# 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 dan! = 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 function**if the following properties are met:- G (1) = 1
- G (x + 1) = x⋅G (x)
- 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

Provider identification: Mathеpеdιa von Тhοmas Stеιnfеld • Dοrfplatz 25 • 17237 Blankеnsее • Tel .: 01734332309 (Vodafone / D2) • Email: cο@maτhepedιa.dе

- What sets billionaires apart from the rest
- Why did you leave your boyfriend
- Indian films are harmful to us
- Do you have QQ
- Which is better Authorize net or PayPal
- Is Lipton tea herbal tea
- How do I pronounce faux in English
- Why do some people experience deja vu
- What is the worst museum in Buffalo
- What are some encrypted streaming sites
- What are the topics in Pulp Fiction
- A single-stroke engine is possible
- Can Prozac Fluoxetine Help People Stop Drinking?
- What are good interviewing habits?
- How can you search for competitor keywords?
- What do Israeli soldiers think of India?
- Why don't more places deliver food
- Where is the Karachi high school
- Can beat whis vados
- CNC milling parts
- Where can you survive if you only speak toisan
- Is there a section break in GATE
- Trump would win another election
- How did Swami Paramhansa die