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 may refer to: Beta function (physics), details the running of the coupling strengths Dirichlet beta function, closely related to the Riemann...

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

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

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...

\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...

where B is the Beta function. The cumulative distribution function is F ( x ; α , β ) = I x 1 + x ( α , β ) , {\displaystyle F(x;\alpha ,\beta )=I_{\frac...

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...

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...

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...

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

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...

^{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...

{\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...

{\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...

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...

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 +...

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...

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...

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...

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...

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...

continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural...

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

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...

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

In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron)...

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...