infinite%20series%20expansions
(0.003 seconds)
1—10 of 13 matching pages
1: 25.12 Polylogarithms
…
►The cosine series in (25.12.7) has the elementary sum
…
►For real or complex and the polylogarithm
is defined by
…
►For each fixed complex the series defines an analytic function of for .
The series also converges when , provided that .
…
►The notation was used for in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function.
…
2: Bibliography R
…
►
Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions.
ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
…
►
On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
…
►
Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
…
►
Elliptic hypergeometric series on root systems.
Adv. Math. 181 (2), pp. 417–447.
…
►
Sources in the development of mathematics.
Cambridge University Press, Cambridge.
…
3: 8.17 Incomplete Beta Functions
…
►
8.17.5
positive integers; .
…
►The expansion (8.17.22) converges rapidly for .
…
►For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62).
…
►
8.17.24
positive integers; .
…
4: Bibliography S
…
►
FGH, a code for the calculation of Coulomb radial wave functions from series expansions.
Comput. Phys. Comm. 146 (2), pp. 250–253.
…
►
On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
►
On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
…
►
The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc. 62, pp. 441–452.
…
►
Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
…
5: Bibliography M
…
►
Siegel’s modular forms and Dirichlet series.
Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
…
►
Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions.
Ramanujan J. 6 (1), pp. 7–149.
…
►
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function.
Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
…
►
The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
…
►
On the evaluation of some multiple series.
J. London Math. Soc. (2) 33, pp. 368–371.
…
6: Bibliography D
…
►
Chebyshev series for the spherical Bessel function
.
Comput. Phys. Comm. 18 (1), pp. 73–86.
…
►
The Taylor Series.
Oxford University Press, Oxford.
…
►
Infinite integrals in the theory of Bessel functions.
Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.
…
►
Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
…
►
Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
…
7: Bibliography V
…
►
On the series expansion method for computing incomplete elliptic integrals of the first and second kinds.
Math. Comp. 23 (105), pp. 61–69.
…
►
An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
…
►
Some novel infinite series of spherical Bessel functions.
Quart. Appl. Math. 42 (3), pp. 321–324.
…
►
Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
…
►
Fourier series representation of Ferrers function
.
…
8: Bibliography C
…
►
Power series for inverse Jacobian elliptic functions.
Math. Comp. 77 (263), pp. 1615–1621.
…
►
Introduction to the Theory of Fourier’s Series and Integrals.
3rd edition, Macmillan, London.
…
►
Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
…
►
Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
…
►
Advanced Combinatorics: The Art of Finite and Infinite Expansions.
enlarged edition, D. Reidel Publishing Co., Dordrecht.
…
9: Bibliography B
…
►
Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
…
►
Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
…
►
Infinitely many Stokes smoothings in the gamma function.
Proc. Roy. Soc. London Ser. A 434, pp. 465–472.
…
►
A short table of the functions , from to
.
Phil. Mag. Series 7 20, pp. 343–347.
►
Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
…
10: 19.36 Methods of Computation
…
►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
…
►If the iteration of (19.36.6) and (19.36.12) is stopped when ( and being approximated by and , and the infinite series being truncated), then the relative error in and is less than if we neglect terms of order .
…
►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
…
►For series expansions of Legendre’s integrals see §19.5.
Faster convergence of power series for and can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12).
…