Jacobian elliptic-function form

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11: 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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22.13.1 $${\left(\frac{d}{dz}\mathrm{sn}(z,k)\right)}^2=\left(1-{\mathrm{sn}}^2(z,k)\right)\left(1-k^2{\mathrm{sn}}^2(z,k)\right),$$

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Symbols:

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

Referenced by:

§22.13(iii), §22.18(iv)

Permalink:

http://dlmf.nist.gov/22.13.E1

Encodings:

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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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12: 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

13: Bille C. Carlson

… ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

14: 22.20 Methods of Computation

§22.20 Methods of Computation

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§22.20(iii) Landen Transformations

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§22.20(iv) Lattice Calculations

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§22.20(v) Inverse Functions

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15: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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16: 22.6 Elementary Identities

§22.6 Elementary Identities

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§22.6(ii) Double Argument

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§22.6(iii) Half Argument

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§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

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17: 22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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22.7.3 $$\mathrm{cn}(z,k)=\frac{\mathrm{cn}(z/(1+k_1),k_1)\mathrm{dn}(z/(1+k_1),k_1)}{1+k_1{\mathrm{sn}}^2(z/(1+k_1),k_1)},$$

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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 $k_1$: change of variable

A&S Ref:

16.12.3

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(i), §22.7 and Ch.22

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§22.7(ii) Ascending Landen Transformation

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22.7.6 $$\mathrm{sn}(z,k)=\frac{(1+k_2^{\prime })\mathrm{sn}(z/(1+k_2^{\prime }),k_2)\mathrm{cn}(z/(1+k_2^{\prime }),k_2)}{\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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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, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.2

Permalink:

http://dlmf.nist.gov/22.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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§22.7(iii) Generalized Landen Transformations

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18: 19.25 Relations to Other Functions

… ►Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … ►

§19.25(v) Jacobian Elliptic Functions

… ►For the use of $R$-functions with $\mathrm{\Delta }(\mathrm{p},\mathrm{q})$ in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008). ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. … …

19: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

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29.2.3 $$\xi ={\mathrm{sn}}^2(z,k).$$

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Defines:

$\xi$: variable (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/29.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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20: 31.2 Differential Equations

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(iii) Trigonometric Form

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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