relation%20to%20theta%20functions
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11—18 of 18 matching pages
11: Bibliography D
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Computational properties of three-term recurrence relations for Kummer functions.
J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions.
Methods Appl. Anal. 6 (3), pp. 21–56.
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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions.
SIAM J. Math. Anal. 32 (5), pp. 987–1013.
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12: 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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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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Tata Lectures on Theta. II.
Birkhäuser Boston Inc., Boston, MA.
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13: Bibliography I
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Theta Functions.
Springer-Verlag, New York.
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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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14: 14.30 Spherical and Spheroidal Harmonics
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are known as spherical
harmonics.
…Sometimes is denoted by ; also the definition of can differ from (14.30.1), for example, by inclusion of a factor .
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►Most mathematical properties of can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter.
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►See also (34.3.22), and for further related integrals see Askey et al. (1986).
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►has solutions , which are everywhere one-valued and continuous.
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15: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Cyclic identities for Jacobi elliptic and related functions.
J. Math. Phys. 44 (4), pp. 1822–1841.
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Theta relations and projective normality of Abelian varieties.
Amer. J. Math. 98 (4), pp. 865–889.
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The structure relation for Askey-Wilson polynomials.
J. Comput. Appl. Math. 207 (2), pp. 214–226.
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16: 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
…17: Bibliography C
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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions.
Math. Comp. 75 (255), pp. 1309–1318.
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Permutation symmetry for theta functions.
J. Math. Anal. Appl. 378 (1), pp. 42–48.
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Performance evaluation of programs related to the real gamma function.
ACM Trans. Math. Software 17 (1), pp. 46–54.
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An efficient algorithm for the Hurwitz zeta and related functions.
J. Comput. Appl. Math. 225 (2), pp. 338–346.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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18: 1.11 Zeros of Polynomials
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►A similar relation holds for the changes in sign of the coefficients of , and hence for the number of negative zeros of .
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►Addition of
to each of these roots gives the roots of .
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►For the roots of and the roots of the resolvent cubic equation
…Add
to the roots of
to get those of .
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►Resolvent cubic is with roots , , , and , , .
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