Jacobian elliptic functions

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11: 22.5 Special Values

§22.5 Special Values

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Table 22.5.2: Other special values of Jacobian elliptic functions.

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	$z$
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§22.5(ii) Limiting Values of $k$

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12: 22.13 Derivatives and Differential Equations

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§22.13(i) Derivatives

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Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

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$\frac{d}{dz}(\mathrm{sn}z)$ $=$	$\mathrm{cn}z\mathrm{dn}z$	$\frac{d}{dz}(\mathrm{dc}z)$ =	$k_{}^{\prime }{}_{}{}^2\mathrm{sc}z\mathrm{nc}z$
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§22.13(ii) First-Order Differential Equations

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22.13.7 $${\left(\frac{d}{dz}\mathrm{dc}(z,k)\right)}^2=\left({\mathrm{dc}}^2(z,k)-1\right)\left({\mathrm{dc}}^2(z,k)-k^2\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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§22.13(iii) Second-Order Differential Equations

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13: 22 Jacobian Elliptic Functions

Chapter 22 Jacobian Elliptic Functions

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14: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

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§22.19(ii) Classical Dynamics: The Quartic Oscillator

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§22.19(iii) Nonlinear ODEs and PDEs

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§22.19(v) Other Applications

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15: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

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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, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: 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, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.11 $$2K\mathrm{ns}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-n\tau }\right),$$

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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, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: 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, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

16: 22.10 Maclaurin Series

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§22.10(i) Maclaurin Series in $z$

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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

Encodings:

pMML, png, TeX

See also:

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

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§22.10(ii) Maclaurin Series in $k$ and $k^{\prime }$

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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:

pMML, png, TeX

See also:

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

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17: 22.3 Graphics

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Figure 22.3.24: $\mathrm{sn}(x+\mathrm{i}y,k)$ for $-4\le x\le 4$, $0\le y\le 8$, $k=1+\frac{1}{2}\mathrm{i}$. … Magnify 3D Help

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Figure 22.3.25: $\mathrm{sn}(5,k)$ as a function of complex $k^2$, $-1\le \mathrm{\Re }\left(k^2\right)\le 3.5$, $-1\le \mathrm{\Im }\left(k^2\right)\le 1$. … Magnify 3D Help

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Figure 22.3.26: Density plot of $|\mathrm{sn}(5,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

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Figure 22.3.27: Density plot of $|\mathrm{sn}(10,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

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Figure 22.3.28: Density plot of $|\mathrm{sn}(20,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

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18: 22.21 Tables

§22.21 Tables

… ►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.

19: 22.16 Related Functions

… ►See Figure 22.16.2. … ►

22.16.18 $$ℰ(x,k)=-k^2{\int }_0^x{\mathrm{cd}}^2(t,k)dt+x+k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.4

Permalink:

http://dlmf.nist.gov/22.16.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.19 $$ℰ(x,k)=k^{2}k_{}^{\prime }{}_{}{}^2{\int }_0^x{\mathrm{sd}}^2(t,k)dt+k_{}^{\prime }{}_{}{}^{2}x+k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.26.5

Permalink:

http://dlmf.nist.gov/22.16.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.21 $$ℰ(x,k)=-{\int }_0^x{\mathrm{dc}}^2(t,k)dt+x+\mathrm{sn}(x,k)\mathrm{dc}(x,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.7

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.26 $$ℰ(x,k)=-{\int }_0^x\left({\mathrm{cs}}^2(t,k)-t^{-2}\right)dt+x^{-1}-\mathrm{cn}(x,k)\mathrm{ds}(x,k).$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.12

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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20: 22.11 Fourier and Hyperbolic Series

§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

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 …