theta%20functions
(0.005 seconds)
11—20 of 31 matching pages
11: Bibliography F
…
►
Theta Functions on Riemann Surfaces.
Springer-Verlag, Berlin.
►
…
►
Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
…
►
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.
…
►
On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
…
12: 9.18 Tables
…
►
•
…
Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
13: 19.36 Methods of Computation
…
►If and , so that , then this procedure reduces to the AGM method for the complete integral.
►The step from to is an ascending Landen transformation if (leading ultimately to a hyperbolic case of ) or a descending Gauss transformation if (leading to a circular case of ).
…
►
§19.36(iii) Via Theta Functions
►Lee (1990) compares the use of theta functions for computation of , , and , , with four other methods. …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). …14: 25.12 Polylogarithms
…
►When , , (25.12.1) becomes
…
►The special case is the Riemann zeta function: .
…
►Further properties include
…and
…
►In terms of polylogarithms
…
15: 14.30 Spherical and Spheroidal Harmonics
…
►
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 .
…
►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.
…
►where and .
…
►has solutions , which are everywhere one-valued and continuous.
…
16: Bibliography
…
►
Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
…
►
On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
…
►
Chapter of Ramanujan’s second notebook: Theta-functions and -series.
Mem. Amer. Math. Soc. 53 (315), pp. v+85.
…
►
Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
…
►
On basic hypergeometric series, mock theta functions, and partitions. II.
Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.
…
17: Bibliography L
…
►
Algorithm 917: complex double-precision evaluation of the Wright
function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
…
►
Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds.
Comput. Phys. Comm. 60 (3), pp. 319–327.
…
►
An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
…
►
Monotonicity of the differences of zeros of Bessel functions as a function of order.
Proc. Amer. Math. Soc. 15 (1), pp. 91–96.
…
►
Integral representation of the Hankel function in terms of parabolic cylinder functions.
Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
…
18: 10.73 Physical Applications
…
►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
…
►
…
►The functions
, , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)):
…
►
§10.73(iii) Kelvin Functions
… ►§10.73(iv) Bickley Functions
…19: 1.11 Zeros of Polynomials
…
►The elementary
symmetric functions of the zeros are (with )
…
►For the roots of and the roots of the resolvent cubic equation
…
►
…
►
1.11.20
…
►Resolvent cubic is with roots , , , and , , .
…
20: Bibliography M
…
►
Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
…
►
Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
…
►
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.
…
►
The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
…
►
Tata Lectures on Theta. II.
Birkhäuser Boston Inc., Boston, MA.
…