% Lecture notes for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/10/28

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\begin{document}

\section{Gamma function}

\subsection{Definition and holomorphic structure}

The \href{http://en.wikipedia.org/wiki/Gamma_function}{\keyword{gamma function}} $\fn{\Gamma}{z}$ is usually represented as the analytic continuation of
\begin{equation} \label{gf}
\fn{\Gamma}{z} = \int_0^\infty t^{z-1} e^{-t} \d{t}
\qquad \left( \Re z > 0 \right)
\end{equation}
Integrating by parts gives
\begin{equation} \label{prop}
\fn{\Gamma}{z} = \frac{\fn{\Gamma}{z+1}}{z}
\qquad \left( z \in \C \right)
\end{equation}
Note that although \eq{prop} was derived from \eq{gf} under the condition $\Re z > 0$, it can be extended to the whole complex plane by analytic continuation.
The particular value $\fn{\Gamma}{1} = 1$ combined with \eq{prop} gives
\begin{equation}
\fn{\Gamma}{n+1} = n!
\qquad \left( n \in \N \right)
\end{equation}
The gamma function is the natural generalization of the factorial function to a holomorphic function.
The singularities of the gamma function are simple poles at zero and the negative integers, as can be seen using \eqs{gf}{prop}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[x=0.03\textwidth,y=0.03\textwidth]
% axes
\draw[->] (-3.5,0) -- (3.5,0) node[right]{$\Re z$};
\draw[->] (0,-3.5) -- (0,3.5) node[above]{$\Im z$};
% poles
\foreach \x in {0,-1,-2,-3} \fill[red] (\x,0) circle (0.15);
\end{tikzpicture}
\caption{
$\fn{\Gamma}{z}$ has simple poles at zero and the negative integers.
}
\end{figure}

\subsection{Saddle point approximation}

The asymptotic behavior of the gamma function as $|z| \to \infty$ can be derived from \eq{gf} using the \keyword{saddle point approximation}.
Changing the variable of integration to $s = t/z$ gives
\begin{equation} \label{gammasaddle}
\fn{\Gamma}{z} = z^z \int_0^{z^{-1}\infty} s^{-1} e^{z(\ln s - s)} \d{s}
\qquad \left( \Re z > 0 \right)
\end{equation}
which is an integral of the form
\begin{equation}
\fn{I}{z} = \int_C \fn{g}{s} e^{z \fn{f}{s}} \d{s}
\end{equation}
with $C = [0,z^{-1}\infty)$, $\fn{g}{s} = s^{-1}$ and $\fn{f}{s} = \ln s - s$.
For large $|z|$, the integral will be dominated by contributions from points with maximal $\Re(z f)$, to give the maximal value, and along a path with stationary $\Im(z f)$, to avoid cancellations. Using Eq.~(Q2.2.1), the conditions of maximal $\Re(z f)$ and stationary $\Im(z f)$ are satisfied at an extremum\footnote{Or possibly at an endpoint of $C$, in which case a slightly different method should be used.}, $df/ds = 0$, and from Eq.~(2.1.13) all extrema are saddle points. Taylor expanding about the relevant saddle point $s = s_0$, we approximate
\begin{equation}
\fn{I}{z} \sim \fn{g}{s_0} e^{z \fn{f}{s_0}} \int_C \exp \left[ \frac{1}{2} z \fn{f''}{s_0} \left( s - s_0 \right)^2 \right] \d{s}
\end{equation}
\begin{figure}
\centering
\begin{tikzpicture}[x=0.01\textwidth,y=0.01\textwidth]
% axes
\draw[->] (-10,0) -- (20,0) node[right]{$\Re s$};
\draw[->] (0,-20) -- (0,10) node[above]{$\Im s$};
% saddle
\begin{scope}[shift={(10,0)}]
\begin{scope}[rotate=-30]
\clip (0,0) circle(3);
 % $\Re f$ contours
 \foreach \a/\ashade in {0/55,1/65,2/75,3/85,4/95}
 \fill[variable=\x,domain=-4:4,samples=50,smooth,blue!\ashade] plot (\x,{sqrt((\x)^2 + 2*\a)}) plot (\x,{-sqrt((\x)^2 + 2*\a)});
 \foreach \a/\ashade in {0/45,-1/35,-2/25,-3/15,-4/5}
 \fill[variable=\y,domain=-4:4,samples=50,smooth,blue!\ashade] plot ({sqrt((\y)^2 - 2*\a)},\y) plot ({-sqrt((\y)^2 - 2*\a)},\y);
 % $\Im f$ contours
 \draw[variable=\z,domain=-4:4,smooth,red!50] plot (\z,0) plot (0,\z);
 \foreach \a/\ashade in {4/90,3/80,2/70,1/60,-1/40,-2/30,-3/20,-4/10}
 \draw[variable=\x,domain=0.2:4,smooth,red!\ashade] plot (\x,{\a/(-\x)}) plot (-\x,{\a/(\x)});
\end{scope}
\end{scope}
% contour
\draw[thick,green] (0,0) to[out=0,in=150] ($(10,0) + (-30:-3)$);
\draw[thick,green,->] ($(10,0) + (-30:3)$) to[out=-30,in=120] (-60:20) node[below right,black]{$z^{-1}\infty$};
\end{tikzpicture}
\caption{ \label{saddle}
The curve {\color{green} $C$} is deformed to take the steepest path over the saddle. The integral is then dominated by contributions from the neighborhood of the saddle point. {\color{blue} $\Re(z f)$} contours are shaded blue, and {\color{red} $\Im(z f)$} contours are drawn in red, with a stronger color indicating a larger value.
}
\end{figure}
Taking the steepest path over the saddle
\begin{equation}
\left( s - s_0 \right)^2 = - \frac{u^2}{z \fn{f''}{s_0}}
\qquad \left( u \in \R \right)
\end{equation}
see Figure~\ref{saddle}, and using the Gaussian integral $\int_{-\infty}^\infty e^{-\frac{1}{2}u^2} \d{u} = \sqrt{2\pi}$, gives
\begin{equation}
\fn{I}{z} \sim \pm \fn{g}{s_0} e^{z \fn{f}{s_0}} \sqrt{\frac{2\pi}{-z \fn{f''}{s_0}}}
\end{equation}
where the sign of the square root is determined by the direction of the path over the saddle.

