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1: 19.37 Tables
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 7S by Beli͡akov et al. (1962).
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675).
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►Tabulated (with different notation) for , , to 5D by Abramowitz and Stegun (1964, Chapter 17), and for , , to 7D by Zhang and Jin (1996, pp. 676–677).
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2: 20.15 Tables
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►This reference gives , , and their logarithmic -derivatives to 4D for , , where is the modular angle given by
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►Spenceley and Spenceley (1947) tabulates , , , to 12D for , , where and is defined by (20.15.1), together with the corresponding values of and .
►Lawden (1989, pp. 270–279) tabulates , , to 5D for , , and also to 5D for .
►Tables of Neville’s theta functions , , , (see §20.1) and their logarithmic -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for , where (in radian measure) , and is defined by (20.15.1).
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3: 5.16 Sums
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5.16.1
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►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
4: 14.33 Tables
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Abramowitz and Stegun (1964, Chapter 8) tabulates for , , 5–8D; for , , 5–7D; and for , , 6–8D; and for , , 6S; and for , , 6S. (Here primes denote derivatives with respect to .)
Zhang and Jin (1996, Chapter 4) tabulates for , , 7D; for , , 8D; for , , 8S; for , , 8D; for , , , , 8S; for , , 8S; for , , , 5D; for , , 7S; for , , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 -zeros of and of its derivative for , .
Belousov (1962) tabulates (normalized) for , , , 6D.
5: 33.26 Software
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►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web.
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Noble (2004). Fortran 90.
6: Bibliography E
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Some recent results on the zeros of Bessel functions and orthogonal polynomials.
J. Comput. Appl. Math. 133 (1-2), pp. 65–83.
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The Fuchsian equation of second order with four singularities.
Duke Math. J. 9 (1), pp. 48–58.
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Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter.
ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.
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Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm.
ACM Trans. Math. Software 32 (4), pp. 622–634.
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On the representations of a number as a sum of three squares.
Proc. London Math. Soc. (3) 9, pp. 575–594.
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7: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…( is defined to be 0.)
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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8: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function for .
See Calkin et al. (2007), and Lehmer (1941, pp. 5–83).
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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9: 4.48 Software
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►These will not be listed here.
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►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web.
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Kearfott (1996). Fortran 90.
10: Bibliography S
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Calculator function approximation.
Amer. Math. Monthly 90 (5), pp. 317–325.
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A new Fortran 90 program to compute regular and irregular associated Legendre functions.
Comput. Phys. Comm. 181 (12), pp. 2091–2097.
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The elliptical microstrip antenna with circular polarization.
IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.
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Représentation asymptotique des fonctions de Mathieu et des fonctions sphéroidales. II.
Trans. Amer. Math. Soc. 90 (2), pp. 340–368.
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Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions.
ACM Trans. Math. Software 27 (4), pp. 377–387.
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