integral]]
<math display="block"> \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\, dt</math>
[[absolute convergence|converges absolutely]], and is known as the '''Euler integral of the second kind'''. (Euler's integral of the first kind is the [[beta function]].<ref name="Davis" />) Using [[integration by parts]], one sees that:
[[File:Plot of gamma function in complex plane in 3D with color and legend and 1000 plot points created with Mathematica.svg|alt=Absolute value (vertical) and argument (color) of the gamma function on the complex plane|thumb|Absolute value (vertical) and argument (color) of the gamma function on the complex plane]]
<math display="block">\begin{align}
   \Gamma(z+1) & = \int_0^\infty t^{z} e^{-t} \, dt \\
&= \Bigl[-t^z e^{-t}\Bigr]_0^\infty + \int_0^\infty z t^{z-1} e^{-t}\, dt \\
&= \lim_{t\to \infty}\left(-t^z e^{-t}\right) - \left(-0^z e^{-0}\right) + z\int_0^\infty t^{z-1} e^{-t}\, dt.
\end{align}</math>

Recognizing that <math>-t^z e^{-t}\to 0</math> as <math>t\to \infty,</math>
<math display="block">\begin{align}
   \Gamma(z+1) & = z\int_0^\infty t^{z-1} e^{-t}\, dt \\
 &= z\Gamma(z).
\end{align}</math>

Then {{nowrap|<math>\Gamma(1)</math>}} can be calculated as:
<math display="block">\begin{align}
   \Gamma(1) & = \int_0^\infty t^{1-1} e^{-t}\,dt \\
   & = \int_0^\infty  e^{-t} \, dt \\
   & = 1.
\end{align}</math>

Thus we can show that <math>\Gamma(n) = (n-1)!</math> for any positive integer {{mvar|n}} by [[proof by induction|induction]].  Specifically, the base case is that <math>\Gamma(1) = 1 = 0!</math>, and the induction step is that <math>\Gamma(n+1) = n\Gamma(n) = n(n-1)! = n!.</math>

The identity <math display="inline">\Gamma(z) = \frac {\Gamma(z + 1)} {z}</math> can be used (or, yielding the same result, [[analytic continuation]] can be used) to uniquely extend the integral formulation for <math>\Gamma (z)</math> to a [[meromorphic function]] defined for all complex numbers {{mvar|z}}, except integers less than or equal to zero.<ref name="Davis" />  It is this extended version that is commonly referred to as the gamma function.<ref name="Davis" />