Jacobi nome

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1: 19.5 Maclaurin and Related Expansions

… ► ►For Jacobi’s nome $q$: ►

19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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2: 22.2 Definitions

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$k={\displaystyle \frac{{\theta }_2^2(0,q)}{{\theta }_3^2(0,q)}},$

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$k^{\prime }={\displaystyle \frac{{\theta }_4^2(0,q)}{{\theta }_3^2(0,q)}},$

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22.2.4 $$\mathrm{sn}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_2(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_4(\zeta ,q)}=\frac{1}{\mathrm{ns}(z,k)},$$

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Defines:

$\mathrm{ns}(z,k)$: Jacobian elliptic function and $\mathrm{sn}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

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22.2.7 $$\mathrm{sd}(z,k)=\frac{{\theta }_3^2(0,q)}{{\theta }_2(0,q){\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_3(\zeta ,q)}=\frac{1}{\mathrm{ds}(z,k)},$$

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Defines:

$\mathrm{ds}(z,k)$: Jacobian elliptic function and $\mathrm{sd}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

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22.2.11 $$\mathrm{p}\mathrm{q}(z,k)={\theta }_p\left(z\right|\tau )/{\theta }_q\left(z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $q$: nome, $z$: complex, $k$: modulus and $\tau$: ratio

A&S Ref:

16.36.3

Referenced by:

§22.2

Permalink:

http://dlmf.nist.gov/22.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

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3: 22.16 Related Functions

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

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

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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 $\xi =x/{\theta }_3^2(0,q)$. …

4: 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. …

5: 20.4 Values at $z$ = 0

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20.4.1 $${\theta }_1(0,q)={\theta }_2^{\prime }(0,q)={\theta }_3^{\prime }(0,q)={\theta }_4^{\prime }(0,q)=0,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.3 $${\theta }_2(0,q)=2q^{1/4}\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n}\right)}^2,$$

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Symbols:

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

Referenced by:

(20.10.3)

Permalink:

http://dlmf.nist.gov/20.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.4 $${\theta }_3(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n-1}\right)}^2,$$

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Symbols:

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

Referenced by:

(20.10.2)

Permalink:

http://dlmf.nist.gov/20.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.5 $${\theta }_4(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1-q^{2n-1}\right)}^2.$$

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Symbols:

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

Referenced by:

(20.10.1)

Permalink:

http://dlmf.nist.gov/20.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.6 $${\theta }_1^{\prime }(0,q)={\theta }_2(0,q){\theta }_3(0,q){\theta }_4(0,q).$$

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Symbols:

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

A&S Ref:

16.28.6

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

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6: 22.11 Fourier and Hyperbolic Series

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22.11.1 $$\mathrm{sn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic 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, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.1

Permalink:

http://dlmf.nist.gov/22.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.2 $$\mathrm{cn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic 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, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.2

Permalink:

http://dlmf.nist.gov/22.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic 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, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.4 $$\mathrm{cd}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic 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, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.4

Permalink:

http://dlmf.nist.gov/22.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic 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, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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7: 20.7 Identities

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

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

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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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20.7.3 $${\theta }_2^2(0,q){\theta }_4^2(z,q)={\theta }_3^2(0,q){\theta }_1^2(z,q)+{\theta }_4^2(0,q){\theta }_2^2(z,q),$$

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Symbols:

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

A&S Ref:

16.28.1

Permalink:

http://dlmf.nist.gov/20.7.E3

Encodings:

pMML, png, TeX

See also:

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

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20.7.4 $${\theta }_2^2(0,q){\theta }_3^2(z,q)={\theta }_4^2(0,q){\theta }_1^2(z,q)+{\theta }_3^2(0,q){\theta }_2^2(z,q).$$

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Symbols:

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

A&S Ref:

16.28.4

Permalink:

http://dlmf.nist.gov/20.7.E4

Encodings:

pMML, png, TeX

See also:

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

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

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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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8: 20.3 Graphics

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

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Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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►►

Figure 20.3.3: ${\theta }_2(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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

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►

Figure 20.3.18: ${\theta }_1\left(0.1\right|u+\mathrm{i}v)$, $-1\le u\le 1$, $0.005\le v\le 0.5$. …1 of $z$ is chosen arbitrarily since ${\theta }_1$ vanishes identically when $z=0$. Magnify 3D Help

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9: 20.8 Watson’s Expansions

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20.8.1 $$\frac{{\theta }_2(0,q){\theta }_3(z,q){\theta }_4(z,q)}{{\theta }_2(z,q)}=2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n^2}{\mathrm{e}}^{\mathrm{i}2nz}}{q^{-n}{\mathrm{e}}^{-\mathrm{i}z}+q^n{\mathrm{e}}^{\mathrm{i}z}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.8 and Ch.20

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10: 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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