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11: 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

… ►

§20.7(iv) Reduction Formulas for Products

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§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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12: 20.5 Infinite Products and Related Results

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

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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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13: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
…
$={\mathrm{e}}^{\mathrm{i}\pi \tau }$	nome.
…

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14: 23.12 Asymptotic Approximations

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23.12.1 $$\mathrm{\wp }\left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(-\frac{1}{3}+{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)+8\left(1-\cos \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

►

23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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23.12.3 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Permalink:

http://dlmf.nist.gov/23.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

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23.12.4 $${\eta }_1=\frac{{\pi }^2}{4{\omega }_1}\left(\frac{1}{3}-8q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: 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:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

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15: 20.2 Definitions and Periodic Properties

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

ⓘ

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

►

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

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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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16: 20.11 Generalizations and Analogs

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20.11.6 $${\phi }_{m,1}(z,q)=\frac{{\theta }_1^{\prime }(0,q){\theta }_m(z,q)}{{\theta }_m(0,q){\theta }_1(z,q)},$$ $m=2,3,4$,

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(v), §20.11 and Ch.20

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20.11.7 $${\phi }_{1,n}(z,q)=\frac{{\theta }_n(0,q){\theta }_1(z,q)}{{\theta }_1^{\prime }(0,q){\theta }_n(z,q)},$$ $n=2,3,4$,

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\phi }_{n,m}(z,q)$: combined theta function, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(v), §20.11 and Ch.20

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20.11.8 $${\phi }_{m,n}(z,q)=\frac{{\theta }_n(0,q){\theta }_m(z,q)}{{\theta }_m(0,q){\theta }_n(z,q)},$$ $m,n=2,3,4$.

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(v), §20.11 and Ch.20

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20.11.9 $${\phi }_{m,n}(z,q)={\phi }_{m,1}(z,q){\phi }_{1,n}(z,q)=\frac{1}{{\phi }_{n,m}(z,q)}=\frac{{\phi }_{m,1}(z,q)}{{\phi }_{n,1}(z,q)}=\frac{{\phi }_{1,n}(z,q)}{{\phi }_{1,m}(z,q)}.$$

ⓘ

Symbols:

${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(v), §20.11 and Ch.20

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17: 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. …

18: 22.16 Related Functions

… ►With $q$ as in (22.2.1) and $\zeta =\pi x/(2K)$, ►

22.16.9 $$\mathrm{am}(x,k)=\frac{\pi }{2K}x+2\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\sin \left(2n\zeta \right)}{n(1+q^{2n})}.$$

ⓘ

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $\sin z$: sine function, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E9

Encodings:

pMML, png, TeX

See also:

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

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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

►where $\xi =x/{\theta }_3^2(0,q)$. …

19: 22.20 Methods of Computation

… ►If either $\tau$ or $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ is given, then we use $k={\theta }_2^2(0,q)/{\theta }_3^2(0,q)$, $k^{\prime }={\theta }_4^2(0,q)/{\theta }_3^2(0,q)$, $K=\frac{1}{2}\pi {\theta }_3^2(0,q)$, and $K^{\prime }=-\mathrm{i}\tau K$, obtaining the values of the theta functions as in §20.14. …

20: 20.9 Relations to Other Functions

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

ⓘ

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

►

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

►

20.9.5 $$\exp \left(-\frac{\pi R_F(0,k^2,1)}{R_F(0,k_{}^{\prime }{}_{}{}^2,1)}\right)=q.$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $q$: nome and $k$: modulus

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E5

Encodings:

pMML, png, TeX

See also:

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

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