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21: 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

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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

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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

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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

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Annotations for §22.11 and Ch.22

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22.11.5 $$\mathrm{sd}(z,k)=\frac{2\pi }{Kk{k}^{\prime }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

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Symbols:

$\mathrm{sd}(z,k)$: 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, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.5

Permalink:

http://dlmf.nist.gov/22.11.E5

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See also:

Annotations for §22.11 and Ch.22

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22: 22.8 Addition Theorems

… ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►

22.8.21 $$k_{}^{\prime }{}_{}{}^2-k_{}^{\prime }{}_{}{}^2{k}^2\mathrm{sn}{z}_1\mathrm{sn}{z}_2\mathrm{sn}{z}_3\mathrm{sn}{z}_4+k^2\mathrm{cn}{z}_1\mathrm{cn}{z}_2\mathrm{cn}{z}_3\mathrm{cn}{z}_4-\mathrm{dn}{z}_1\mathrm{dn}{z}_2\mathrm{dn}{z}_3\mathrm{dn}{z}_4=0.$$

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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, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.8.E21

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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22.8.22 $$z_1+z_2+z_3+z_4=2K\left(k\right).$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E22

Encodings:

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Annotations for §22.8(iii), §22.8 and Ch.22

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22.8.27 $$\mathrm{dn}{z}_1\mathrm{dn}{z}_3=\mathrm{dn}{z}_2\mathrm{dn}{z}_4=k^{\prime }.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.8.E27

Encodings:

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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23: 19.39 Software

… ►For research software see Bulirsch (1969b, function $\mathrm{cel}$), Herndon (1961a, b), Merner (1962), Morita (1978, complex modulus $k$), and Thacher Jr. (1963). …

24: 22.10 Maclaurin Series

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22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

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Annotations for §22.10(i), §22.10 and Ch.22

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22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.7 $$\mathrm{sn}(z,k)=\tanh z-\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z){\mathrm{sech}}^{2}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.1

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E7

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.9 $$\mathrm{dn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z+\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.3

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E9

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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25: 19.33 Triaxial Ellipsoids

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19.33.2 $$\frac{S}{2\pi }=c^2+\frac{ab}{\sin \varphi }\left(E(\varphi ,k){\sin }^2\varphi +F(\varphi ,k){\cos }^2\varphi \right),$$ $a\ge b\ge c$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $S$: area

Permalink:

http://dlmf.nist.gov/19.33.E2

Encodings:

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See also:

Annotations for §19.33(i), §19.33 and Ch.19

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26: 22.2 Definitions

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22.2.3 $$\zeta =\frac{\pi z}{2K\left(k\right)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E3

Encodings:

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See also:

Annotations for §22.2 and Ch.22

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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:

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Annotations for §22.2 and Ch.22

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22.2.5 $$\mathrm{cn}(z,k)=\frac{{\theta }_4(0,q)}{{\theta }_2(0,q)}\frac{{\theta }_2(\zeta ,q)}{{\theta }_4(\zeta ,q)}=\frac{1}{\mathrm{nc}(z,k)},$$

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Defines:

$\mathrm{cn}(z,k)$: Jacobian elliptic function and $\mathrm{nc}(z,k)$: 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.E5

Encodings:

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Annotations for §22.2 and Ch.22

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22.2.6 $$\mathrm{dn}(z,k)=\frac{{\theta }_4(0,q)}{{\theta }_3(0,q)}\frac{{\theta }_3(\zeta ,q)}{{\theta }_4(\zeta ,q)}=\frac{1}{\mathrm{nd}(z,k)},$$

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Defines:

$\mathrm{dn}(z,k)$: Jacobian elliptic function and $\mathrm{nd}(z,k)$: Jacobian elliptic function

Symbols:

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

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.2.E6

Encodings:

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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:

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See also:

Annotations for §22.2, §22.2 and Ch.22

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27: 23.20 Mathematical Applications

… ►It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). …

28: 22.9 Cyclic Identities

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22.9.1 $$s_{m,p}^{(2)}=\mathrm{sn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E1

Encodings:

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See also:

Annotations for §22.9(i), §22.9 and Ch.22

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22.9.2 $$c_{m,p}^{(2)}=\mathrm{cn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E2

Encodings:

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See also:

Annotations for §22.9(i), §22.9 and Ch.22

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22.9.3 $$d_{m,p}^{(2)}=\mathrm{dn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E3

Encodings:

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See also:

Annotations for §22.9(i), §22.9 and Ch.22

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22.9.4 $$s_{m,p}^{(4)}=\mathrm{sn}(z+4p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E4

Encodings:

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See also:

Annotations for §22.9(i), §22.9 and Ch.22

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22.9.5 $$c_{m,p}^{(4)}=\mathrm{cn}(z+4p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E5

Encodings:

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See also:

Annotations for §22.9(i), §22.9 and Ch.22

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29: 22.1 Special Notation

… ► ►►►►►►

$x,y$	real variables.
$z$	complex variable.
$k$	modulus. Except in §§22.3(iv), 22.17, and 22.19, $0\le k\le 1$.
$k^{\prime }$	complementary modulus, $k^2+k_{}^{\prime }{}_{}{}^2=1$. If $k\in [0,1]$, then $k^{\prime }\in [0,1]$.
…
$\tau$	$\mathrm{i}{K}^{\prime }/K$.

… ►Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

30: Bibliography M

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Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.

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Notes:

An Algol procedure for calculating the complete elliptic integrals of the first and the second kind with complex modulus $k$ is included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.39(ii).

Encodings:

BibTeX

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