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31: 18.1 Notation

§18.1 Notation

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§18.1(i) Special Notation

►(For other notation see Notation for the Special Functions.) … ►

Classical OP’s

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§18.1(iii) Other Notations

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32: 21.1 Special Notation

§21.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Uppercase boldface letters are $g\times g$ real or complex matrices. …

33: 1.1 Special Notation

§1.1 Special Notation

►(For other notation see Notation for the Special Functions.) …

34: 23.1 Special Notation

§23.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►

$𝕃$	lattice in $ℂ$.
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►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. ►

Other Notations

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35: 16.1 Special Notation

§16.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.

36: Mathematical Introduction

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Notation for the Special Functions

… ►This section may also include important alternative notations that have appeared in the literature. … ►The exceptions are ones for which the existing notations have drawbacks. … ►The Notations section includes all the notations for the special functions adopted in this Handbook. … ►

Common Notations and Definitions

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37: Need Help?

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• How do I search within DLMF? See Guide to Searching the DLMF.

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• Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

38: 19 Elliptic Integrals

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39: 26.13 Permutations: Cycle Notation

§26.13 Permutations: Cycle Notation

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26.13.1

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Symbols:

$n$: nonnegative integer and $\sigma$: permutation

Permalink:

http://dlmf.nist.gov/26.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that $\sigma (j)$ immediately follows $j$ or, if $j$ is the last listed element of the cycle, then $\sigma (j)$ is the first element of the cycle. … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …They are often dropped from the cycle notation. …

40: 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

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27.12.4 $$\pi \left(x\right)\sim \sum _{k=1}^{\mathrm{\infty }}\frac{(k-1)!x}{{(\ln x)}^k}.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $k$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.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 §27.12 and Ch.27

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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

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27.12.7

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{li}\left(x\right)$: 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:

§27.12

Permalink:

http://dlmf.nist.gov/27.12.E7

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, §27.12 and Ch.27

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