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21: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

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22: 13.2 Definitions and Basic Properties

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§13.2(vi) Wronskians

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Kummer’s Transformations

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23: 27.12 Asymptotic Formulas: Primes

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27.12.1 $$\underset{n\to \mathrm{\infty }}{lim}\frac{p_n}{n\ln n}=1,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/27.12.E1

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 §27.12 and Ch.27

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27.12.2 $$p_n>n\ln n,$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/27.12.E2

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 §27.12 and Ch.27

… ►where the series terminates when the product of the first $r$ primes exceeds $x$. … ►

27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

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

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►where $\lambda (\alpha )$ depends only on $\alpha$, and $\varphi \left(m\right)$ is the Euler totient function (§27.2). …

24: 11.9 Lommel Functions

§11.9 Lommel Functions

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§11.9(ii) Expansions in Series of Bessel Functions

… ►If either of $\mu \pm \nu$ equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents $S_{\mu ,\nu }\left(z\right)$ exactly. … ►

25: 13.4 Integral Representations

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§13.4(i) Integrals Along the Real Line

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§13.4(ii) Contour Integrals

… ►The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. … … ►

26: 1.10 Functions of a Complex Variable

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Analytic Functions

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§1.10(vi) Multivalued Functions

… ►(Or more generally, a simple contour that starts at the center and terminates on the boundary.) … ►

§1.10(vii) Inverse Functions

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§1.10(xi) Generating Functions

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27: 15.6 Integral Representations

§15.6 Integral Representations

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. … ►In (15.6.1) all functions in the integrand assume their principal values. … ►In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. … ►In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …

28: 4.6 Power Series

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§4.6(i) Logarithms

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4.6.1 $$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$ $|z|\le 1$, $z\ne -1$,

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.24

Referenced by:

§4.45(i), §4.6(i), (8.15.2)

Permalink:

http://dlmf.nist.gov/4.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

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4.6.2 $$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}{\left(\frac{z-1}{z}\right)}^2+\frac{1}{3}{\left(\frac{z-1}{z}\right)}^3+\mathrm{\cdots },$$ $\mathrm{\Re }z\ge \frac{1}{2}$,

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Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.25

Permalink:

http://dlmf.nist.gov/4.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

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4.6.3 $$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$ $|z-1|\le 1$, $z\ne 0$,

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.26

Permalink:

http://dlmf.nist.gov/4.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►If $a=0,1,2,\mathrm{\dots }$, then the series terminates and $z$ is unrestricted. …

29: 2.7 Differential Equations

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. … ►such that …Here $F(x)$ is the error-control function …

30: 18.39 Applications in the Physical Sciences

… ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

►The functions ${\psi }_{p,l}(r)$ satisfy the equation, … ►

c) Spherical Radial Coulomb Wave Functions

… ►The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with $\lambda =l+1$, $a=-b=2Z/s$, and $c=0$, and terminates for $x=x_i^N$ being a zero of the polynomial of order $N$. …