relations%20to%20other%20functions
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1: 25.6 Integer Arguments
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§25.6(i) Function Values
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25.6.3
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§25.6(ii) Derivative Values
… ►§25.6(iii) Recursion Formulas
… ►For related results see Basu and Apostol (2000).2: 25.12 Polylogarithms
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►Other notations and names for include (Kölbig et al. (1970)), Spence function
(’t Hooft and Veltman (1979)), and (Maximon (2003)).
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►The special case is the Riemann zeta function: .
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►Further properties include
…and
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►In terms of polylogarithms
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3: Bibliography F
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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I.
Technical report
Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
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Numerical calculation of singular integrals related to Hankel transform.
Comput. Math. Appl. 21 (2-3), pp. 87–94.
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Guide to tables of elliptic functions.
Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.
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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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4: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When , (25.11.35) reduces to (25.2.3). … ►Similarly, as in the sector , …5: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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Special Functions, -Series and Related Topics.
Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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Bounds for the small real and purely imaginary zeros of Bessel and related functions.
Methods Appl. Anal. 2 (1), pp. 1–21.
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From Gauss to Painlevé: A Modern Theory of Special Functions.
Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
6: Bibliography M
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Uniform computation of the error function and other related functions.
Math. Comp. 25 (114), pp. 339–344.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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Evaluation of complex logarithms and related functions.
SIAM J. Numer. Anal. 18 (4), pp. 744–750.
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A new symmetry related to
for classical basic hypergeometric series.
Adv. in Math. 57 (1), pp. 71–90.
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7: Bibliography
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Some series of the zeta and related functions.
Analysis (Munich) 18 (2), pp. 131–144.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Integral equations and relations for Lamé functions.
Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
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8: 18.5 Explicit Representations
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Chebyshev
… ►Related formula: … ►§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
… ►Laguerre
… ►Hermite
…9: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
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Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions.
J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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10: 18.39 Applications in the Physical Sciences
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is referred to as the ground state, all others, in order of increasing energy being excited states.
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