Gamma function
19th century: Gauss, Weierstrass and Legendre

Carl Friedrich Gauss rewrote Euler's product as Γ (z) = lim m → ∞ m z m ! z (z + 1) (z + 2) ⋯ (z + m) {\displaystyle \Gamma (z)=\lim _{m\to \infty }{\frac {m^{z}m!}{z(z+1)(z+2)\cdots (z+m)}}} and used this 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 instead 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, Γ (z) = e − γ z z ∏ k = 1 ∞ (1 + z k) − 1 e z k, {\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{k=1}^{\infty }\left(1+{\frac {z}{k}}\right)^{-1}e^{\frac {z}{k}},} where γ is the Euler–Mascheroni constant. Weierstrass originally wrote his product as one for 1 / Γ, 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 gamma function and the symbol Γ were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition in its modern form. Although the symbol is an upper-case Greek gamma, there is no accepted standard for whether the function name should be written "gamma function" or "Gamma function" (some authors simply write " Γ -function"). The alternative "pi function" notation Π(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 Γ(n + 1) = n ! instead of simply using " Γ(n) = n ! ". Consider that the notation for exponents, x, has been generalized from integers to complex numbers x without any change. 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 some formulae, but complicates others. From a modern point of view, the Legendre normalization of the gamma function is the integral of the additive character e against the multiplicative character x with respect to the Haar measure d x x {\textstyle {\frac {dx}{x}}} on the Lie group R. Thus this normalization makes it clearer that the gamma function is a continuous analogue of a Gauss sum.
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