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1: Foreword

… ►Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. … ►November 20, 2009 …

2: 12.11 Zeros

… ►

12.11.5 $$p_0(\zeta )=t(\zeta ),$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.8 $$q_0(\zeta )=t(\zeta ).$$

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Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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3: Errata

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Changes

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Changes

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Changes

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Changes

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Changes

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4: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.4 $$p_n\sim n\ln n.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

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Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

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5: 9.7 Asymptotic Expansions

… ►

9.7.1 $$\zeta =\frac{2}{3}{z}^{3/2}.$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Permalink:

http://dlmf.nist.gov/9.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(i), §9.7 and Ch.9

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

… ►

9.7.20 $$R_n(z)={(-1)}^n\sum _{k=0}^{m-1}{(-1)}^k{u}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+R_{m,n}(z),$$

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Defines:

$R_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $u_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

►

9.7.21 $$S_n(z)={(-1)}^{n-1}\sum _{k=0}^{m-1}{(-1)}^k{v}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+S_{m,n}(z),$$

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Defines:

$S_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $v_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

…

6: 20.11 Generalizations and Analogs

… ►

§20.11(iii) Ramanujan’s Change of Base

… ►These results are called Ramanujan’s changes of base. …

7: Bibliography D

… ►

Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

►

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

…

8: 19.36 Methods of Computation

… ►If (19.36.1) is used instead of its first five terms, then the factor ${(3r)}^{-1/6}$ in Carlson (1995, (2.2)) is changed to ${(3r)}^{-1/8}$. ►For both $R_D$ and $R_J$ the factor ${(r/4)}^{-1/6}$ in Carlson (1995, (2.18)) is changed to ${(r/5)}^{-1/8}$ when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: ►

19.36.2 $$1-\frac{3}{14}{E}_2+\frac{1}{6}{E}_3+\frac{9}{88}{E}_2^2-\frac{3}{22}{E}_4-\frac{9}{52}{E}_2{E}_3+\frac{3}{26}{E}_5-\frac{1}{16}{E}_2^3+\frac{3}{40}{E}_3^2+\frac{3}{20}{E}_2{E}_4+\frac{45}{272}{E}_2^2{E}_3-\frac{9}{68}(E_3{E}_4+E_2{E}_5).$$

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Symbols:

$E_s(𝐳)$: symmetric function

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(i), §19.36 and Ch.19

… ►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). …

9: 5.11 Asymptotic Expansions

… ►

5.11.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)\sim \left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.40

Referenced by:

§5.11(i), §5.11(i), §5.11(ii), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

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

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►

5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

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

…

10: 3.8 Nonlinear Equations

… ►

$f(x_0)f^{\prime \prime }(x_0)>0$ and $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign between $x_0$ and $\xi$ (monotonic convergence).

►

, $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign in the interval $(x_0,x_1)$, and $\xi \in [x_0,x_1]$ (monotonic convergence after the first iteration).

… ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …