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1: 35.11 Tables

… ►Tables of zonal polynomials are given in James (1964) for $|\kappa |\le 6$, Parkhurst and James (1974) for $|\kappa |\le 12$, and Muirhead (1982, p. 238) for $|\kappa |\le 5$. …

2: 10.37 Inequalities; Monotonicity

… ►If and $|\mathrm{ph}z|\le \frac{1}{2}\pi$, then ►

10.37.1

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Symbols:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.37, §10.37, Erratum (V1.0.17) for Section 10.37

Permalink:

http://dlmf.nist.gov/10.37.E1

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Annotations for §10.37 and Ch.10

►Note that previously we did mention that (10.37.1) holds for . …

3: 20.1 Special Notation

… ► ►►►

$m$, $n$	integers.
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$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
…

►The main functions treated in this chapter are the theta functions ${\theta }_j\left(z\right|\tau )={\theta }_j(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. … ►Jacobi’s original notation: $\mathrm{\Theta }(z|\tau )$, ${\mathrm{\Theta }}_1(z|\tau )$, $\mathrm{H}(z|\tau )$, ${\mathrm{H}}_1(z|\tau )$, respectively, for ${\theta }_4\left(u\right|\tau )$, ${\theta }_3\left(u\right|\tau )$, ${\theta }_1\left(u\right|\tau )$, ${\theta }_2\left(u\right|\tau )$, where $u=z/{\theta }_3^2\left(0\right|\tau )$. … ►Neville’s notation: ${\theta }_s(z|\tau )$, ${\theta }_c(z|\tau )$, ${\theta }_d(z|\tau )$, ${\theta }_n(z|\tau )$, respectively, for ${\theta }_3^2\left(0\right|\tau ){\theta }_1\left(u\right|\tau )/{\theta }_1^{\prime }\left(0\right|\tau )$, ${\theta }_2\left(u\right|\tau )/{\theta }_2\left(0\right|\tau )$, ${\theta }_3\left(u\right|\tau )/{\theta }_3\left(0\right|\tau )$, ${\theta }_4\left(u\right|\tau )/{\theta }_4\left(0\right|\tau )$, where again $u=z/{\theta }_3^2\left(0\right|\tau )$. … ►McKean and Moll’s notation: ${\vartheta }_j(z|\tau )={\theta }_j\left(\pi z\right|\tau )$, $j=1,2,3,4$. …

4: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The intensity distribution follows ${|ℱ\left(x\right)|}^2$, where $ℱ$ is the Fresnel integral (See 7.3.4). …

5: 4.18 Inequalities

… ►

4.18.7 $$|\csc z|\le \mathrm{csch}|y|,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.85

Permalink:

http://dlmf.nist.gov/4.18.E7

Encodings:

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Annotations for §4.18, §4.18 and Ch.4

►

4.18.8 $$|\cos z|\le \cosh |z|,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.3.86

Permalink:

http://dlmf.nist.gov/4.18.E8

Encodings:

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Annotations for §4.18, §4.18 and Ch.4

►

4.18.9 $$|\sin z|\le \sinh |z|,$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.87

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E9

Encodings:

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Annotations for §4.18, §4.18 and Ch.4

►

►

$|\sin z|\le \frac{6}{5}|z|$ , .

…

6: 27.9 Quadratic Characters

… ►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$, then the value of $(n|p)$ is $0$. …The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

7: 20.2 Definitions and Periodic Properties

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … ►For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. … ►

…

Figure 20.2.1: $z$-plane. … $\bullet$ zeros of ${\theta }_1\left(z\right|\tau )$, $\mathrm{■}$ zeros of ${\theta }_2\left(z\right|\tau )$, $\mathrm{▲}$ zeros of ${\theta }_3\left(z\right|\tau )$, $\mathrm{◆}$ zeros of ${\theta }_4\left(z\right|\tau )$.

… ►For $m,n\in \mathbb{Z}$, the $z$-zeros of ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, are $(m+n\tau )\pi$, $(m+\frac{1}{2}+n\tau )\pi$, $(m+\frac{1}{2}+(n+\frac{1}{2})\tau )\pi$, $(m+(n+\frac{1}{2})\tau )\pi$ respectively.

8: 27.6 Divisor Sums

… ►

27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

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Annotations for §27.6 and Ch.27

… ►

27.6.2 $$\sum _{d|n}\mu \left(d\right)f(d)=\prod _{p|n}(1-f(p)),$$ $n>1$.

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.6.E2

Encodings:

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Annotations for §27.6 and Ch.27

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27.6.3 $$\sum _{d|n}|\mu \left(d\right)|=2^{\nu \left(n\right)},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\nu \left(n\right)$: number of distinct primes dividing a number, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E3

Encodings:

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Annotations for §27.6 and Ch.27

►

27.6.4 $$\sum _{d^2|n}\mu \left(d\right)=|\mu \left(n\right)|,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E4

Encodings:

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Annotations for §27.6 and Ch.27

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27.6.5 $$\sum _{d|n}\frac{|\mu \left(d\right)|}{\varphi \left(d\right)}=\frac{n}{\varphi \left(n\right)},$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E5

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Annotations for §27.6 and Ch.27

…

9: 27.7 Lambert Series as Generating Functions

… ►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: ►

27.7.2 $$\sum _{n=1}^{\mathrm{\infty }}f(n)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}\sum _{d|n}f(d)x^n.$$

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Symbols:

$d$: positive integer, $n$: positive integer and $x$: real number

Referenced by:

§27.7

Permalink:

http://dlmf.nist.gov/27.7.E2

Encodings:

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Annotations for §27.7 and Ch.27

►Again with , special cases of (27.7.2) include: ►

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

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Annotations for §27.7 and Ch.27

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27.7.4 $$\sum _{n=1}^{\mathrm{\infty }}\varphi \left(n\right)\frac{x^n}{1-x^n}=\frac{x}{{(1-x)}^2},$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $n$: positive integer and $x$: real number

A&S Ref:

24.3.2 I.B

Permalink:

http://dlmf.nist.gov/27.7.E4

Encodings:

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Annotations for §27.7 and Ch.27

…

10: 7.3 Graphics

… ►

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. …He observed that the intensity distribution is given by ${\left|ℱ\left(x\right)\right|}^2$. Magnify

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Figure 7.3.5: $|\mathrm{erf}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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Figure 7.3.6: $|\mathrm{erfc}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help