modified%20functions
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11—20 of 21 matching pages
11: 11.6 Asymptotic Expansions
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§11.6(i) Large , Fixed
… βΊFor the corresponding expansions for and combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1). … βΊ§11.6(ii) Large , Fixed
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Complex zeros of the modified Bessel function
.
Math. Comp. 26 (120), pp. 949–953.
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Bounds for ratios of modified Bessel functions.
Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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Modified Clenshaw-Curtis method for the computation of Bessel function integrals.
BIT 23 (3), pp. 370–381.
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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations.
J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
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13: 28.1 Special Notation
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βΊThe main functions treated in this chapter are the Mathieu functions
…and the modified Mathieu functions
…The functions
and are also known as the radial Mathieu functions.
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Abramowitz and Stegun (1964, Chapter 20)
… βΊThe radial functions and are denoted by and , respectively.14: Bibliography B
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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order.
SIAM J. Appl. Math. 15, pp. 1315–1323.
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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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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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A short table of the functions
, from to
.
Phil. Mag. Series 7 20, pp. 343–347.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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15: Bibliography R
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On the partition function p(n).
Proc. London Math. Soc. (2) 43 (4), pp. 241–254.
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Tablitsy modifitsirovannykh funktsii Besselya
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“Nauka”, Moscow (Russian).
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Elliptic Functions, Theta Functions, and Riemann Surfaces.
The Williams & Wilkins Co., Baltimore, MD.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
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16: Bibliography C
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On Temme’s algorithm for the modified Bessel function of the third kind.
ACM Trans. Math. Software 6 (4), pp. 581–586.
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Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics.
IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.
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Algorithm 597: Sequence of modified Bessel functions of the first kind.
ACM Trans. Math. Software 9 (2), pp. 242–245.
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Derivatives with respect to the degree and order of associated Legendre functions for using modified Bessel functions.
Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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17: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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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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Computation of modified Bessel functions and their ratios.
Math. Comp. 28 (125), pp. 239–251.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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18: Bibliography F
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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 Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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Algorithm 435: Modified incomplete gamma function.
Comm. ACM 15 (11), pp. 993–995.
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19: 25.12 Polylogarithms
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βΊThe notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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βΊThe special case is the Riemann zeta function: .
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βΊFurther properties include
…and
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βΊIn terms of polylogarithms
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20: Bibliography L
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Bounds for modified Bessel functions.
J. Comput. Appl. Math. 34 (3), pp. 263–267.
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Algorithm 917: complex double-precision evaluation of the Wright
function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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An application of the finite element approximation method to find the complex zeros of the modified Bessel function
.
Math. Comp. 33 (148), pp. 1299–1306.
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Monotonicity of the differences of zeros of Bessel functions as a function of order.
Proc. Amer. Math. Soc. 15 (1), pp. 91–96.
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