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1: 35.1 Special Notation
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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 .
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βΊRelated 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
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).3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
βΊ§35.8(i) Definition
… βΊConvergence Properties
… βΊConfluence
… βΊInvariance
…4: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
βΊ§35.6(i) Definitions
… βΊLaguerre Form
… βΊ§35.6(ii) Properties
… βΊ§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
βΊ§35.7(i) Definition
… βΊJacobi Form
… βΊConfluent Form
… βΊIntegral Representation
…6: 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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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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Principles and Techniques of Applied Mathematics.
Dover, New York.
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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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7: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral.
Ε½. VyΔisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
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Mathieu functions of integral orders and real arguments.
IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.
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Total positivity properties of generalized hypergeometric functions of matrix argument.
J. Statist. Phys. 116 (1-4), pp. 907–922.
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Principles of Mathematical Analysis.
3rd edition, McGraw-Hill Book Co., New York.
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8: Bibliography B
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New tables of Bessel functions of complex argument.
Comput. Math. Math. Phys. 37 (12), pp. 1480–1482.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials.
Math. Comp. 15 (73), pp. 7–11.
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Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light.
7th edition, Cambridge University Press, Cambridge.
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Generalized hypergeometric functions at unit argument.
Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
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9: Bibliography K
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Optical Waveguides.
Quantum Electronics - Principles and Applications, Academic Press, New York.
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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument.
ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument.
ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
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The efficient evaluation of the hypergeometric function of a matrix argument.
Math. Comp. 75 (254), pp. 833–846.
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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument.
Comput. Phys. Comm. 4, pp. 221–226.
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