Extension of the partial derivatives of the incomplete beta function for complex values

Sun, Zhongfeng; Qin, Huizeng; Li, Aijuan

doi:10.1016/j.amc.2015.11.054

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…In this section, we shall present some important lemmas that will be frequently used and their proof may be found in [15,16].…”

Section: Lemmasmentioning

confidence: 99%

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Some Notes of the Fractional Integral and Derivatives and Its Applications

Qin

2022

Journal of Mathematics

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In this paper, the definitions of the fractional derivative and integrals are given by the neutrix limit. Not only is it consistent with the classical results but also the representations of the fractional derivative and integrals are obtained for x − a − k , x − a − k ln p x − a , x − a − k f x , and x − a − k ln p x − a f x , where k , p = 1,2 , … , f x is the analytic function.

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“…In this section, we shall present some important lemmas that will be frequently used and their proof may be found in [15,16].…”

Section: Lemmasmentioning

confidence: 99%

“…We use neutrix limit [9][10][11][12][13][14][15][16] to defne the frational derivatives for (1)- (6).…”

Section: Introductionmentioning

confidence: 99%

Some Notes of the Fractional Integral and Derivatives and Its Applications

Qin

2022

Journal of Mathematics

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“…In some applications, it is required to extend the arguments p, q to negative numbers, which has been done by Özçaḡ et al in [12,13]. They can also be extended to complex numbers [15]. In such cases, the integrals (1.1)-(1.4) do not exist in a usual sense, but they can be regularized to finite values in the neutrix sense [12,13], or equivalently, in the Hadamard finite-part sense [6].…”

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confidence: 99%

The series expansions and Gauss-Legendre rule for computing arbitrary derivatives of the Beta-type functions

Li¹,

Wang²,

Hao³

2020

etna

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The beta-type functions play an important role in many applied sciences. The partial derivatives of the beta function and the incomplete beta function are integrals involving algebraic and logarithmic endpoint singularities. In this paper, some series expansions for these beta-type functions are found, which are easily used to evaluate these functions with prescribed precision. On the other hand, an accurate Gauss-Legendre quadrature formula is designed to compute these beta-type functions and their partial derivatives based on the Puiseux series for the integrands at their singularities. Some numerical examples confirm the high accuracy and high efficiency of the two algorithms, and also show that the algorithms can be used to effectively evaluate the generalized beta-type functions.

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The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

Bakhet

et al. 2023

Linear and Multilinear Algebra

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Extension of the partial derivatives of the incomplete beta function for complex values

Cited by 3 publications

References 8 publications

Some Notes of the Fractional Integral and Derivatives and Its Applications

Some Notes of the Fractional Integral and Derivatives and Its Applications

The series expansions and Gauss-Legendre rule for computing arbitrary derivatives of the Beta-type functions

The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

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