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21: 25.12 Polylogarithms

… ►The notation

\operatorname{Li}_{2}\left(z\right)

was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►The special case

z=1

is the Riemann zeta function:

\zeta\left(s\right)=\operatorname{Li}_{s}\left(1\right)

. … ►Further properties include …and … ►In terms of polylogarithms …

22: 12.11 Zeros

… ►

§12.11(i) Distribution of Real Zeros

… ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When

a=\tfrac{1}{2}

these zeros are the same as the zeros of the complementary error function

\operatorname{erfc}(z/\sqrt{2})

; compare (12.7.5). … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

… ►For further information, including associated functions, see Olver (1959).

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

24: 27.15 Chinese Remainder Theorem

… ►Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

25: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

…

26: Staff

… ►

William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

… ►

William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

…

27: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

… ►

•

Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

… ►

•

Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

28: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

►

§25.11(i) Definition

… ►The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

… ►

§25.11(vi) Derivatives

…

29: 3.8 Nonlinear Equations

… ►For multiple zeros the convergence is linear, but if the multiplicity

m

is known then quadratic convergence can be restored by multiplying the ratio

f(z_{n})/f^{\prime}(z_{n})

in (3.8.4) by

m

. … ►However, to guard against the accumulation of rounding errors, a final iteration for each zero should also be performed on the original polynomial

p(z)

. … ►

§3.8(v) Zeros of Analytic Functions

… ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. …

30: 6.16 Mathematical Applications

… ►

6.16.1

\sin x+\tfrac{1}{3}\sin\left(3x\right)+\tfrac{1}{5}\sin\left(5x\right)+\dots=% \begin{cases}\frac{1}{4}\pi,&0<x<\pi,\\ 0,&x=0,\\ -\frac{1}{4}\pi,&-\pi<x<0.\end{cases}

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/6.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.16(i), §6.16 and Ch.6

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

6.16.5

\operatorname{li}\left(x\right)-\pi(x)=O\left(\sqrt{x}\ln x\right),

x\to\infty

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\pi(x)$ : number of primes $\leq x$
A&S Ref:: 5.1.50
Permalink:: http://dlmf.nist.gov/6.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §6.16(ii), §6.16 and Ch.6

… ►

See accompanying text — Figure 6.16.2: The logarithmic integral $\operatorname{li}\left(x\right)$ , together with vertical bars indicating the value of $\pi(x)$ for $x=10,20,\dots,1000$ . Magnify