elliptic functions

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1: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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2: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►are denoted respectively by …The principal values satisfy … ►

3: 22.2 Definitions

§22.2 Definitions

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22.2.4

\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},

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Defines:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : 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:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

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22.2.9

\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.

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Defines:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : 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.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►As a function of

z

, with fixed

k

, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …

4: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

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Definition

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Quasi-Periodicity

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Integral Representation

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Relation to Elliptic Integrals

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5: 19.16 Definitions

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19.16.6

R_{C}\left(x,y\right)=R_{F}\left(x,y,y\right),

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.19, §19.24(ii), §19.6(ii)
Permalink:: http://dlmf.nist.gov/19.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.16(i), §19.16 and Ch.19

… ►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The

R

-function is often used to make a unified statement of a property of several elliptic integrals. … ►

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

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19.16.21

R_{D}\left(0,y,z\right)=\tfrac{3}{4}\pi R_{-\frac{3}{2}}\left(\tfrac{1}{2},% \tfrac{3}{2};y,z\right),

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.9(i)
Permalink:: http://dlmf.nist.gov/19.16.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.16(iii), §19.16 and Ch.19

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6: 19.2 Definitions

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19.2.4

F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{% \sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{% 1-k^{2}t^{2}}},

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Defines:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.5, §19.6(ii)
Permalink:: http://dlmf.nist.gov/19.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.6

D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\,\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.

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Defines:: $D\left(\NVar{\phi},\NVar{k}\right)$ : incomplete elliptic integral of Legendre’s type
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

… ►The principal values of

K\left(k\right)

and

E\left(k\right)

are even functions. … ►

19.2.11_5

\operatorname{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{% 1}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\,\mathrm{d}\theta,

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Defines:: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind
Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $k_{c}$ : real or complex variable not equal to zero and $x$ : real or complex variable
Referenced by:: §19.2(iii), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/19.2.E11_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added and made the definition for $\operatorname{el1}\left(x,k_{c}\right)$ (was previously (19.2.15)).
Suggested 2019-09-26 by Jan Mangaldan
See also:: Annotations for §19.2(iii), §19.2 and Ch.19

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19.2.16

\operatorname{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}% \frac{\,\mathrm{d}\theta}{({\cos}^{2}\theta+p{\sin}^{2}\theta)\sqrt{{\cos}^{2}% \theta+k_{c}^{2}{\sin}^{2}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),

x^{2}\neq-1/p

ⓘ

Defines:: $\operatorname{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)$ : Bulirsch’s incomplete elliptic integral of the third kind
Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $k$ : real or complex modulus, $p$ : real parameter not equal to zero, $k_{c}$ : real or complex variable not equal to zero and $x$ : real or complex variable
Permalink:: http://dlmf.nist.gov/19.2.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(iii), §19.2 and Ch.19

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7: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

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§22.17(i) Real or Purely Imaginary Moduli

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§22.17(ii) Complex Moduli

►When

z

is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of

k^{2}

. …For proofs of these results and further information see Walker (2003).

8: 23.13 Zeros

§23.13 Zeros

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9: 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.

10: 22.8 Addition Theorems

§22.8 Addition Theorems

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22.8.1

\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},

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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, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.1
Referenced by:: §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

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22.8.2

\operatorname{cn}(u+v)=\frac{\operatorname{cn}u\operatorname{cn}v-% \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^% {2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},

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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, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

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§22.8(iii) Special Relations Between Arguments

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