The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

It appears as a factor in various probability-distribution functions and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics.

is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex,[5] meaning that

for which the real part is integer or half-integer quickly follow by induction using the recurrence relation in the positive and negative directions.

When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function: [12]

One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,[1]

The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand.

The complex-valued gamma function is undefined for non-positive integers, but in these cases the value can be defined in the Riemann sphere as ∞.

Following an indication of Carl Friedrich Gauss, Rocktaeschel (1922) proposed for logΓ(z) an approximation for large Re(z):

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A.

[56] If interpolation tables are not desirable, then the Lanczos approximation mentioned above works well for 1 to 2 digits of accuracy for small, commonly used values of z.

"[57] The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics.

Integrals of such expressions can occasionally be solved in terms of the gamma function when no elementary solution exists.

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.

In particular, the arc lengths of ellipses and of the lemniscate, which are curves defined by algebraic equations, are given by elliptic integrals that in special cases can be evaluated in terms of the gamma function.

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.

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural.

is the factorial of a negative integer and hence infinite if we use the gamma function definition of factorials—dividing by infinity gives the expected value of 0.

The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether b − a equals 5 or 105.

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well.

Its history, notably documented by Philip J. Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century.

"[1] The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s.

Leonhard Euler later gave two different definitions: the first was not his integral but an infinite product that is well defined for all complex numbers n other than the negative integers,

Karl Weierstrass further established the role of the gamma function in complex analysis, starting from yet another product representation,

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.

Consider that the notation for exponents, xn, has been generalized from integers to complex numbers xz without any change.

Legendre's motivation for the normalization is not 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!).

Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given by Charles Hermite in 1900.

[66] Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.

According to Michael Berry, "the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status.