Perhaps the next generation will also." === 18th century: Euler and Stirling === 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 Leonhard Euler.
Euler gave two different definitions: the first was not his integral but an infinite product, n! = \prod_{k=1}^\infty \frac{\left(1+\frac{1}{k}\right)^n}{1+\frac{n}{k}}\,, of which he informed Goldbach in a letter dated 13 October 1729.
Petersburg Academy on 28 November 1729.
He wrote to Goldbach again on 8 January 1730, to announce his discovery of the integral representation n!=\int_0^1 (-\ln s)^n\, ds\,, which is valid for .
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 gamma function and the symbol were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition in its modern form.
Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825. Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in Tables of Higher Functions by Jahnke and , first published in Germany in 1909.
Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825. Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in Tables of Higher Functions by Jahnke and , first published in Germany in 1909.
However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842. === Raabe's formula === In 1840 Joseph Ludwig Raabe proved that \int_a^{a+1}\ln\Gamma(z)\, dz = \tfrac12\ln2\pi + a\ln a - a,\quad a>0. In particular, if a = 0 then \int_0^1\ln\Gamma(z)\, dz = \tfrac12\ln2\pi. The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand.
However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842. === Raabe's formula === In 1840 Joseph Ludwig Raabe proved that \int_a^{a+1}\ln\Gamma(z)\, dz = \tfrac12\ln2\pi + a\ln a - a,\quad a>0. In particular, if a = 0 then \int_0^1\ln\Gamma(z)\, dz = \tfrac12\ln2\pi. The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand.
Another champion for that title might be \zeta(s) \; \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}. Both formulas were derived by Bernhard Riemann in his seminal 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Prime Numbers less than a Given Quantity"), one of the milestones in the development of analytic number theory—the branch of mathematics that studies prime numbers using the tools of mathematical analysis.
Otto Hölder 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, making it a transcendentally transcendental function.
Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given by Charles Hermite in 1900.
Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825. Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in Tables of Higher Functions by Jahnke and , first published in Germany in 1909.
This result is known as Hölder's theorem. A definite and generally applicable characterization of the gamma function was not given until 1922.
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." There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics.
As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S.
Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century.
National Bureau of Standards. Abramowitz and Stegun became the standard reference for this and many other special functions after its publication in 1964. Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example TK Solver, Matlab, GNU Octave, and the GNU Scientific Library.
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