Section 2.1 The Gamma Function
Subsection 2.1.1 Interpolation
An interpolation problem is one where we are given input data \(z_1, \ldots, z_n\) and output data \(w_1, \ldots, w_n\text{,}\) and we are asked to find an interpolating function \(f\) subject to some kind of conditions (e.g. continuity, bounded, differentiable, etc) so that
\begin{equation*}
f(z_i) = w_i \text{ for all } i=1, \ldots n.
\end{equation*}
As an example, consider the problem of constructing a polynomial that interpolates the points \((z_1, w_1), (z_2, w_2), (z_3, w_3) \in \R^2\text{.}\) If these points arent collinear, one approach to solving the problem is to use elbow grease - we can write a generic parabola \(f(z) = az^2 + bz + c\) and solve the system of three equations that arise from plugging in each of the points. Another approach, based on noticing patterns, is called Lagrange interpolation. We notice that
\begin{equation*}
f_1(z) = \frac{(z - z_2)(z-z_3)}{(z_1 - z_2)(z_1 - z_3)}
\end{equation*}
is a function so that \(f_1(z_2) = f_1(z_3) = 0\) and that \(f_1(z_1) = 1\text{.}\) We can construct similar functions \(f_2\) and \(f_3\text{.}\) Then we can assemble these into a interpolating function by multiplying the correct piece by the desired output value -
\begin{equation*}
f(z) = w_1 f_1(z) + w_2 f_2(z) + w_3 f_3(z).
\end{equation*}
It turns out that the Lagrange interpolating function will be the unique polynomial of lowest degree that interpolates the data.
Of course, we can construct more exotic interpolation problems. We're going to begin with a problem where we wish to interpolate an infinite number of data points.
Subsection 2.1.2 The Gamma Function
We wish to construct a function so that \(f(n) = n!\) - that is, we want a function that interpolates the factorial function. (One way of thinking about the objective is to give an expression like \((1/2)!\) meaning, as formulas involving factorials can often be extended to take non-integer values). We'll want our function to be continuous and to be defined on the largest domain we can construct.
Mathematicians of the 18th century, in the course of working out the basics of the theory of integral transforms, noticed that something special happened to the integral
\begin{equation*}
\int_0^\infty f(t) e^{-t}\, dt
\end{equation*}
when \(f(t) = t^n\text{.}\)
Following Bernoulli (and Legendre), make the following definition involving our interesting integral.
\begin{equation*}
\Gamma(n) = \int_0^\infty t^{n-1} e^{it} \, dt.
\end{equation*}
Using integration by parts, we can derive
\begin{align*}
\Gamma(n) \amp= \int_0^\infty t^{n-1} e^{it} \, dt\\
\amp= -t^{n-1}e^{it}\bigg\vert_0^\infty + \int_0^\infty(n-1)t^{n-2}e^{-t}\, dt\\
\amp= (n-1) \int_0^\infty t^{n-2} e^{-t}\, dt\\
\amp= (n-1)\Gamma(n-1).
\end{align*}
Coupled with the observation that \(\Gamma(1) = 1\text{,}\) we get that
\begin{equation*}
\Gamma(n+1) = n!.
\end{equation*}
Now, the nice thing about the form of the integral in the definition of \(\Gamma\) is that we can apply it to any \(x \gt 0\text{,}\) and in turn \(\{z: \RE z \gt 0\}\) and get a convergent integral. Thus, the form of the gamma function allows the continuation of the function off of the positive integers and onto the entire right half plane.
The equation
\begin{equation*}
\Gamma(z+1) = z \Gamma(z)
\end{equation*}
is an example of a functional equation (or an equation where the unknowns are functions), and it turns out that we can use this equation to extend the domain of the gamma function beyond the right half plane.
Suppose that we are given a point \(z_0\) with \(0 > \RE z_0 > -1\text{.}\) The integral that defines \(\Gamma\) on the right half plane isn't convergent on this value. It is reasonable to expect that if \(\Gamma(z_0)\) is to be made meaningful, then the functional equation should hold. That is, we desire
\begin{equation*}
z_0 \Gamma(z_0) = \Gamma(z_0 + 1).
\end{equation*}
Accordingly, we can define \(\Gamma\) at \(z_0\) via
\begin{equation*}
\Gamma(z_0) := \frac{\Gamma(z_0 +1)}{z_0}.
\end{equation*}
Notice that this definition works for any \(z\) with \(\RE z > -1\) with the exception of the point \(z = 0\text{,}\) which is a pole of our extension. Thus, we've extended the domain of \(\Gamma\) to the strip \(-1 \lt \RE z \leq 0\) with the exception of the pole at \(z = 0\text{.}\) Proceeding inductively, we can continue to define further strips in terms of the previous using the functional equation. The resulting function is the unique meromorphic function defined this way on all of \(\C - \{0, -1, -2, -3, \ldots\}\text{.}\)
