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1: 25.6 Integer Arguments

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25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.6 $$\zeta \left(2k+1\right)=\frac{{(-1)}^{k+1}{(2\pi )}^{2k+1}}{2(2k+1)!}{\int }_0^1{B}_{2k+1}\left(t\right)\cot \left(\pi t\right)dt,$$ $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral and $k$: nonnegative integer

Source:

Nörlund (1924, (81*), p. 66); with $\nu =k$

A&S Ref:

23.2.17

Permalink:

http://dlmf.nist.gov/25.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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2: 28.8 Asymptotic Expansions for Large $q$

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28.8.1 $$\begin{array}{c}{a}_m\left(h^2\right)\\ b_{m+1}\left(h^2\right)\end{array}\}\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter

A&S Ref:

20.2.30 (in different form)

Referenced by:

§28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(i), §28.8 and Ch.28

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►

28.8.9 $$W_m^{\pm }(x)=\frac{{\mathrm{e}}^{\pm 2h\sin x}}{{(\cos x)}^{m+1}}\{\begin{array}{cc}{\left(\cos \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \\ {\left(\sin \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \end{array}$$

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Defines:

$W_m^{\pm }$: function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.13 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

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Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

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Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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3: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►on assuming a time dependence of the form ${\mathrm{e}}^{\pm \mathrm{i}kt}$. …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). … ►

10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

4: Software Index

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	Open Source													With Book						Commercial
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6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$	✓	✓				✓			✓			✓	✓	✓							✓	✓	✓
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7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
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20 Theta Functions
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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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5: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►If $z$ is complex, then the remainder terms are bounded in magnitude by ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms. … ►

5.11.11 $$\left|R_K(z)\right|\le \frac{(1+\zeta \left(K\right))\mathrm{\Gamma }\left(K\right)}{2{(2\pi )}^{K+1}{\left|z\right|}^K}\left(1+\min (\sec \left(\mathrm{ph}z\right),2K^{\frac{1}{2}})\right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi$,

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Defines:

$R_K(z)$: remainder (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\sec z$: secant function and $z$: complex variable

Referenced by:

§5.11(ii)

Permalink:

http://dlmf.nist.gov/5.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(ii), §5.11 and Ch.5

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6: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right)$$

Originally the symbol $\pm$ was missing after the second equal sign.

Reported 2012-09-27 by Dennis Heim.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Originally the limiting form for $\mathrm{sc}(z,k)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
$\mathrm{cn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{sd}(z,k)$ $\to$	$\sinh z$	$\mathrm{nc}(z,k)$ $\to$	$\cosh z$	$\mathrm{ds}(z,k)$ $\to$	$\mathrm{csch}z$
$\mathrm{dn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{nd}(z,k)$ $\to$	$\cosh z$	$\mathrm{sc}(z,k)$ $\to$	$\sinh z$	$\mathrm{cs}(z,k)$ $\to$	$\mathrm{csch}z$

Reported 2010-11-23.

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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7: 25.12 Polylogarithms

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25.12.7 $${\mathrm{Li}}_2\left({\mathrm{e}}^{\mathrm{i}\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\theta \right)}{n^2}+\mathrm{i}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\theta \right)}{n^2}.$$

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Symbols:

$\cos z$: cosine function, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $n$: nonnegative integer and $\theta$: phase

Keywords:

infinite series, series representation

Source:

Maximon (2003, (8.7), p. 2813)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.8 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\theta \right)}{n^2}=\frac{{\pi }^2}{6}-\frac{\pi \theta }{2}+\frac{{\theta }^2}{4}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: nonnegative integer and $\theta$: phase

Keywords:

infinite series

Source:

Maximon (2003, (8.7), p. 2813)

A&S Ref:

27.8.6 (second series)

Permalink:

http://dlmf.nist.gov/25.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.9 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\theta \right)}{n^2}=-{\int }_0^{\theta }\ln \left(2\sin \left(\frac{1}{2}x\right)\right)dx.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: nonnegative integer, $x$: real variable and $\theta$: phase

Keywords:

definite integral, infinite series

Source:

Maximon (2003, (8.7), (8.8), p. 2813)

A&S Ref:

27.8.6 (integration of first series)

Referenced by:

1st item

Permalink:

http://dlmf.nist.gov/25.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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8: Bibliography S

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

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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

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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

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

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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

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