In the case of the gamma function, \eq{gammasaddle}, substituting $s_0 = 1$, $\fn{g}{s_0} = 1$, $\fn{f}{s_0} = -1$ and $\fn{f''}{s_0} = -1$ gives Stirling's formula
\begin{equation} \label{stirling}
\fn{\Gamma}{z} = \sqrt{2\pi} \, e^{-z} z^{z-\frac{1}{2}} \left[ 1 + \order{\frac{1}{z}} \right]
\qquad \left( |z| \to \infty , |\arg z| < \pi \right)
\end{equation}
Note that the asymptotic form of the gamma function has a branch cut corresponding to the infinite series of poles of the gamma function along the negative real axis.

\subsection{Asymptotic series}

\eq{stirling} gives the first term in the \keyword{asymptotic series}
\begin{equation} \label{lnGamma}
\ln \fn{\Gamma}{z} = z \ln z - z - \frac{1}{2} \ln z + \ln \sqrt{2\pi}
+ \sum_{n=1}^N \frac{B_{2n}}{2n(2n-1)z^{2n-1}}
+ \order{\frac{1}{z^{2N+1}}}
\end{equation}
as $|z| \to \infty$ with $|\arg z| < \pi$.
The \href{http://en.wikipedia.org/wiki/Bernoulli_number}{\keyword{Bernoulli numbers}}
\begin{equation}
B_{2n} = (-1)^{n+1} \frac{2(2n)!}{(2\pi)^{2n}} \fn{\zeta}{2n}
\end{equation}
and the \href{http://en.wikipedia.org/wiki/Riemann_zeta_function}{\keyword{Riemann zeta function}}
\begin{equation}
\fn{\zeta}{s} = \sum_{n=1}^\infty \frac{1}{n^s}
\qquad \left( \Re s > 1 \right)
\end{equation}
The error in \eq{lnGamma} is $\order{z^{-(2N+1)}}$ as $|z| \to \infty$ for fixed $N$, although the series diverges as $N \to \infty$ for fixed $|z|$.
For finite $|z|$, there will be an optimum $N$ that minimizes the error, and this optimum $N$ increases as $|z|$ increases.
In some sense, although the series is divergent, it converges better for larger $|z|$.

Specifically, for large $n$ the ratio of successive terms is
\begin{equation}
\frac{2n(2n-1)B_{2n+2}}{(2n+2)(2n+1)B_{2n}z^2} \sim - \frac{n^2}{\pi^2 z^2}
\end{equation}
and so the optimum $N$ is
\begin{equation}
N \sim \pi |z|
\end{equation}

\end{document}

