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21: 10.72 Mathematical Applications

… ►The canonical form of differential equation for these problems is given by … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). …

22: 18.2 General Orthogonal Polynomials

… ►

First Form

… ►

Second Form

… ►

Monic and Orthonormal Forms

… ►The recurrence relations (18.2.10) can be equivalently written as … ►

Confluent Form

…

23: 28.29 Definitions and Basic Properties

… ►Equivalently, …It has the form … ►The case $c=0$ is equivalent to …

24: 3.7 Ordinary Differential Equations

… ►The remaining two equations are supplied by boundary conditions of the form … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. … ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

25: 19.18 Derivatives and Differential Equations

… ►or equivalently, … ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). …

26: 25.5 Integral Representations

… ►

25.5.20 $$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}-1}dz,$$ $s\ne 1,2,\mathrm{\dots }$,

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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, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Sources:

Apostol (1976, (6), p. 253); with $a=1$; Erdélyi et al. (1953a, (1.12.9), p. 32); with $z=-t$

A&S Ref:

23.2.4 (in slightly different form)

Referenced by:

§25.11(vii)

Permalink:

http://dlmf.nist.gov/25.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(iii), §25.5 and Ch.25

… ►Equivalently, …

27: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

… ►An equivalent formulation is given by …

28: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

►

§33.5(i) Small $\rho$

… ►Equivalently, … ►

§33.5(iii) Small $|\eta |$

… ►

§33.5(iv) Large $\mathrm{\ell }$

…

29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ► … ►compare (1.18.30) and (1.18.45), and the eigenfunction expansions are of the form … ►Consider formally self-adjoint operators of the form …

30: 27.11 Asymptotic Formulas: Partial Sums

… ►It is more fruitful to study partial sums and seek asymptotic formulas of the form … ►

27.11.11 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{\ln p}{p}=\frac{1}{\varphi \left(k\right)}\ln x+O\left(1\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E11

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 §27.11 and Ch.27

… ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.