A gamma function makes integration easier

Gamma function

Γ (x) = 0∫∞ tx − 1e − tdt
Γ (n) = (n − 1) !,
what results from the functional equation
Γ (x + 1) = x⋅Γ (x)

Forms of representation

The representation of the Gamma function according to Gauss:
Γ (x) = limn → ∞ x (x + 1) (x + 2) ⋯ (x + n) n! Nx
for x∈R \ {0, −1, −2, ...}
That of Karl Weierstraß is derived directly from the Gaussian form of representation:
Γ (x) = [x⋅eγx⋅k = 1∏∞ (1 + kx) e − x / k] −1,
γ = limn → ∞ (k = 1∑n k1 −lnn)
Approximate values ​​of the Gamma function for x> 0, Stirling's formula gives da
n! = Γ (x + 1) = 2πx ex − μ (x) xx + 1/2
follows for the gamma function.
Γ (x) = 2πx ex − μ (x) xx − 1/2
with 0 <μ (x) <12x1

Bohr-Mollerup's theorem

The Bohr-Mollerup's theorem (H. Bohr and J. Mollerup, 1922) allows a surprisingly simple characterization of the Gamma function:
A function G: R> 0 → R> 0 is equal to that in this range Gamma functionif the following properties are met:
  1. G (1) = 1
  2. G (x + 1) = x⋅G (x)
  3. G is logarithmic convex, i.e. x↦logG (x) is a convex function.

Functional equations

The supplementary sentence of the Gamma function
Γ (x) Γ (1 − x) = sinπxπ
facilitates the calculation of values ​​of the Gamma function from already known function values ​​as well as the Legendary doubling formula
Γ (2x) Γ (2x + 1) = 2x − 1π Γ (x)

Generalization: Incomplete gamma function

If the upper limit of integration of the integral is a fixed, finite value, one speaks of the incomplete gamma function:
γ (x, y) = 0∫y tx − 1e − tdt

History

Γ (x) = 0∫1 [ln (t1)] x − 1dt
(this function definition goes over into the above form by the substitution u = ln (1 / t))
Euler discovered this integral while investigating a mechanical problem in which the acceleration of a particle is considered.

Measure everything that can be measured and make everything measurable that cannot be measured.

Galileo Galilei

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