Mobius%20inversion%20formulas

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11: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

12: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

13: 8 Incomplete Gamma and Related
Functions

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14: 28 Mathieu Functions and Hill’s Equation

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15: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 23 Weierstrass Elliptic and Modular
Functions

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17: 27.4 Euler Products and Dirichlet Series

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27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.8 $$\sum _{n=1}^{\mathrm{\infty }}|\mu \left(n\right)|n^{-s}=\frac{\zeta \left(s\right)}{\zeta \left(2s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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18: 20.11 Generalizations and Analogs

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

19: 3.8 Nonlinear Equations

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Regula Falsi

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►

§3.8(iv) Zeros of Polynomials

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

20: 27.7 Lambert Series as Generating Functions

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27.7.3 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)\frac{x^n}{1-x^n}=x,$$

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Symbols:

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

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

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