inverse Jacobian elliptic functions

SearchAdvancedHelp

(0.009 seconds)

1—10 of 17 matching pages

1: 22.15 Inverse Functions

§22.15 Inverse Functions

►

§22.15(i) Definitions

… ►are denoted respectively by … ►

§22.15(ii) Representations as Elliptic Integrals

… ►

2: 22.18 Mathematical Applications

… ►

Lemniscate

… ►

22.18.4 $$l(r)=(1/\sqrt{2})\mathrm{arccn}(r,1/\sqrt{2}).$$

ⓘ

Symbols:

$\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

…

3: 22.20 Methods of Computation

… ►

§22.20(v) Inverse Functions

…

4: Bibliography C

… ►

Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

… ►

Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

ⓘ

External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

…

5: 22.14 Integrals

… ►

22.14.2 $$\int \mathrm{cn}(x,k)dx=k^{-1}\mathrm{Arccos}\left(\mathrm{dn}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real and $k$: modulus

A&S Ref:

16.24.2

Permalink:

http://dlmf.nist.gov/22.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

ⓘ

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.6 $$\int \mathrm{nd}(x,k)dx=k_{}^{\prime }{}_{}{}^{-1}\mathrm{Arccos}\left(\mathrm{cd}(x,k)\right).$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.6

Permalink:

http://dlmf.nist.gov/22.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

…

6: 22.19 Physical Applications

… ►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …

7: 22.16 Related Functions

… ►

22.16.1 $$\mathrm{am}(x,k)=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right),$$ $x\in ℝ$,

ⓘ

Defines:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\in$: element of, $\mathrm{Arcsin}z$: general arcsine function, $ℝ$: real line, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E1

Encodings:

pMML, png, TeX

See also:

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

…

8: 19.25 Relations to Other Functions

… ►

19.25.30 $$\mathrm{am}(u,k)=R_C({\mathrm{cs}}^2(u,k),{\mathrm{ns}}^2(u,k)),$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function and $k$: real or complex modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/19.25.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

… ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. …

9: 20.9 Relations to Other Functions

… ►

§20.9(i) Elliptic Integrals

… ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). ►As a function of $\tau$, $k^2$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …

10: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. … ►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.