Beta functionLegendre duplication formulaGauss multiplication formulaWinckler theoremStirling theoremHankel integral theoremHankel contourLogarithmic derivativeIn this chapter, we introduce the definitions and theorems for the Euler gamma function, Pochhammer symbols, and Euler beta function. We present the ...
The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a)
组合恒等式Based on the theorem 1, series of combinatorial identities are derived by using the properties and relationship of Beta function and Gamma function.% 以定理1为基础,用 Beta 函数和 Gamma 函数性质及其之间的关系,导出了一系列组合恒等式张秦新乡学院...
The beta function can be defined in terms of the gamma function and thus, calculated using the Stata functions exp() and lngamma(). On the latter, see the section "Mathematical functions" of the [FN] Stata Functions Reference Manual. Given positive arguments a and b, the beta function ...
A generalization of the completebeta functiondefined by (1) sometimes also denoted . The so-calledChebyshev integralis given by (2) The incomplete beta function is implemented in theWolfram LanguageasBeta[z,a,b]. It is given in terms ofhypergeometric functionsby ...
These incomplete gamma functions may also be expressed quite elegantly in terms of confluent hypergeometric functions (compare Section 18.6). Incomplete Beta Function Just as there are incomplete gamma functions, there is also an incomplete beta function, customarily defined for 0≤x≤1, p > 0(an...
The utility of the Beta function is often overshadowed by that of the Gamma function, partly perhaps because it can be evaluated in terms of the Gamma function. However, since it occurs so frequently in practice, a special designation for it is widely accepted. It is obvious that the Beta ...
0 (1.2) Various well-known formulas for Gamma function are also given in [6]. For example, ∞Γ(x) = sx e−sttx−1dt, x, s > 0. 0 The Beta function is given by (1.3) 1 B(x, y) = tx−1(1 − t)y−1dt, x > 0, y > 0, 0 and its connection to Gamma ...
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(GO) molecular function pathway enrichment analysis of the predicted key drivers revealed similar biological processes of “chromatin-binding” (AdjP = 0.005) among the most significantly enriched terms along with and “SMAD-binding” (AdjP = 1.6 × 10−6) (Fig.1d, Supplementary...