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11: 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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20.2.4 $${\theta }_4\left(z\right|\tau )={\theta }_4(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n{q}^{n^2}\cos \left(2nz\right).$$

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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:

16.27.4

Referenced by:

§20.13, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E4

Encodings:

pMML, png, TeX

See also:

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

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20.2.6 $${\theta }_1\left(z+(m+n\tau )\pi \right|\tau )={(-1)}^{m+n}{q}^{-n^2}{\mathrm{e}}^{-2\mathrm{i}nz}{\theta }_1\left(z\right|\tau ),$$

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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, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.33.2 (in different notation)

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/20.2.E6

Encodings:

pMML, png, TeX

See also:

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

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12: 20.5 Infinite Products and Related Results

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20.5.1 $${\theta }_1(z,q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i), §23.17(iii)

Permalink:

http://dlmf.nist.gov/20.5.E1

Encodings:

pMML, png, TeX

See also:

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

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20.5.2 $${\theta }_2(z,q)=2q^{1/4}\cos z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.5.E2

Encodings:

pMML, png, TeX

See also:

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

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20.5.3 $${\theta }_3(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§23.15(ii), §23.17(iii), §23.19

Permalink:

http://dlmf.nist.gov/20.5.E3

Encodings:

pMML, png, TeX

See also:

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

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20.5.4 $${\theta }_4(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right).$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E4

Encodings:

pMML, png, TeX

See also:

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

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20.5.12 $$\frac{{\theta }_3^{\prime }(z,q)}{{\theta }_3(z,q)}=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^n}{1-q^{2n}}\sin \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.3

Permalink:

http://dlmf.nist.gov/20.5.E12

Encodings:

pMML, png, TeX

See also:

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

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13: 23.15 Definitions

… ►In §§23.15–23.19, $k$ and $k^{\prime }$ $(\in ℂ)$ denote the Jacobi modulus and complementary modulus, respectively, and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ ($\mathrm{\Im }\tau >0$) denotes the nome; compare §§20.1 and 22.1. … ►

23.15.6 $$\lambda \left(\tau \right)=\frac{{\theta }_2^4(0,q)}{{\theta }_3^4(0,q)};$$

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Defines:

$\lambda \left(\tau \right)$: elliptic modular function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Referenced by:

§20.9(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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23.15.7 $$J\left(\tau \right)=\frac{{\left({\theta }_2^8(0,q)+{\theta }_3^8(0,q)+{\theta }_4^8(0,q)\right)}^3}{54{\left({\theta }_1^{\prime }(0,q)\right)}^8},$$

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Defines:

$J\left(\tau \right)$: Klein’s complete invariant

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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23.15.8 $${\theta }_1^{\prime }(0,q)={\partial {\theta }_1(z,q)/\partial z|}_{z=0}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q$: nome and $z$: complex

Permalink:

http://dlmf.nist.gov/23.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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23.15.9 $$\eta \left(\tau \right)={\left(\frac{1}{2}{\theta }_1^{\prime }(0,q)\right)}^{1/3}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}{\theta }_3\left(\frac{1}{2}\pi (1+\tau )\right|3\tau ).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), ${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nome and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.15(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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14: 20.15 Tables

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20.15.1 $$\sin \alpha ={\theta }_2^2(0,q)/{\theta }_3^2(0,q)=k.$$

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Defines:

$\alpha$: modular angle (locally)

Symbols:

${\theta }_j(z,q)$: theta function, $\sin z$: sine function and $q$: nome

Referenced by:

§20.15, §20.15

Permalink:

http://dlmf.nist.gov/20.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.15 and Ch.20

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15: 22.21 Tables

… ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. …

16: 20.9 Relations to Other Functions

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20.9.3 $$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§19.25(v), §20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E3

Encodings:

pMML, png, TeX

See also:

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

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20.9.4 $$R_F(0,{\theta }_3^4(0,q),{\theta }_4^4(0,q))=\frac{1}{2}\pi ,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter and $q$: nome

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E4

Encodings:

pMML, png, TeX

See also:

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

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17: 22.1 Special Notation

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$x,y$	real variables.
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$q$	nome. except in §22.17; see also §20.1.
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… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. … ►The notation $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

18: 23.6 Relations to Other Functions

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23.6.2 $$e_1=\frac{{\pi }^2}{12{\omega }_1^2}\left({\theta }_2^4(0,q)+2{\theta }_4^4(0,q)\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $e_j$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $𝕃$: lattice, $q$: nome and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

A&S Ref:

18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11

Referenced by:

§23.22(ii), §23.22(i), §23.6(i), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.3 $$e_2=\frac{{\pi }^2}{12{\omega }_1^2}\left({\theta }_2^4(0,q)-{\theta }_4^4(0,q)\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $e_j$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $𝕃$: lattice, $q$: nome and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

A&S Ref:

18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11

Permalink:

http://dlmf.nist.gov/23.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.4 $$e_3=-\frac{{\pi }^2}{12{\omega }_1^2}\left(2{\theta }_2^4(0,q)+{\theta }_4^4(0,q)\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $e_j$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $𝕃$: lattice, $q$: nome and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

A&S Ref:

18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11

Referenced by:

§23.22(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.8 $${\eta }_1=-\frac{{\pi }^2}{12{\omega }_1}\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.15 $$\frac{\sigma \left(u+{\omega }_j\right)}{\sigma \left({\omega }_j\right)}=\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)\frac{{\theta }_{j+1}(z,q)}{{\theta }_{j+1}(0,q)},$$ $j=1,2,3$.

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Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.6(i), Erratum (V1.1.1) for Equation (23.6.15)

Permalink:

http://dlmf.nist.gov/23.6.E15

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The factor $\exp \left({\eta }_{j}u+\frac{{\eta }_j{u}^2}{2{\omega }_1}\right)$ has been corrected to be $\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)$.

Suggested 2021-02-10 by Jan Felipe van Diejen

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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19: 20.1 Special Notation

… ►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 $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. When $\tau$ is fixed the notation is often abbreviated in the literature as ${\theta }_j(z)$, or even as simply ${\theta }_j$, it being then understood that the argument is the primary variable. … ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. … ►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 $u=z/{\theta }_3^2\left(0\right|\tau )$. …