nome

SearchAdvancedHelp

(0.001 seconds)

21—25 of 25 matching pages

21: 20.15 Tables

… ►

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

ⓘ

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

…

22: 23.6 Relations to Other Functions

… ►

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

ⓘ

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

►

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

ⓘ

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

►

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

… ►

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

ⓘ

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

►

23.6.9 $$\sigma \left(z\right)=2{\omega }_1\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\frac{{\theta }_1(\pi z/(2{\omega }_1),q)}{\pi {\theta }_1^{\prime }(0,q)},$$

ⓘ

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

A&S Ref:

18.10.8

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E9

Encodings:

pMML, png, TeX

See also:

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

…

23: 23.8 Trigonometric Series and Products

… ►

23.8.1 $$\mathrm{\wp }\left(z\right)+\frac{{\eta }_1}{{\omega }_1}-\frac{{\pi }^2}{4{\omega }_1^2}{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)=-\frac{2{\pi }^2}{{\omega }_1^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^{2n}}{1-q^{2n}}\cos \left(\frac{n\pi z}{{\omega }_1}\right),$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): 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, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(i), §23.8 and Ch.23

►

23.8.2 $$\zeta \left(z\right)-\frac{{\eta }_{1}z}{{\omega }_1}-\frac{\pi }{2{\omega }_1}\cot \left(\frac{\pi z}{2{\omega }_1}\right)=\frac{2\pi }{{\omega }_1}\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(\frac{n\pi z}{{\omega }_1}\right).$$

ⓘ

Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): 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, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(i), §23.8 and Ch.23

… ►

23.8.6 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\prod _{n=1}^{\mathrm{\infty }}\frac{1-2q^{2n}\cos \left(\pi z/{\omega }_1\right)+q^{4n}}{{(1-q^{2n})}^2},$$

ⓘ

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(iii), §23.8 and Ch.23

…

24: 23.10 Addition Theorems and Other Identities

… ►

23.10.15 $$A_n={\left(\frac{{\pi }^2{G}^2}{{\omega }_1}\right)}^{n^2-1}\frac{q^{n(n-1)/2}}{{\mathrm{i}}^{n-1}}\exp \left(-\frac{(n-1){\eta }_1}{3{\omega }_1}\left((2n-1)({\omega }_1^2+{\omega }_3^2)+3(n-1){\omega }_1{\omega }_3\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $q$: nome, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $A_n$: coefficients, $G$ and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

…

25: 23.22 Methods of Computation

… ►Given ${\omega }_1$ and ${\omega }_3$, with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, the nome $q$ is computed from $q={\mathrm{e}}^{\mathrm{i}\pi {\omega }_3/{\omega }_1}$. …