theta functions

SearchAdvancedHelp

(0.008 seconds)

41—50 of 176 matching pages

41: 23.6 Relations to Other Functions

… ►

§23.6(i) Theta 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

…

42: 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

…

43: 25.10 Zeros

… ►

25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

►where ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

… ►

25.10.3 $$Z(t)=2\sum _{n=1}^m\frac{\cos \left(\vartheta (t)-t\ln n\right)}{n^{1/2}}+R(t),$$ $m=\lfloor \sqrt{t/(2\pi )}\rfloor$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $n$: nonnegative integer, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

Riemann–Siegel formula

Source:

Titchmarsh (1986b, (4.17.4), p. 89)

Referenced by:

§25.18(i), Erratum (V1.1.4) for Equation (25.10.3)

Permalink:

http://dlmf.nist.gov/25.10.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint $m=\lfloor \sqrt{t/(2\pi )}\rfloor$ was added.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

…

44: 10.18 Modulus and Phase Functions

… ►

10.18.1 $$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}=H_{\nu }^{(1)}\left(x\right),$$

ⓘ

Defines:

$M_{\nu }\left(x\right)$: modulus of Bessel functions

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(i), §10.18 and Ch.10

… ►where $M_{\nu }\left(x\right)$ $(>0)$, $N_{\nu }\left(x\right)$ $(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ fixed by … ►

10.18.9 $$N_{\nu }^2\left(x\right)=M_{\nu }^{\prime }{}_{}{}^2\left(x\right)+M_{\nu }^2\left(x\right)\theta _{\nu }^{\prime }{}_{}{}^2\left(x\right)=M_{\nu }^{\prime }{}_{}{}^2\left(x\right)+\frac{4}{{(\pi x{M}_{\nu }\left(x\right))}^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.22

Permalink:

http://dlmf.nist.gov/10.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(ii), §10.18 and Ch.10

… ►

10.18.11 $$\tan \left({\varphi }_{\nu }\left(x\right)-{\theta }_{\nu }\left(x\right)\right)=\frac{M_{\nu }\left(x\right){\theta }_{\nu }^{\prime }\left(x\right)}{M_{\nu }^{\prime }\left(x\right)}=\frac{2}{\pi x{M}_{\nu }\left(x\right)M_{\nu }^{\prime }\left(x\right)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\tan z$: tangent function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.24

Permalink:

http://dlmf.nist.gov/10.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(ii), §10.18 and Ch.10

►

10.18.12 $$M_{\nu }\left(x\right)N_{\nu }\left(x\right)\sin \left({\varphi }_{\nu }\left(x\right)-{\theta }_{\nu }\left(x\right)\right)=\frac{2}{\pi x}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.24

Permalink:

http://dlmf.nist.gov/10.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(ii), §10.18 and Ch.10

…

45: 32.6 Hamiltonian Structure

… ►

32.6.21 $${(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2\left({({\sigma }^{\prime })}^2-{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2\right)(z{\sigma }^{\prime }-2\sigma )+8{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_0{\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }=4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2({\theta }_0^2+{\theta }_{\mathrm{\infty }}^2)z^2.$$

ⓘ

Symbols:

$z$: real, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.22 $$q=\frac{{\kappa }_0\left(z{\sigma }^{\prime \prime }-(2{\theta }_0+1){\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}z\right)}{{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2-{({\sigma }^{\prime })}^2},$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Permalink:

http://dlmf.nist.gov/32.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.30 $$q=\frac{{\eta }_0\left(\zeta {\sigma }^{\prime \prime }-2{\theta }_0{\sigma }^{\prime }+{\eta }_0{\eta }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}\right)}{{\eta }_0^2{\eta }_{\mathrm{\infty }}^2-4{({\sigma }^{\prime })}^2},$$

ⓘ

Symbols:

$q$: coordinate, $\sigma$: function and $\theta$: parameter

Permalink:

http://dlmf.nist.gov/32.6.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.37 $${(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2{({\sigma }^{\prime })}^2(z{\sigma }^{\prime }-2\sigma )-4{\kappa }_0{\kappa }_{\mathrm{\infty }}(\theta +1){\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }=4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2.$$

ⓘ

Symbols:

$z$: real, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.38 $$q={\kappa }_0\left(z{\sigma }^{\prime \prime }-\theta {\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}z\right)/{({\sigma }^{\prime })}^2,$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Permalink:

http://dlmf.nist.gov/32.6.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

…

46: 17.3 $q$-Elementary and $q$-Special Functions

… ►

§17.3(iv) Theta Functions

…

47: Bibliography I

… ►

Theta Functions. Springer-Verlag, New York.

ⓘ

Notes:

Die Grundlehren der mathematischen Wissenschaften, Band 194

External Links:

ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt

Referenced by:

§21.5(ii), §21.8, Ch.21.

Encodings:

BibTeX

…

48: 23.17 Elementary Properties

§23.17 Elementary Properties

►

§23.17(i) Special Values

… ►

§23.17(ii) Power and Laurent Series

… ►

23.17.6 $$\eta \left(\tau \right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{{(6n+1)}^2/12}.$$

ⓘ

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

… ►

§23.17(iii) Infinite Products

…

49: Bibliography R

… ►

Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

ⓘ

External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

BibTeX

…

50: 19.25 Relations to Other Functions

… ►

§19.25(iv) Theta Functions

►For relations of symmetric integrals to theta functions, see §20.9(i). …