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1: 35.1 Special Notation
a , b complex variables.
f ( X ) complex-valued function with X Ω .
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( T ) and (of the second kind) B ν ( T ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; T ) or F 1 1 ( a b ; T ) and (of the second kind) Ψ ( a ; b ; T ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; T ) or F 1 2 ( a 1 , a 2 b ; T ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; T ) or F q p ( a 1 , , a p b 1 , , b q ; T ) . … Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( T ) = A ν ( T ) / A ν ( 0 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | S , T ) = | T | ν B ν ( S T ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( T ) = | T | ν B ν ( S T ) (Faraut and Korányi (1994, pp. 357–358)).
2: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
§35.5(i) Definitions
§35.5(ii) Properties
§35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
Convergence Properties
Confluence
Invariance
4: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(i) Definitions
Laguerre Form
§35.6(ii) Properties
§35.6(iii) Relations to Bessel Functions of Matrix Argument
5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(i) Definition
Jacobi Form
Confluent Form
Integral Representation
6: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
For properties of Struve functions of argument x e ± 3 π i / 4 see McLachlan and Meyers (1936).
7: 4.3 Graphics
§4.3(i) Real Arguments
See accompanying text
Figure 4.3.1: ln x and e x . … Magnify
§4.3(ii) Complex Arguments: Conformal Maps
Figure 4.3.2 illustrates the conformal mapping of the strip - π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …
§4.3(iii) Complex Arguments: Surfaces
8: 4.15 Graphics
§4.15(i) Real Arguments
§4.15(ii) Complex Arguments: Conformal Maps
See accompanying text
A B C C ¯ D D ¯ E E ¯ F
Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify
§4.15(iii) Complex Arguments: Surfaces
The corresponding surfaces for arccos ( x + i y ) , arccot ( x + i y ) , arcsec ( x + i y ) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).
9: 35.9 Applications
§35.9 Applications
In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument F q p , with p 2 and q 1 . See James (1964), Muirhead (1982), Takemura (1984), Farrell (1985), and Chikuse (2003) for extensive treatments. … These references all use results related to the integral formulas (35.4.7) and (35.5.8). … In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …
10: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
13.19.1 M κ , μ ( x ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 x x - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! x - s , μ - κ - 1 2 , - 3 2 , .
13.19.2 M κ , μ ( z ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 z z - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! z - s + Γ ( 1 + 2 μ ) Γ ( 1 2 + μ + κ ) e - 1 2 z ± ( 1 2 + μ - κ ) π i z κ s = 0 ( 1 2 + μ - κ ) s ( 1 2 - μ - κ ) s s ! ( - z ) - s , - 1 2 π + δ ± ph z 3 2 π - δ ,