Gamma function in the context of Bohr–Mollerup theorem

⭐ Core Definition: Gamma function

In mathematics, the gamma function (represented by $\Gamma$ , capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function $\Gamma (z)$ is defined for all complex numbers $z$ except non-positive integers, and $\Gamma (n)=(n-1)!$ for every positive integer $n$ . The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.$ The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.

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👉 Gamma function in the context of Bohr–Mollerup theorem

In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for $x > 0$ by

as the only positive function $f$ , with domain on the interval $x > 0$ , that simultaneously has the following three properties:

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⭐ Core Definition: Gamma function
👉 Gamma function in the context of Bohr–Mollerup theorem
Gamma function in the context of Functional equation

Gamma function in the context of Functional equation

In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation ⁠ $\log(xy)=\log(x)+\log(y)$ ⁠.

If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have highly irregular solutions. For example, the gamma function is a function that satisfies the functional equation $f(x+1)=xf(x)$ and the initial value $f(1)=1.$ There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for $x$ real and positive (Bohr–Mollerup theorem).

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Gamma function in the context of Bohr–Mollerup theorem

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⭐ Core Definition: Gamma function

👉 Gamma function in the context of Bohr–Mollerup theorem

Gamma function in the context of Functional equation