Gammawerte

Gammawerte refers to the values of the Gamma function, denoted Γ(z), which generalizes factorials to real and complex numbers. For positive real numbers n, Γ(n) = (n−1)!, and the function extends to a meromorphic function on the complex plane, with simple poles at z = 0, −1, −2, ….

The Gamma function is defined by the integral Γ(z) = ∫0^∞ t^{z−1} e^{−t} dt for Re(z) > 0 and then analytically continued to other values of z. It satisfies the fundamental relation Γ(z+1) = z Γ(z), which links consecutive values and implies Γ(n) = (n−1)! for natural n. The reflection formula Γ(z) Γ(1−z) = π / sin(π z) connects values at z and 1−z. Special values include Γ(1/2) = √π and, for integers n ≥ 1, Γ(n) = (n−1)!.

Gammawerte appear in many areas of mathematics and applied sciences. They underpin the beta function B(x,y) = Γ(x) Γ(y) / Γ(x+y), appear in probability distributions such as the gamma distribution, and arise in various integrals and series in complex analysis. They also allow the extension of factorial-based identities to non-integer arguments, enabling, for example, the computation of moments and densities in statistical models.

Numerical evaluation of Γ(z) relies on recurrence relations, the reflection formula, and asymptotic approximations such as the Lanczos method. In practice, Gammawerte are tabulated or computed by computer algebra systems and scientific libraries, reflecting their central role in special-function theory and its applications.