Jacobian elliptic-function form

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11: 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)

… ►See §22.17.

12: 22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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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13: 22.14 Integrals

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§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

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§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

… ►The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. … ► ►

§22.14(iv) Definite Integrals

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14: 29.15 Fourier Series and Chebyshev Series

… ►Since (29.2.5) implies that

\cos\phi=\operatorname{sn}\left(z,k\right)

, (29.15.1) can be rewritten in the form … ►

29.15.45

\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E45
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.46

\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left% (z,k\right)\right)\right),

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E46
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.49

\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(% \operatorname{sn}\left(z,k\right)\right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E49
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.50

\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(% \operatorname{sn}\left(z,k\right)\right).

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E50
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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15: 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={\operatorname{sn}}^{2}\left(z,k\right).

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Defines:: $\xi$ : variable (locally)
Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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

… ►we have … ►

16: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions

\operatorname{sn}\left(z,k\right)

\operatorname{cn}\left(z,k\right)

\operatorname{dn}\left(z,k\right)

; the nine subsidiary Jacobian elliptic functions

\operatorname{cd}\left(z,k\right)

\operatorname{sd}\left(z,k\right)

\operatorname{nd}\left(z,k\right)

\operatorname{dc}\left(z,k\right)

\operatorname{nc}\left(z,k\right)

\operatorname{sc}\left(z,k\right)

\operatorname{ns}\left(z,k\right)

\operatorname{ds}\left(z,k\right)

\operatorname{cs}\left(z,k\right)

; the amplitude function

\operatorname{am}\left(x,k\right)

; Jacobi’s epsilon and zeta functions

\mathcal{E}\left(x,k\right)

and

\mathrm{Z}\left(x|k\right)

. … ►Other notations for

\operatorname{sn}\left(z,k\right)

are

\mathrm{sn}(z\mathpunct{|}m)

and

\mathrm{sn}(z,m)

with

m=k^{2}

; see Abramowitz and Stegun (1964) and Walker (1996). …

17: 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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18: 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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$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sn}z)$ $\;=\;$	$\operatorname{cn}z\operatorname{dn}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{dc}z)$ =	${k^{\prime}}^{2}\operatorname{sc}z\operatorname{nc}z$
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► … ►

§22.13(ii) First-Order Differential Equations

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22.13.7

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{% 2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{% dc}}^{2}\left(z,k\right)-k^{2}\right),

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Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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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19: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►The prefixes

\mathit{u}

\mathit{s}

\mathit{c}

\mathit{d}

\mathit{sc}

\mathit{sd}

\mathit{cd}

\mathit{scd}

indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable

z

and modulus

k

of the Jacobian elliptic functions have been suppressed, and

P({\operatorname{sn}}^{2})

denotes a polynomial of degree

n

{\operatorname{sn}}^{2}\left(z,k\right)

(different for each type). … ►

§29.12(ii) Algebraic Form

►With the substitution

\xi={\operatorname{sn}}^{2}\left(z,k\right)

every Lamé polynomial in Table 29.12.1 can be written in the form …

20: 22.10 Maclaurin Series

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§22.10(i) Maclaurin Series in $z$

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22.10.1

\operatorname{sn}\left(z,k\right)=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:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\sin\NVar{z}$ : 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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