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1: 17.14 Constant Term Identities

… ►

17.14.2

\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1}q^{2};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z% ^{-1}q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ % coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q% ;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1}q;q\right% )_{\infty}}=\frac{H(q)}{\left(-q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.3

\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z^{-1}% q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ coeff.% of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1};q\right)_{% \infty}}=\frac{G(q)}{\left(-q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.4

\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{2}% \right)_{n}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{% \infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{% \left(-z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(z^{-1};q% ^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}\mbox{ coeff. of % }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-2};q^{4}\right)_{% \infty}}=\frac{G(q^{4})}{\left(q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.5

\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{% 2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right% )_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty% }}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(% z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}% \mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-% z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(q^{4}z^{% -2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{\left(q;q^{2}\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

…

2: 4.31 Special Values and Limits

… ►

Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

.

► ►►►►►►►

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
…
$\cosh z$	$1$	$0$	$-1$	$0$	$\infty$
$\tanh z$	$0$	$\infty\mathrm{i}$	$0$	$-\infty\mathrm{i}$	$1$
$\operatorname{csch}z$	$\infty$	$-\mathrm{i}$	$\infty$	$\mathrm{i}$	$0$
$\operatorname{sech}z$	$1$	$\infty$	$-1$	$\infty$	$0$
$\coth z$	$\infty$	$0$	$\infty$	$0$	$1$

► …

3: 17.13 Integrals

… ►

17.13.1

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}\,{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},

ⓘ

Symbols:: $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.2

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.3

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $q$ : complex base
Referenced by:: Erratum (V1.0.1) for Equation (17.13.3)
Permalink:: http://dlmf.nist.gov/17.13.E3
Encodings:: pMML, png, TeX
Errata (effective with 1.0.1):: Originally the differential was identified incorrectly as a $q$ -differential; the correct differential is $\,\mathrm{d}t$ .
See also:: Annotations for §17.13, §17.13 and Ch.17

►

17.13.4

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13, §17.13 and Ch.17

…

4: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.2

2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{% \sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(% \sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.12, §22.12
Permalink:: http://dlmf.nist.gov/22.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

►

22.12.3

2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n% }\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}% \left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}% \right),

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

… ►

22.12.8

2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=% -\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

… ►

22.12.11

2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t-m-n\tau}\right),

ⓘ

Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

►

22.12.12

2K\operatorname{ds}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}% \pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-% \infty}^{\infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right),

ⓘ

Symbols:: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

…

5: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

,

m=1(1)100

,

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

,

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

,

\sum_{k=0}^{\infty}(2k+1)^{-n}

,

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

,

n=1,2,\dotsc

, 18D. …

6: 4.22 Infinite Products and Partial Fractions

… ►

4.22.1

\sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.89
Permalink:: http://dlmf.nist.gov/4.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.22 and Ch.4

►

4.22.2

\cos z=\prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.90
Permalink:: http://dlmf.nist.gov/4.22.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.22 and Ch.4

… ►

4.22.3

\cot z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.91
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/4.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.22 and Ch.4

►

4.22.4

{\csc}^{2}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi)^{2}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.92
Permalink:: http://dlmf.nist.gov/4.22.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.22 and Ch.4

►

4.22.5

\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.93
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/4.22.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.22 and Ch.4

7: 4.36 Infinite Products and Partial Fractions

… ►

4.36.1

\sinh z=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sinh\NVar{z}$ : hyperbolic sine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.5.68
Permalink:: http://dlmf.nist.gov/4.36.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.36 and Ch.4

►

4.36.2

\cosh z=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.5.69
Permalink:: http://dlmf.nist.gov/4.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.36 and Ch.4

… ►

4.36.3

\coth z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\coth\NVar{z}$ : hyperbolic cotangent function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.36 and Ch.4

►

4.36.4

{\operatorname{csch}}^{2}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\mathrm{i}$ : imaginary unit, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.36 and Ch.4

►

4.36.5

\operatorname{csch}z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^% {2}\pi^{2}}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.36 and Ch.4

8: 27.7 Lambert Series as Generating Functions

… ►

27.7.1

\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.

ⓘ

Symbols:: $n$ : positive integer and $x$ : real number
Referenced by:: §27.7
Permalink:: http://dlmf.nist.gov/27.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.7 and Ch.27

… ►

27.7.2

\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \mathbin{|}n}f(d)x^{n}.

