Mobius inversion formulas

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1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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§4.37(iii) Reflection Formulas

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Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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Inverse Hyperbolic Tangent

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2: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(iii) Reflection Formulas

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Inverse Sine

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Inverse Cosine

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Inverse Tangent

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3: 22.15 Inverse Functions

§22.15 Inverse Functions

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§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

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

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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

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

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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

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Annotations for §27.6 and Ch.27

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

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Symbols:

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

Permalink:

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

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Annotations for §27.6 and Ch.27

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

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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

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Annotations for §27.6 and Ch.27

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

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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

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Annotations for §27.11 and Ch.27

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

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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

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Annotations for §27.11 and Ch.27

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8: Bibliography R

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

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External Links:

ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

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

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§34.6.

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§27.5.

Encodings:

BibTeX

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9: 4.24 Inverse Trigonometric Functions: Further Properties

§4.24 Inverse Trigonometric Functions: Further Properties

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§4.24(i) Power Series

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§4.24(ii) Derivatives

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§4.24(iii) Addition Formulas

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4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

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Annotations for §4.24(iii), §4.24 and Ch.4

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10: 4.38 Inverse Hyperbolic Functions: Further Properties

§4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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§4.38(ii) Derivatives

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§4.38(iii) Addition Formulas

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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

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Annotations for §4.38(iii), §4.38 and Ch.4

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