gamma function - Definition. Was ist gamma function
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Was (wer) ist gamma function - definition


Gamma function         
  • [[Daniel Bernoulli]]'s letter to [[Christian Goldbach]], October 6, 1729
  • The first page of Euler's paper
  • The gamma function interpolates the factorial function to non-integer values.
  • reproduction of a famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) of the gamma function from -4.5-2.5i to 4.5+2.5i
  • 3-dimensional plot of the absolute value of the complex gamma function
  • </math> is also displayed.
  • Γ(''z'') + sin(π''z'')}} in green. Notice the intersection at positive integers. Both are valid analytic continuations of the factorials to the non-integers.
  • lt=Emde}}.
  • log Γ(''z'')}}
  • Comparison gamma (blue line)  with the factorial (blue dots) and Stirling's approximation (red line)
  • Plot of gamma function in complex plane in 3D with color and legend and 1000 plot points created with Mathematica
  • 6+2''i''}} with colors created in Mathematica
  • Plot of logarithmic gamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
EXTENSION OF THE FACTORIAL FUNCTION, WITH ITS ARGUMENT SHIFTED DOWN BY 1, TO REAL AND COMPLEX NUMBERS
Gamma-function; Complete gamma function; Gamma Function; Gamma integral; Euler Gamma Function; Γ(x); Γ function; Raabe's formula; Approximations of the gamma function; Weierstrass definition of the gamma function; Complex number factorial
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.
Multiple gamma function         
  • Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
GENERALIZATION OF THE EULER GAMMA FUNCTION AND THE BARNES G-FUNCTION
Barnes gamma function; Double gamma function; Triple gamma function; Barnes multiple gamma function; Multiple Gamma function
In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by .
Q-gamma function         
Q-ANALOG OF THE GAMMA FUNCTION
Q-Gamma function; Basic gamma function; Q gamma function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by .