Jacobi theta functions

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11: 20.9 Relations to Other Functions

… ►

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

ⓘ

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

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

12: 27.13 Functions

… ►Jacobi (1829) notes that $r_2\left(n\right)$ is the coefficient of $x^n$ in the square of the theta function $\vartheta \left(x\right)$: ►

27.13.4 $$\vartheta \left(x\right)=1+2\sum _{m=1}^{\mathrm{\infty }}{x}^{m^2},$$ .

ⓘ

Defines:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation

Symbols:

${\theta }_j(z,q)$: theta function, $m$: positive integer and $x$: real number

A&S Ref:

16.27.3 (with $z=0$, $q=x$)

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►(In §20.2(i), $\vartheta \left(x\right)$ is denoted by ${\theta }_3(0,x)$.) … ►

27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

ⓘ

Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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27.13.6 $${(\vartheta \left(x\right))}^2=1+4\sum _{n=1}^{\mathrm{\infty }}\left({\delta }_1(n)-{\delta }_3(n)\right)x^n,$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $n$: positive integer, $x$: real number and ${\delta }_j(n)$: number of divisors

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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13: 20.3 Graphics

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§20.3(i) $\theta$-Functions: Real Variable and Real Nome

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§20.3(ii) $\theta$-Functions: Complex Variable and Real Nome

►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … ►

§20.3(iii) $\theta$-Functions: Real Variable and Complex Lattice Parameter

►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

14: 22.16 Related Functions

… ►

Relation to Theta Functions

►

22.16.30 $$ℰ(x,k)=\frac{1}{{\theta }_3^2(0,q){\theta }_4(\xi ,q)}\frac{d}{d\xi }{\theta }_4(\xi ,q)+\frac{E\left(k\right)}{K\left(k\right)}x,$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, ${\theta }_j(z,q)$: theta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $q$: nome, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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15: 20.5 Infinite Products and Related Results

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

ⓘ

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

►

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

►

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

ⓘ

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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Jacobi’s Triple Product

…

16: 20.12 Mathematical Applications

§20.12 Mathematical Applications

►

§20.12(i) Number Theory

… ►

§20.12(ii) Uniformization and Embedding of Complex Tori

… ►Thus theta functions “uniformize” the complex torus. …

17: 20.7 Identities

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20.7.1 $${\theta }_3^2(0,q){\theta }_3^2(z,q)={\theta }_4^2(0,q){\theta }_4^2(z,q)+{\theta }_2^2(0,q){\theta }_2^2(z,q),$$

ⓘ

Symbols:

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

A&S Ref:

16.28.3

Permalink:

http://dlmf.nist.gov/20.7.E1

Encodings:

pMML, png, TeX

See also:

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

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20.7.2 $${\theta }_3^2(0,q){\theta }_4^2(z,q)={\theta }_2^2(0,q){\theta }_1^2(z,q)+{\theta }_4^2(0,q){\theta }_3^2(z,q),$$

ⓘ

Symbols:

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

A&S Ref:

16.28.2

Permalink:

http://dlmf.nist.gov/20.7.E2

Encodings:

pMML, png, TeX

See also:

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

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20.7.5 $${\theta }_3^4(0,q)={\theta }_2^4(0,q)+{\theta }_4^4(0,q).$$

ⓘ

Symbols:

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

A&S Ref:

16.28.5

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/20.7.E5

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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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20.7.29 $${\theta }_4\left(z\right|\tau +1)={\theta }_3\left(z\right|\tau ).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E29

Encodings:

pMML, png, TeX

See also:

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

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18: 22.19 Physical Applications

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right)),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $\theta (t)$: angular displacement and $\alpha$: initial displacement

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.2)

Permalink:

http://dlmf.nist.gov/22.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E}),$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\theta (t)$: angular displacement and $E$: energy

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.3)

Permalink:

http://dlmf.nist.gov/22.19.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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19: 20.6 Power Series

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

ⓘ

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 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

ⓘ

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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20.6.4 $${\theta }_3\left(\pi z\right|\tau )={\theta }_3\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\beta }_{2j}(\tau )z^{2j}\right),$$

ⓘ

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 ${\beta }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.5 $${\theta }_4\left(\pi z\right|\tau )={\theta }_4\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\gamma }_{2j}(\tau )z^{2j}\right).$$

ⓘ

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 ${\gamma }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

…

20: 21.2 Definitions

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21.2.8 $$\theta \left(z\right|\mathrm{\Omega })={\theta }_3\left(\pi z\right|\mathrm{\Omega }),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left(𝐳\right|𝛀)$: Riemann theta function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

►

21.2.9 $${\theta }_1\left(\pi z\right|\mathrm{\Omega })=-\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}\right]\left(z\right|\mathrm{\Omega }),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.10 $${\theta }_2\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{0}\right]\left(z\right|\mathrm{\Omega }),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.11 $${\theta }_3\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{0}{0}\right]\left(z\right|\mathrm{\Omega }),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.12 $${\theta }_4\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{0}{\frac{1}{2}}\right]\left(z\right|\mathrm{\Omega }).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21