mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial...

\beta )} is the incomplete beta function and I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )} is the regularized incomplete beta function. The beta...

Beta function, also called the Euler beta function or the Euler integral of the first kind, is a special function in mathematics. Beta function may also...

the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular...

In theoretical physics, specifically quantum field theory, a beta function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale...

\operatorname {swish} _{\beta }(x)=x\operatorname {sigmoid} (\beta x)={\frac {x}{1+e^{-\beta x}}}.} where β {\displaystyle \beta } can be constant (usually...

\beta _{2}={\frac {(\alpha +\beta )^{2}(1+\alpha +\beta )}{n\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)(\alpha +\beta +n)}}\left[(\alpha +\beta )(\alpha...

The beta function in accelerator physics is a function related to the transverse size of the particle beam at the location s along the nominal beam trajectory...

Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree. It...

analogue. Digamma function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization...

integral of the second kind. (Euler's integral of the first kind is the beta function.) Using integration by parts, one sees that: Γ ( z + 1 ) = ∫ 0 ∞ t z...

j-invariant, a modular function, is a rational function in λ ( τ ) {\displaystyle \lambda (\tau )} . Incomplete beta functions Bx(p,q) are related by...

If the function is scaled with the parameter β {\displaystyle \beta } , then these expressions must be multiplied by β {\displaystyle \beta } . See multinomial...

F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),} where I is the regularized incomplete beta function. While the related beta distribution...

^{k}\operatorname {B} (1-k/\beta ,1+k/\beta )\\[5pt]&=\alpha ^{k}\,{k\pi /\beta \over \sin(k\pi /\beta )}\end{aligned}}} where B is the beta function. Expressions for...

It can also be represented in terms of the regularized incomplete beta function, as follows: F ( k ; n , p ) = Pr ( X ≤ k ) = I 1 − p ( n − k , k +...

In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous. If a beta function is positive, the corresponding...

Look up Beta, beta, béta, or bêta in Wiktionary, the free dictionary. Beta (B, β) is the second letter of the Greek alphabet. Beta or BETA may also refer...

Pancreatic beta cell function (synonyms Gβ or, if calculated from fasting concentrations of insulin and glucose, HOMA-Beta or SPINA-GBeta) is one of the...

{\textstyle \beta _{2}} yields an optimal value function β 2 ( β 1 ) = ( X 2 T X 2 ) − 1 X 2 T ( y − X 1 β 1 ) {\textstyle \beta _{2}(\beta _{1})=\left(\mathbf...

and discrete, the canonical partition function is defined as Z = ∑ i e − β E i , {\displaystyle Z=\sum _{i}e^{-\beta E_{i}},} where i {\displaystyle i} is...

islets, beta cells play a vital role in maintaining blood glucose levels. Problems with beta cells can lead to disorders such as diabetes. The function of...

a single good with two factors, the function is given by: Y ( L , K ) = A L β K α {\displaystyle Y(L,K)=AL^{\beta }K^{\alpha }} where: Y = total production...

(\alpha ,\beta )} is the Beta function acting as a normalising constant. In this context, α {\displaystyle \alpha } and β {\displaystyle \beta } are called...

X-\log(1-X)} is the logit function. If X ∼ G u m b e l ( μ X , β ) {\displaystyle X\sim \mathrm {Gumbel} (\mu _{X},\beta )} and Y ∼ G u m b e l ( μ Y...

{\displaystyle p} , and q {\displaystyle q} positive. The function B(p,q) is the beta function. The parameter b {\displaystyle b} is the scale parameter...

moment generating function M X ( t ) {\displaystyle M_{X}(t)} , then α X + β {\displaystyle \alpha X+\beta } has moment generating function M α X + β ( t...

following, three important theta function values are to be derived as examples: This is how the Euler beta function is defined in its reduced form: β...

parametrization of the beta prime distribution, which is also called the beta distribution of the second kind. The characteristic function is listed incorrectly...

Beta (UK: /ˈbiːtə/, US: /ˈbeɪtə/; uppercase Β, lowercase β, or cursive ϐ; Ancient Greek: βῆτα, romanized: bē̂ta or Greek: βήτα, romanized: víta) is the...