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1: 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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complex-valued function with . | |
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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: Bibliography N
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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes.
ACM Trans. Math. Software 18 (3), pp. 345–349.
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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes.
J. Comput. Appl. Math. 39 (2), pp. 193–200.
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On the large argument asymptotics of the Lommel function via Stieltjes transforms.
Asymptot. Anal. 91 (3-4), pp. 265–281.
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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions.
Acta Appl. Math. 150, pp. 141–177.
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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions.
Stud. Appl. Math. 140 (4), pp. 508–541.
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7: 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 , with and . 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. …8: 35.10 Methods of Computation
§35.10 Methods of Computation
… ►See Yan (1992) for the and functions of matrix argument in the case , and Bingham et al. (1992) for Monte Carlo simulation on applied to a generalization of the integral (35.5.8). …9: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments.
National Physical Laboratory Mathematical Tables, Vol. 4.
Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument.
SIAM J. Math. Anal. 23 (2), pp. 505–511.
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Fast computation of incomplete elliptic integral of first kind by half argument transformation.
Numer. Math. 116 (4), pp. 687–719.
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