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1: 22.15 Inverse Functions

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§22.15(ii) Representations as Elliptic Integrals

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2: 19.26 Addition Theorems

… ►Equivalent forms of (19.26.2) are given by …Equivalent forms of (19.26.11) are given by … ►Equivalent forms are given by (19.22.22). …

3: 13.3 Recurrence Relations and Derivatives

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13.3.14 $$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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4: 27.2 Functions

… ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: …

5: 18.35 Pollaczek Polynomials

… ►or, equivalently in second form (18.2.10), …

6: Guide to Searching the DLMF

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. …

7: 27.5 Inversion Formulas

… ►The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). …For example, the equation $\zeta \left(s\right)\cdot (1/\zeta \left(s\right))=1$ is equivalent to the identity ►

27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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8: 27.16 Cryptography

… ►To code a piece $x$, raise $x$ to the power $r$ and reduce $x^r$ modulo $n$ to obtain an integer $y$ (the coded form of $x$) between $1$ and $n$. Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. But $y^s\equiv x^{rs}\equiv x^{1+t\varphi \left(n\right)}\equiv x(modn)$, so $y^s$ is the same as $x$ modulo $n$. …

9: 30.3 Eigenvalues

… ►With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form ${(1-x^2)}^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. …

10: 28.2 Definitions and Basic Properties

… ►The standard form of Mathieu’s equation with parameters $(a,q)$ is …With $\zeta =\cos z$ we obtain another algebraic form: … ►Equivalently, …If $\widehat{\nu }=0$ or $1$, or equivalently, $\nu =n$, then $\nu$ is a double root of the characteristic equation, otherwise it is a simple root. … ►An equivalent formulation is given by …