Mobius%20inversion%20formulas

SearchAdvancedHelp

(0.003 seconds)

1—10 of 397 matching pages

1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►

§4.37(iii) Reflection Formulas

… ►

Inverse Hyperbolic Sine

… ►

Inverse Hyperbolic Cosine

… ►

Inverse Hyperbolic Tangent

…

2: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►

§4.23(iii) Reflection Formulas

… ►

Inverse Sine

… ►

Inverse Cosine

… ►

Inverse Tangent

…

3: 22.15 Inverse Functions

§22.15 Inverse Functions

►

§22.15(i) Definitions

… ►Each of these inverse functions is multivalued. The principal values satisfy … ►

4: 27.5 Inversion Formulas

… ►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: … ►For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964).

5: 27.6 Divisor Sums

… ►

27.6.2 $$\sum _{d|n}\mu \left(d\right)f(d)=\prod _{p|n}(1-f(p)),$$ $n>1$.

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. … ►

27.6.3 $$\sum _{d|n}|\mu \left(d\right)|=2^{\nu \left(n\right)},$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $\nu \left(n\right)$: number of distinct primes dividing a number, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

►

27.6.4 $$\sum _{d^2|n}\mu \left(d\right)=|\mu \left(n\right)|,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

… ►

27.6.7 $$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^k=J_k\left(n\right),$$

ⓘ

Symbols:

$J_k\left(n\right)$: Jordan’s function, $\mu \left(n\right)$: Möbius function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

…

6: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

7: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

27.2.12

ⓘ

Defines:

$\mu \left(n\right)$: Möbius function

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a$: primitive root

A&S Ref:

24.3.1 I.A

Permalink:

http://dlmf.nist.gov/27.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►This is the Möbius function. … ►

Table 27.2.2: Functions related to division.

► ►►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

►

8: Bibliography R

… ►

On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

… ►

Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

… ►

Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

ⓘ

Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

… ►

Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

BibTeX

… ►

On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.

ⓘ

External Links:

Document, MathReview (M. A. Harrison), ZentralBlatt

Referenced by:

§27.5.

Encodings:

BibTeX

…

9: 20 Theta Functions

Chapter 20 Theta Functions

…

10: 27.11 Asymptotic Formulas: Partial Sums

§27.11 Asymptotic Formulas: Partial Sums

… ►It is more fruitful to study partial sums and seek asymptotic formulas of the form … ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. … ►

27.11.13 $$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}\sum _{n\le x}\mu \left(n\right)=0,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

►

27.11.14 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)}{n}=0,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

…