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21—30 of 423 matching pages

21: Peter L. Walker

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20 Theta Functions

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22: Staff

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

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

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

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

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23: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes $p\le x$: … ►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

24: 25.12 Polylogarithms

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25.12.7 $${\mathrm{Li}}_2\left({\mathrm{e}}^{\mathrm{i}\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\theta \right)}{n^2}+\mathrm{i}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\theta \right)}{n^2}.$$

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Symbols:

$\cos z$: cosine function, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $n$: nonnegative integer and $\theta$: phase

Keywords:

infinite series, series representation

Source:

Maximon (2003, (8.7), p. 2813)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

►The cosine series in (25.12.7) has the elementary sum … ►

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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25.12.10 $${\mathrm{Li}}_s\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$$

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Defines:

${\mathrm{Li}}_s\left(z\right)$: polylogarithm

Symbols:

$n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Lewin (1981, p. 236)

Referenced by:

(25.14.3)

Permalink:

http://dlmf.nist.gov/25.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12 and Ch.25

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25: 10.75 Tables

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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.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.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.

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

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

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

26: 17.18 Methods of Computation

§17.18 Methods of Computation

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27: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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28: Foreword

… ►November 20, 2009 …

29: 13.30 Tables

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Zhang and Jin (1996, pp. 411–423) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $a=-5(.5)5$, $b=0.5(.5)5$, and $x=0.1,1,5,10,20,30$, 8S (for $M(a,b,x)$) and 7S (for $U(a,b,x)$).

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30: 28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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