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31: 22.11 Fourier and Hyperbolic Series

§22.11 Fourier and Hyperbolic Series

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22.11.7 $$\mathrm{ns}(z,k)-\frac{\pi }{2K}\csc \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{ns}(z,k)$: 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, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.10

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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

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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, $E\left(k\right)$: 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

… ►A related hyperbolic series is …

32: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

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§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). ►As a function of $\tau$, $k^2$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …

33: 23.23 Tables

§23.23 Tables

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34: 23.4 Graphics

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§23.4(i) Real Variables

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Figure 23.4.6: $\sigma (x;0,g_3)$ for $-5\le x\le 5$, $g_3$ = 0. … Magnify

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Figure 23.4.7: $\mathrm{\wp }\left(x\right)$ with ${\omega }_1=K\left(k\right)$, ${\omega }_3=\mathrm{i}{K}^{\prime }\left(k\right)$ for $0\le x\le 9$, $k^2$ = 0. … Magnify

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§23.4(ii) Complex Variables

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Figure 23.4.12: $\mathrm{\wp }(3.7;a+\mathrm{i}b,0)$ for $-5\le a\le 3$, $-4\le b\le 4$. … Magnify 3D Help

35: 23.5 Special Lattices

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§23.5(ii) Rectangular Lattice

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§23.5(iii) Lemniscatic Lattice

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§23.5(iv) Rhombic Lattice

… ►For the case ${\omega }_3={\mathrm{e}}^{\pi \mathrm{i}/3}{\omega }_1$ see §23.5(v). ►

§23.5(v) Equianharmonic Lattice

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36: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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37: William P. Reinhardt

… ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. … ►

22 Jacobian Elliptic Functions

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23 Weierstrass Elliptic and Modular Functions

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38: 29.18 Mathematical Applications

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29.18.6 $$\frac{d^2{u}_2}{{d\beta }^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(\beta ,k))u_2=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\beta$: sphero-conal coordinate and $u_j$: solutions

Referenced by:

§29.18(i)

Permalink:

http://dlmf.nist.gov/29.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(i), §29.18 and Ch.29

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29.18.7 $$\frac{d^2{u}_3}{{d\gamma }^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(\gamma ,k))u_3=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\gamma$: sphero-conal coordinate and $u_j$: solutions

Referenced by:

§29.18(i)

Permalink:

http://dlmf.nist.gov/29.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(i), §29.18 and Ch.29

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39: 23.20 Mathematical Applications

§23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

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§23.20(iii) Factorization

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§23.20(v) Modular Functions and Number Theory

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40: 29.8 Integral Equations

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29.8.1 $$x=k^2\mathrm{sn}(z,k)\mathrm{sn}(z_1,k)\mathrm{sn}(z_2,k)\mathrm{sn}(z_3,k)-\frac{k^2}{k_{}^{\prime }{}_{}{}^2}\mathrm{cn}(z,k)\mathrm{cn}(z_1,k)\mathrm{cn}(z_2,k)\mathrm{cn}(z_3,k)+\frac{1}{k_{}^{\prime }{}_{}{}^2}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k)\mathrm{dn}(z_2,k)\mathrm{dn}(z_3,k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $x$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

►where $z,z_1,z_2,z_3$ are real, and $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ are the Jacobian elliptic functions (§22.2). … ►

29.8.6 $$y=\frac{1}{k^{\prime }}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $y$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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