DLMF: Untitled Document

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the sum of the

\alpha

th powers of the divisors of

n

, where the exponent

\alpha

can be real or complex. … ►where

p^{a}

is a prime power with

a\geq 1

; otherwise

\Lambda\left(n\right)=0

. …

… ►

32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

… ►where the

Q_{n}(z)

are monic polynomials (coefficient of highest power of

z

is

1

) satisfying … ►

Q_{3}(z)=z^{6}+20z^{3}-80,

… ►

32.8.8

\sum_{m=0}^{\infty}p_{m}(z)\lambda^{m}=\exp\left(z\lambda-\tfrac{4}{3}\lambda^% {3}\right).

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $m$ : integer, $z$ : real and $p_{m}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

…

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions

E_{s}

are defined by (19.19.4). … ►

§19.36(iii) Via Theta Functions

… ►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). … ►Faster convergence of power series for

K\left(k\right)

and

E\left(k\right)

can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …

§23.9 Laurent and Other Power Series

… ►

23.9.2

\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{n=2}^{\infty}c_{n}z^{2n-2},

0<|z|<|z_{0}|

,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex and $c_{n}$
A&S Ref:: 18.5.1
Referenced by:: §23.9, §23.9
Permalink:: http://dlmf.nist.gov/23.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

►

23.9.3

\zeta\left(z\right)=\frac{1}{z}-\sum_{n=2}^{\infty}\frac{c_{n}}{2n-1}z^{2n-1},

0<|z|<|z_{0}|

.

ⓘ

Symbols:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex and $c_{n}$
A&S Ref:: 18.5.5
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►

c_{2}=\frac{1}{20}g_{2},

… ►

23.9.7

\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},

ⓘ

Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $!$ : factorial (as in $n!$ ), $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $c_{n}$ and $a_{m,n}$ : coefficients
A&S Ref:: 18.5.6
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

…

… ►

A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

… ►

D. K. Bhaumik and S. K. Sarkar (2002) On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

ⓘ

External Links:: ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

W. G. Bickley and J. Nayler (1935) A short table of the functions

\mathrm{Ki}_{n}(x)

, from

n=1

to

n=16

. Phil. Mag. Series 7 20, pp. 343–347.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.43(iii), 2nd item.
Encodings:: BibTeX

… ►

S. Bochner (1952) 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.

ⓘ

External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

…

… ►

T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

ⓘ

External Links:: ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt
Referenced by:: §28.24, §28.24.
Encodings:: BibTeX

… ►

B. J. Laurenzi (1993) Moment integrals of powers of Airy functions. Z. Angew. Math. Phys. 44 (5), pp. 891–908.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (B. M. Agrawal), ZentralBlatt
Referenced by:: (9.11.15), (9.11.18), §9.11(v), §9.11(v).
Encodings:: BibTeX

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

L. Lorch and P. Szegő (1964) Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

…

… ►

B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

ⓘ

External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

ⓘ

External Links:: ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §14.28(ii), §14.28(ii).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

…

§20.11 Generalizations and Analogs

… ►

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ► … ►

§20.11(iv) Theta Functions with Characteristics

…

… ►

K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

… ►

A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

D. Sornette (1998) Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.

ⓘ

External Links:: Document
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

ⓘ

Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

ⓘ

Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

…

… ►

W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §28.26(i), §28.26(i), §28.8(iii).
Encodings:: BibTeX

… ►

Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

ⓘ

External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

ⓘ

External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

…

power%20function

1—10 of 14 matching pages

1: 27.2 Functions

2: 32.8 Rational Solutions

3: 19.36 Methods of Computation

§19.36(iii) Via Theta Functions

4: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

5: Bibliography B

6: Bibliography L

7: Bibliography C

8: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

§20.11(iv) Theta Functions with Characteristics

9: Bibliography S

10: Bibliography G

power%20function

1—10 of 14 matching pages

1: 27.2 Functions

2: 32.8 Rational Solutions

3: 19.36 Methods of Computation

§19.36(iii) Via Theta Functions

4: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

5: Bibliography B

6: Bibliography L

7: Bibliography C

8: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

§20.11(ii) Ramanujan’s Theta Function and q -Series

§20.11(iv) Theta Functions with Characteristics

9: Bibliography S

10: Bibliography G

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series