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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.

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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.

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.

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