ⓘ

Symbols:: $d$ : positive integer, $n$ : positive integer and $x$ : real number
Referenced by:: §27.7
Permalink:: http://dlmf.nist.gov/27.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.7 and Ch.27

… ►

27.7.3

\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,

ⓘ

Symbols:: $\mu\left(\NVar{n}\right)$ : 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

… ►

27.7.5

\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{% \alpha}\left(n\right)x^{n},

ⓘ

Symbols:: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.3 I.B
Permalink:: http://dlmf.nist.gov/27.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.7 and Ch.27

►

27.7.6

\sum_{n=1}^{\infty}\lambda\left(n\right)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{% \infty}x^{n^{2}}.

ⓘ

Symbols:: $\lambda\left(\NVar{n}\right)$ : Liouville’s function, $n$ : positive integer and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.7 and Ch.27

9: 32.6 Hamiltonian Structure

… ►

32.6.16

z\mathrm{H}_{\mbox{\scriptsize III}}(q,p,z)=q^{2}p^{2}-{\left(\kappa_{\infty}% zq^{2}+(2\theta_{0}+1)q-\kappa_{0}z\right)p}+\kappa_{\infty}(\theta_{0}+\theta% _{\infty})zq,

ⓘ

Symbols:: $z$ : real, $q$ : coordinate, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter, $p$ : momentum and $\kappa_{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.18

zp^{\prime}=-2qp^{2}+2\kappa_{\infty}zqp+(2\theta_{0}+1)p-\kappa_{\infty}(% \theta_{0}+\theta_{\infty})z.

ⓘ

Symbols:: $z$ : real, $q$ : coordinate, $\theta$ : parameter, $p$ : momentum and $\kappa_{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.19

(\alpha,\beta,\gamma,\delta)=\left(-2\kappa_{\infty}\theta_{\infty},2\kappa_{0% }(\theta_{0}+1),\kappa_{\infty}^{2},-\kappa_{0}^{2}\right).

ⓘ

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant, $\theta$ : parameter and $\kappa_{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.21

(z\sigma^{\prime\prime}-\sigma^{\prime})^{2}+2\left((\sigma^{\prime})^{2}-% \kappa_{0}^{2}\kappa_{\infty}^{2}z^{2}\right)(z\sigma^{\prime}-2\sigma)+8% \kappa_{0}\kappa_{\infty}\theta_{0}\theta_{\infty}z\sigma^{\prime}=4\kappa_{0}% ^{2}\kappa_{\infty}^{2}(\theta_{0}^{2}+\theta_{\infty}^{2})z^{2}.

ⓘ

Symbols:: $z$ : real, $\sigma$ : function, $\theta$ : parameter and $\kappa_{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §32.6(iv), §32.6 and Ch.32

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32.6.29

\zeta^{2}(\sigma^{\prime\prime})^{2}+\left(4(\sigma^{\prime})^{2}-\eta_{0}^{2}% \eta_{\infty}^{2}\right)(\zeta\sigma^{\prime}-\sigma)+\eta_{0}\eta_{\infty}% \theta_{0}\theta_{\infty}\sigma^{\prime}=\tfrac{1}{4}\eta_{0}^{2}\eta_{\infty}% ^{2}(\theta_{0}^{2}+\theta_{\infty}^{2}).

ⓘ

Symbols:: $\sigma$ : function and $\theta$ : parameter
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §32.6(iv), §32.6 and Ch.32

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10: 20.6 Power Series

… ►

20.6.2

\theta_{1}\left(\pi z\middle|\tau\right)=\pi z\theta_{1}'\left(0\middle|\tau% \right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\delta_{2j}(\tau)$ : coefficient
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §20.6 and Ch.20

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20.6.6

\delta_{2j}(\tau)=\left.\sum_{n=-\infty}^{\infty}\sum_{\begin{subarray}{c}m=-% \infty\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{\infty}\right.(m+n\tau)^{-% 2j},

ⓘ

Defines:: $\delta_{2j}(\tau)$ : coefficient (locally)
Symbols:: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter
Referenced by:: §20.6, §20.6
Permalink:: http://dlmf.nist.gov/20.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.6 and Ch.20

►

20.6.7

\alpha_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{% 1}{2}+n\tau)^{-2j},

ⓘ

Defines:: $\alpha_{2j}(\tau)$ : coefficient (locally)
Symbols:: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.6 and Ch.20

►

20.6.8

\beta_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1% }{2}+(n-\tfrac{1}{2})\tau)^{-2j},

ⓘ

Defines:: $\beta_{2j}(\tau)$ : coefficient (locally)
Symbols:: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §20.6 and Ch.20

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20.6.9

\gamma_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m+(n-% \tfrac{1}{2})\tau)^{-2j},

ⓘ

Defines:: $\gamma_{2j}(\tau)$ : coefficient (locally)
Symbols:: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.6 and Ch.20

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