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1: 20.2 Definitions and Periodic Properties

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20.2.1 $${\theta }_1\left(z\right|\tau )={\theta }_1(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n{q}^{{(n+\frac{1}{2})}^2}\sin \left((2n+1)z\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\sin z$: sine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.1

Referenced by:

§20.14, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

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20.2.2 $${\theta }_2\left(z\right|\tau )={\theta }_2(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{q}^{{(n+\frac{1}{2})}^2}\cos \left((2n+1)z\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.2

Permalink:

http://dlmf.nist.gov/20.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

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20.2.3 $${\theta }_3\left(z\right|\tau )={\theta }_3(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\cos \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.3

Referenced by:

§20.10(i), §20.13, §20.14, §27.13(iv)

Permalink:

http://dlmf.nist.gov/20.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

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20.2.5 $$z_{m,n}=(m+n\tau )\pi ,$$ $m,n\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathbb{Z}$: set of all integers, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

A&S Ref:

16.33.1 (in different notation)

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(ii), §20.2 and Ch.20

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20.2.10 $$M\equiv M(z|\tau )={\mathrm{e}}^{\mathrm{i}z+(\mathrm{i}\pi \tau /4)},$$

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Defines:

$M$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

A&S Ref:

16.33.1 (in different notation)

Permalink:

http://dlmf.nist.gov/20.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(iii), §20.2 and Ch.20

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2: 20.6 Power Series

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20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: 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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20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.7 $${\alpha }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+n\tau )}^{-2j},$$

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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

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20.6.8 $${\beta }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )}^{-2j},$$

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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 $${\gamma }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m+(n-\frac{1}{2})\tau )}^{-2j},$$

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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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3: 20.7 Identities

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20.7.15 $$A\equiv A(\tau )=1/{\theta }_4\left(0\right|2\tau ),$$

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Defines:

$A$ (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\equiv$: equals by definition and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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20.7.16 $${\theta }_1\left(2z\right|2\tau )=A{\theta }_1\left(z\right|\tau ){\theta }_2\left(z\right|\tau ),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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20.7.19 $${\theta }_4\left(2z\right|2\tau )=A{\theta }_3\left(z\right|\tau ){\theta }_4\left(z\right|\tau ).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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§20.7(viii) Transformations of Lattice Parameter

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20.7.28 $${\theta }_3\left(z\right|\tau +1)={\theta }_4\left(z\right|\tau ),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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4: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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5: 20.1 Special Notation

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$m$, $n$	integers.
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$\tau$ $(\in ℂ)$	the lattice parameter, $\mathrm{\Im }\tau >0$.
$q$ $(\in ℂ)$	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.
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6: 20.5 Infinite Products and Related Results

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20.5.5 $${\theta }_1\left(z\right|\tau )={\theta }_1^{\prime }\left(0\right|\tau )\sin z\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\pi \tau +z\right)\sin \left(n\pi \tau -z\right)}{{\sin }^2\left(n\pi \tau \right)},$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.5(i), §20.5(iii), §20.7(iv)

Permalink:

http://dlmf.nist.gov/20.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.6 $${\theta }_2\left(z\right|\tau )={\theta }_2\left(0\right|\tau )\cos z\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\pi \tau +z\right)\cos \left(n\pi \tau -z\right)}{{\cos }^2\left(n\pi \tau \right)},$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.7 $${\theta }_3\left(z\right|\tau )={\theta }_3\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left((n-\frac{1}{2})\pi \tau +z\right)\cos \left((n-\frac{1}{2})\pi \tau -z\right)}{{\cos }^2\left((n-\frac{1}{2})\pi \tau \right)},$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.8 $${\theta }_4\left(z\right|\tau )={\theta }_4\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left((n-\frac{1}{2})\pi \tau +z\right)\sin \left((n-\frac{1}{2})\pi \tau -z\right)}{{\sin }^2\left((n-\frac{1}{2})\pi \tau \right)}.$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.5(iii), §20.7(iv)

Permalink:

http://dlmf.nist.gov/20.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.9 $${\theta }_3\left(\pi z\right|\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{p}^{2n}{q}^{n^2}=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+q^{2n-1}{p}^2\right)\left(1+q^{2n-1}{p}^{-2}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

Referenced by:

§17.8, §20.12(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5(i), §20.5 and Ch.20

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7: 20.9 Relations to Other Functions

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20.9.1 $$k={\theta }_2^2\left(0\right|\tau )/{\theta }_3^2\left(0\right|\tau )$$

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Defines:

$k$: modulus (locally)

Symbols:

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

Referenced by:

§20.11(iii), §20.9(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/20.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

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8: 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

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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

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9: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$\tau ={\omega }_3/{\omega }_1$	lattice parameter ($\mathrm{\Im }\tau >0$).
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10: 20.13 Physical Applications

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20.13.1 $$\partial \theta (z|\tau )/\partial \tau =\kappa {\partial }^2\theta (z|\tau )/{\partial z}^2,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\tau$: lattice parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

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