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11: 31.11 Expansions in Series of Hypergeometric Functions

… ►Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. … ►Here one of the following four pairs of conditions is satisfied: …

12: 3.5 Quadrature

… ►In the case of Chebyshev weight functions $w(x)={(1-x)}^{\alpha }{(1+x)}^{\beta }$ on $[-1,1]$, with $\left|\alpha \right|=\left|\beta \right|=\frac{1}{2}$, the nodes $x_k$, weights $w_k$, and error constant ${\gamma }_n$, are as follows: … ►The pair …

13: 28.2 Definitions and Basic Properties

… ►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. … ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►

§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

… ►

28.2.16 $$\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ;a,q)=w_{\text{I}}(\pi ;a,-q).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $w_{\text{I}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.10 (in slightly different form)

Referenced by:

§28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), item (c), §28.34(i)

Permalink:

http://dlmf.nist.gov/28.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(iii), §28.2 and Ch.28

… ►Either $\widehat{\nu }$ or $\nu$ is called a characteristic exponent of (28.2.1). …

14: 27.3 Multiplicative Properties

… ►

27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20, §27.3

Permalink:

http://dlmf.nist.gov/27.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

►

27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

27.3.10 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}{\left(f(p_r)\right)}^{a_r}.$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

15: 20 Theta Functions

Chapter 20 Theta Functions

…

16: 17.12 Bailey Pairs

§17.12 Bailey Pairs

… ►

Bailey Pairs

… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: … ►The Bailey pair and Bailey chain concepts have been extended considerably. …

17: 36.6 Scaling Relations

… ►

Table 36.6.1: Special cases of scaling exponents for cuspoids.

► ►►

singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
…

►

►For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).

18: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§28.17 Stability as $x\to \pm \mathrm{\infty }$

►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. All other pairs are unstable. ►For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$. ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …

19: 10.75 Tables

… ►

Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

… ►

Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of $Y_n\left(z\right)$ and $Y_n^{\prime }\left(z\right)$ for $n=0(1)5$, 8D.

… ►

Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

… ►

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

… ►

Zhang and Jin (1996, p. 322) tabulates $\mathrm{ber}x$, ${\mathrm{ber}}^{\prime }x$, $\mathrm{bei}x$, ${\mathrm{bei}}^{\prime }x$, $\ker x$, ${\ker }^{\prime }x$, $\mathrm{kei}x$, ${\mathrm{kei}}^{\prime }x$, $x=0(1)20$, 7S.

…

20: Bibliography B

… ►

Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

… ►

Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10 (12), pp. 2061–2081.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§36.6.

Encodings:

BibTeX

… ►

Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

… ►

The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

…