beta function
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11—20 of 179 matching pages
11: 5.21 Methods of Computation
12: 5.1 Special Notation
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►The main functions treated in this chapter are the gamma function
, the psi function (or digamma function) , the beta function
, and the -gamma function
.
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13: 5.16 Sums
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►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
14: 35.1 Special Notation
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
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►Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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15: 8.28 Software
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§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters
… ►§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters
…16: 19.16 Definitions
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19.16.9
, , ,
►where is the beta function (§5.12) and
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19.16.12
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19.16.19
, .
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19.16.24
, .
17: 19.23 Integral Representations
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►Also, in (19.23.8) and (19.23.10) denotes the beta function (§5.12).
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19.23.8
; .
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19.23.10
; ;
.
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18: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►The function
is defined by
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10.46.1
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10.46.2
►For asymptotic expansions of as in various sectors of the complex -plane for fixed real values of and fixed real or complex values of , see Wright (1935) when , and Wright (1940b) when .
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►The Laplace transform of can be expressed in terms of the Mittag-Leffler function:
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19: Guide to Searching the DLMF
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