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1: 20 Theta Functions

Chapter 20 Theta Functions

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2: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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27.2.4 $$p_n\sim n\ln n.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►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: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
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3: Foreword

… ►Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. … ►November 20, 2009 …

4: 3.8 Nonlinear Equations

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$f(x_0)f^{\prime \prime }(x_0)>0$ and $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign between $x_0$ and $\xi$ (monotonic convergence).

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, $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign in the interval $(x_0,x_1)$, and $\xi \in [x_0,x_1]$ (monotonic convergence after the first iteration).

… ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

5: 8 Incomplete Gamma and Related
Functions

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6: 28 Mathieu Functions and Hill’s Equation

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7: 8.26 Tables

… ►

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.

8: 23 Weierstrass Elliptic and Modular
Functions

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9: 12.11 Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

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12.11.5 $$p_0(\zeta )=t(\zeta ),$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.8 $$q_0(\zeta )=t(\zeta ).$$

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Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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10: 20.11 Generalizations and Analogs

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20.11.4 $$f(a,b)={\theta }_3\left(z\right|\tau ).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(ii), §20.11 and Ch.20

… ►

§20.11(iii) Ramanujan’s Change of Base

… ►

20.11.5 $${}_2{}^{}F_1^{}(\frac{1}{2},\frac{1}{2};1;k^2)={\theta }_3^2\left(0\right|\tau );$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\theta }_j\left(z\right|\tau )$: theta function, $\tau$: lattice parameter and $k$: modulus

Permalink:

http://dlmf.nist.gov/20.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(iii), §20.11 and Ch.20

… ►These results are called Ramanujan’s changes of base. …