Jacobi theta functions

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21: 10.35 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, …

22: 18.16 Zeros

… ►Let ${\theta }_{n,m}={\theta }_{n,m}^{(\alpha ,\beta )}$, $m=1,2,\mathrm{\dots },n$, denote the zeros of $P_n^{(\alpha ,\beta )}\left(\cos \theta \right)$ as function of $\theta$ with …

23: 23.6 Relations to Other Functions

… ►

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

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

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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

… ►

23.6.10 $$\sigma \left({\omega }_1\right)=2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1{\omega }_1\right){\theta }_2(0,q)}{\pi {\theta }_1^{\prime }(0,q)},$$

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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, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E10

Encodings:

pMML, png, TeX

See also:

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

…

24: 22.20 Methods of Computation

… ►If either $k$ or $x$ is complex then (22.2.6) gives the definition of $\mathrm{dn}(x,k)$ as a quotient of theta functions. … ►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. … ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

25: 18.15 Asymptotic Approximations

… ►

18.15.1 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-1}{2}^{2n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1)\left(\sum _{m=0}^{M-1}\frac{f_m(\theta )}{2^m{\left(2n+\alpha +\beta +2\right)}_m}+O\left(n^{-M}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $f_m(\theta )$

Referenced by:

§18.15(i), §18.15(i)

Permalink:

http://dlmf.nist.gov/18.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►

18.15.4_5 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-\frac{1}{2}}{n}^{-\frac{1}{2}}\cos \left(\frac{1}{2}(2n+\alpha +\beta +1)\theta -\frac{1}{4}(2\alpha +1)\pi \right)+O\left(n^{-\frac{3}{2}}\right),$$ $\alpha ,\beta \in ℝ$,

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\in$: element of, $ℝ$: real line, $\sin z$: sine function and $n$: nonnegative integer

Proveds:

Ismail (2009, Theorem 4.3.1); Szegő (1975, Theorem 8.21.8); with attribution to Darboux(proved)

Referenced by:

§18.15(i), Erratum (V1.2.0) Section 18.15

Permalink:

http://dlmf.nist.gov/18.15.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►

18.15.6 $${(\sin \frac{1}{2}\theta )}^{\alpha +\frac{1}{2}}{(\cos \frac{1}{2}\theta )}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)=\frac{\mathrm{\Gamma }\left(n+\alpha +1\right)}{2^{\frac{1}{2}}{\rho }^{\alpha }n!}\left({\theta }^{\frac{1}{2}}{J}_{\alpha }\left(\rho \theta \right)\sum _{m=0}^M\frac{A_m(\theta )}{{\rho }^{2m}}+{\theta }^{\frac{3}{2}}{J}_{\alpha +1}\left(\rho \theta \right)\sum _{m=0}^{M-1}\frac{B_m(\theta )}{{\rho }^{2m+1}}+{\epsilon }_M(\rho ,\theta )\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ , ${\epsilon }_M(\rho ,\theta )$ , $A_m(\theta )$: coefficient and $B_m(\theta )$: coefficient

Permalink:

http://dlmf.nist.gov/18.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

…

26: 25.5 Integral Representations

… ►

25.5.14 $$\omega (x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2\pi x}=\frac{1}{2}\left({\theta }_3\left(0\right|\mathrm{i}x)-1\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 22)

Permalink:

http://dlmf.nist.gov/25.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

…

27: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

►

$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

…

28: 18.10 Integral Representations

… ►

18.10.1 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,$$ , $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.11

Referenced by:

§18.10(i), §18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

… ►

18.10.3 $$\frac{P_n^{(\alpha ,\beta )}\left(\cos \theta \right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\frac{2\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha -\beta \right)\mathrm{\Gamma }\left(\beta +\frac{1}{2}\right)}{\int }_0^1{\int }_0^{\pi }{\left({(\cos \frac{1}{2}\theta )}^2-r^2{(\sin \frac{1}{2}\theta )}^2+ir\sin \theta \cos \varphi \right)}^n{(1-r^2)}^{\alpha -\beta -1}{r}^{2\beta +1}{(\sin \varphi )}^{2\beta }d\varphi dr,$$ $\alpha >\beta >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►

18.10.4 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.10

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

…

29: 22.21 Tables

§22.21 Tables

… ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

30: 20.14 Methods of Computation

§20.14 Methods of Computation

►The Fourier series of §20.2(i) usually converge rapidly because of the factors $q^{{(n+\frac{1}{2})}^2}$ or $q^{n^2}$, and provide a convenient way of calculating values of ${\theta }_j\left(z\right|\tau )$. …For instance, the first three terms of (20.2.1) give the value of ${\theta }_1\left(2-\mathrm{i}\right|\mathrm{i})$ ($={\theta }_1(2-\mathrm{i},{\mathrm{e}}^{-\pi })$) to 12 decimal places. … ►For instance, to find ${\theta }_3(z,0.9)$ we use (20.7.32) with $q=0.9={\mathrm{e}}^{\mathrm{i}\pi \tau }$, $\tau =-\mathrm{i}\ln \left(0.9\right)/\pi$. …Hence the first term of the series (20.2.3) for ${\theta }_3\left(z{\tau }^{\prime }\right|{\tau }^{\prime })$ suffices for most purposes. …