3F2%20functions%20of%20matrix%20argument
(0.005 seconds)
1—10 of 984 matching pages
1: 34.2 Definition: Symbol
§34.2 Definition: Symbol
βΊThe quantities in the symbol are called angular momenta. …The corresponding projective quantum numbers are given by … βΊwhere is defined as in §16.2. βΊFor alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).2: 35.1 Special Notation
…
βΊ(For other notation see Notation for the Special Functions.)
…
βΊ
βΊ
βΊThe 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 .
βΊAn alternative notation for the multivariate gamma function is (Herz (1955, p. 480)).
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. | |
… |
3: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
βΊ§35.5(i) Definitions
… βΊ§35.5(ii) Properties
… βΊ§35.5(iii) Asymptotic Approximations
βΊFor asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).4: 9.1 Special Notation
…
βΊ(For other notation see Notation for the Special Functions.)
βΊ
βΊ
βΊThe main functions treated in this chapter are the Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
βΊOther notations that have been used are as follows: and for and (Jeffreys (1928), later changed to and ); , (Fock (1945)); (SzegΕ (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii). | |
… | |
primes | derivatives with respect to argument. |
5: 9.12 Scorer Functions
§9.12 Scorer Functions
… βΊIf or , and is the modified Bessel function (§10.25(ii)), then … βΊwhere the integration contour separates the poles of from those of . … βΊFunctions and Derivatives
…6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
… βΊ§35.8(iii) Case
βΊKummer Transformation
… βΊPfaff–Saalschütz Formula
… βΊThomae Transformation
…7: 20.2 Definitions and Periodic Properties
…
βΊ
§20.2(i) Fourier Series
… βΊCorresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . … βΊFor fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary . … βΊ§20.2(iii) Translation of the Argument by Half-Periods
… βΊFor , the -zeros of , , are , , , respectively.8: 23.15 Definitions
§23.15 Definitions
βΊ§23.15(i) General Modular Functions
… βΊElliptic Modular Function
… βΊDedekind’s Eta Function (or Dedekind Modular Function)
… βΊ9: 16.13 Appell Functions
§16.13 Appell Functions
βΊThe following four functions of two real or complex variables and cannot be expressed as a product of two functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): βΊ
16.13.1
,
…
βΊ
16.13.3
,
…
βΊ
…