Jordan curve theorem

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11: 26.1 Special Notation

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

12: 27.3 Multiplicative Properties

… ►

27.3.4

J_{k}\left(n\right)=n^{k}\prod_{p\mathbin{|}n}(1-p^{-k}),

ⓘ

Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $k$ : positive integer, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

…

13: 1.6 Vectors and Vector-Valued Functions

… ► ►

Green’s Theorem

… ►

Stokes’s Theorem

… ►

Gauss’s (or Divergence) Theorem

… ►

Green’s Theorem (for Volume)

…

15: 21.10 Methods of Computation

… ►

•

Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

►

•

Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

16: 21.7 Riemann Surfaces

… ►

§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces

… ►Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). …Equation (21.7.1) determines a plane algebraic curve in

{\mathbb{C}}^{2}

, which is made compact by adding its points at infinity. … ►

§21.7(iii) Frobenius’ Identity

… ►These are Riemann surfaces that may be obtained from algebraic curves of the form …

17: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

18: 1.4 Calculus of One Variable

… ►

Mean Value Theorem

… ►

Fundamental Theorem of Calculus

… ►

First Mean Value Theorem

… ►

Second Mean Value Theorem

… ►

§1.4(vi) Taylor’s Theorem for Real Variables

…

19: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

20: 10.44 Sums

… ►

§10.44(i) Multiplication Theorem

… ►

§10.44(ii) Addition Theorems

►

Neumann’s Addition Theorem

… ►

Graf’s and Gegenbauer’s Addition Theorems

…