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Mobius inversion formulas

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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 d | n μ ( d ) f ( d ) = p | n ( 1 - f ( p ) ) , n > 1 .
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
27.6.3 d | n | μ ( d ) | = 2 ν ( n ) ,
27.6.4 d 2 | n μ ( d ) = | μ ( n ) | ,
27.6.7 d | n μ ( d ) ( n d ) k = J k ( n ) ,
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 lim x 1 x n x μ ( n ) = 0 ,
27.11.14 lim x n x μ ( n ) n = 0 ,
8: Bibliography R
  • I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • G. Rota (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.
  • 9: 4.24 Inverse Trigonometric Functions: Further Properties
    §4.24 Inverse Trigonometric Functions: Further Properties
    §4.24(i) Power Series
    §4.24(ii) Derivatives
    §4.24(iii) Addition Formulas
    4.24.17 Arctan u ± Arccot v = Arctan ( u v ± 1 v u ) = Arccot ( v u u v ± 1 ) .
    10: 4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38(i) Power Series
    §4.38(ii) Derivatives
    §4.38(iii) Addition Formulas
    4.38.19 Arctanh u ± Arccoth v = Arctanh ( u v ± 1 v ± u ) = Arccoth ( v ± u u v ± 1 ) .