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1: 19.37 Tables

… ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(1^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 7S by Beli͡akov et al. (1962). … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated (with different notation) for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 5D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 7D by Zhang and Jin (1996, pp. 676–677). …

2: 20.15 Tables

… ►This reference gives ${\theta }_j(x,q)$, $j=1,2,3,4$, and their logarithmic $x$-derivatives to 4D for $x/\pi =0(.1)1$, $\alpha =0(9^{\circ }){90}^{\circ }$, where $\alpha$ is the modular angle given by ►

20.15.1 $$\sin \alpha ={\theta }_2^2(0,q)/{\theta }_3^2(0,q)=k.$$

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Defines:

$\alpha$: modular angle (locally)

Symbols:

${\theta }_j(z,q)$: theta function, $\sin z$: sine function and $q$: nome

Referenced by:

§20.15, §20.15

Permalink:

http://dlmf.nist.gov/20.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.15 and Ch.20

►Spenceley and Spenceley (1947) tabulates ${\theta }_1(x,q)/{\theta }_2(0,q)$, ${\theta }_2(x,q)/{\theta }_2(0,q)$, ${\theta }_3(x,q)/{\theta }_4(0,q)$, ${\theta }_4(x,q)/{\theta }_4(0,q)$ to 12D for $u=0(1^{\circ }){90}^{\circ }$, $\alpha =0(1^{\circ }){89}^{\circ }$, where $u=2x/(\pi {\theta }_3^2(0,q))$ and $\alpha$ is defined by (20.15.1), together with the corresponding values of ${\theta }_2(0,q)$ and ${\theta }_4(0,q)$. ►Lawden (1989, pp. 270–279) tabulates ${\theta }_j(x,q)$, $j=1,2,3,4$, to 5D for $x=0(1^{\circ }){90}^{\circ }$, $q=0.1(.1)0.9$, and also $q$ to 5D for $k^2=0(.01)1$. ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

3: 5.16 Sums

… ►

5.16.1 $$\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{\psi }^{\prime }\left(k\right)=-\frac{{\pi }^2}{8},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

►

5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

4: 14.33 Tables

… ►

Abramowitz and Stegun (1964, Chapter 8) tabulates ${𝖯}_n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; ${𝖯}_n^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${𝖰}_n\left(x\right)$ and ${𝖰}_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P_n\left(x\right)$ and $P_n^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q_n\left(x\right)$ and $Q_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

►

Zhang and Jin (1996, Chapter 4) tabulates ${𝖯}_n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${𝖯}_n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖰}_n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${𝖰}_n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖯}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${𝖰}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${𝖯}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P_n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${𝖯}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

►

Belousov (1962) tabulates ${𝖯}_n^m\left(\cos \theta \right)$ (normalized) for $m=0(1)36$, $n-m=0(1)56$, $\theta =0({2.5}^{\circ }){90}^{\circ }$, 6D.

►

Žurina and Karmazina (1964, 1965) tabulate the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

►

Žurina and Karmazina (1963) tabulates the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

…

5: 33.26 Software

… ►

Noble (2004). Fortran 90.

…

6: Bibliography E

… ►

Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

… ►

The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

ⓘ

External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

Encodings:

BibTeX

… ►

Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter. ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.

ⓘ

External Links:

ISSN 0098-3500, FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §28.36(iii).

Encodings:

BibTeX

►

Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §28.36(iii).

Encodings:

BibTeX

… ►

On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

ⓘ

External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

…

7: 4.48 Software

… ►

Kearfott (1996). Fortran 90.

…

8: 27.2 Functions

… ►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, …where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. …

9: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. …

10: 25.19 Tables

… ►

Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

►

Morris (1979) tabulates ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

►

Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

…