The Gamma Function

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The Gamma Function
The Gamma function is an extension
of the factorial function, with its
argument shifted down by 1, to real
and complex numbers. The Gamma
function is defined by an improper
integral that converges for all real
numbers except the non-positive
integers, and converges for all
complex numbers with nonzero
imaginary part. The factorial is
extended by analytic continuation to
all complex numbers except the non-
positive integers (where the integral
function has simple poles), yielding
the meromorphic function we know as
the Gamma function.
The Gamma function has very many extremely important applications
in probability theory, combinatorics, statistical and quantum
mechanics, solid-state physics, plasma physics, nuclear physics, and
in the decades-long quest to unify quantum mechanics with the
theory of relativity - the development of the theory of quantum
gravity - the objective of string theory.
The problem of extending the factorial to non-integer arguments was
apparently first considered by Daniel Bernoulli and Christian Goldbach
in the 1720s, and was solved at the end of the same decade by
Leonard Euler. Euler gave two different definitions: the first was an
infinite product, of which he informed Goldbach in a letter dated
October 13, 1729. He wrote to Goldbach again on January 8, 1730, to
announce his discovery of the integral representation. Euler further
discovered some of the Gamma function's important functional
properties, notably the reflection formula.
Carl Friedrich Gauss rewrote Euler's product and then used his
formula to discover new properties of the Gamma function. Although

Euler was a pioneer in the theory of complex variables, he does not
appear to have considered the factorial of a complex number, as
Gauss first did. Gauss also proved the multiplication theorem of the
Gamma function and investigated the connection between the Gamma
function and elliptic integrals.
Karl Weierstrass further established the role of the Gamma function in
complex analysis, starting from yet another product representation.
Weierstrass originally wrote his product as one for 1/, in which case
it is taken over the function's zeros rather than its poles. Inspired by
this result, he proved what is known as the Weierstrass factorization
theorem - that any entire function can be written as a product over
its zeros in the complex plane; a generalization of the fundamental
theorem of algebra.
The name of the Gamma function and its symbol were introduced by
Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's
integral definition in its modern form. The alternative "Pi function"
notation (z) = z! due to Gauss is sometimes encountered in older
literature, but Legendre's notation is dominant in modern works. It is
justified to ask why we distinguish between the "ordinary factorial"
and the Gamma function by using distinct symbols, and particularly
why the Gamma function should be normalized to (n + 1) = n!
instead of simply using "(n) = n!". Legendre's motivation for the
normalization does not appear to be known, and has been criticized
as cumbersome by some (the 20th-century mathematician Cornelius
Lanczos, for example, called it "void of any rationality" and would
instead use z!). Legendre's normalization does simplify a few
formulas, but complicates most others.
A large number of definitions have been given for the Gamma
function. Although they describe the same function, it is not entirely
straightforward to prove their equivalence. Instead of having to find a
specialized proof for each formula, it would be highly desirable to
have a general method of identifying the Gamma function given any
particular form.
One way to prove equivalence would be to find a differential equation
that characterizes the Gamma function. Most special functions in
applied mathematics arise as solutions to differential equations,

whose solutions are unique. However, the Gamma function does not
appear to satisfy any simple differential equation. Otto Hlder proved
in 1887 that the Gamma function at least does not satisfy any
algebraic differential equation by showing that a solution to such an
equation could not satisfy the Gamma function's recurrence formula.
This result is known as Hlder's theorem.
A definite and generally applicable characterization of the Gamma
function was not given until 1922. Harald Bohr and Johannes Mollerup
then proved what is known as the Bohr-Mollerup theorem: that the
Gamma function is the unique solution to the factorial recurrence
relation that is positive and logarithmically convex for positive z and
whose value at 1 is 1 (a function is logarithmically convex if its
logarithm is convex).
The Bohr-Mollerup theorem is useful because it is relatively easy to
prove logarithmic convexity for any of the different formulas used to
define the Gamma function. Taking things further, instead of defining
the Gamma function by any particular formula, we can choose the
conditions of the Bohr-Mollerup theorem as the definition, and then
pick any formula we like that satisfies the conditions as a starting
point for studying the Gamma function. This approach was used by
the Bourbaki group.
G.P. Michon describes the Gamma function as "Arguably, the most
common special function, or the least 'special' of them. The other
transcendental functions . . . are called 'special' because you could
conceivably avoid some of them by staying away from many
specialized mathematical topics. On the other hand, the Gamma
function is most difficult to avoid."
The Gamma function finds application in diverse areas such as
quantum physics, statistical mechanics and fluid dynamics. The
Gamma distribution, which is formulated in terms of the Gamma
function, is used in statistics to model a wide range of processes; for
example, the time between occurrences of time-series events.
The primary reason for the Gamma function's usefulness is the
prevalence of expressions of the type f (t) exp(-g(t)) which describe
processes that decay exponentially in time or space. Integrals of such

expressions can often be solved in terms of the Gamma function when
no elementary solution exists. For example, if f is a power function
and g is a linear function, a simple change of variables yields

The fact that the integration is performed along the entire positive
real line might signify that the Gamma function describes the
cumulation of a time-dependent process that continues indefinitely, or
the value might be the total of a distribution in an infinite space. It is
of course frequently useful to take limits of integration other than 0
and to describe the cumulation of a finite process, in which case
the ordinary Gamma function is no longer a solution; the solution is
then called an incomplete Gamma function. (The ordinary Gamma
function, obtained by integrating across the entire positive real line, is
sometimes called the complete Gamma function for contrast.)
The Gamma function's ability to generalize factorial products
immediately leads to applications in many areas of mathematics; in
combinatorics, and by extension in areas such as probability theory
and the calculation of power series. Many expressions involving
products of successive integers can be written as some combination of
factorials, the most important example perhaps being that of the
binomial coefficient.
By taking limits, certain rational products with infinitely many factors
can be evaluated in terms of the Gamma function as well. Due to the
Weierstrass factorization theorem, analytic functions can be written as
infinite products, and these can sometimes be represented as finite
products or quotients of the Gamma function. For one example, the
reflection formula essentially represents the sine function as the
product of two Gamma functions. Starting from this formula, the
exponential function as well as all the trigonometric and and
hyperbolic functions can be expressed in terms of the Gamma
function. More functions yet, including the hypergeometric function
and special cases thereof, can be represented by means of complex
contour integrals of products and quotients of the Gamma function,
called Mellin-Barnes integrals.

The Gamma function can also be used to calculate the "volume" and
"area" of n-dimensional hyperspheres.
An elegant and deep application of the Gamma function is in the study
of the Riemann zeta function. A fundamental property of the Riemann
zeta function is its functional equation. Among other things, it
provides an explicit form for the analytic continuation of the zeta
function to a meromorphic function in the complex plane and leads to
an immediate proof that the zeta function has infinitely many so-
called "trivial" zeros on the real line. Borwein et. al call this formula
"one of the most beautiful findings in mathematics".
The Gamma function has caught the interest of some of the most
prominent mathematicians of all time. In the words of Philip J. Davis,
"each generation has found something of interest to say about the
Gamma function. Perhaps the next generation will also." Its history
reflects many of the major developments within mathematics since
the 18th century.