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Jacobi Elliptic Functions

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The Jacobi elliptic functions are standard forms ofelliptic functions. The three basic functions are denotedcn(u,k),dn(u,k), andsn(u,k), wherek is known as theelliptic modulus. They arise from the inversion of theelliptic integral of the first kind,

u=F(phi,k)=int_0^phi(dt)/(sqrt(1-k^2sin^2t)),
(1)

where0<k^2<1,k=modu is theelliptic modulus, andphi=am(u,k)=am(u) is theJacobi amplitude, giving

phi=F^(-1)(u,k)=am(u,k).
(2)

From this, it follows that

sinphi=sin(am(u,k))
(3)
=sn(u,k)
(4)
cosphi=cos(am(u,k))
(5)
=cn(u,k)
(6)
sqrt(1-k^2sin^2phi)=sqrt(1-k^2sin^2(am(u,k)))
(7)
=dn(u,k).
(8)

These functions are doubly periodic generalizations of the trigonometric functions satisfying

sn(u,0)=sinu
(9)
cn(u,0)=cosu
(10)
dn(u,0)=1.
(11)

In terms ofJacobi theta functions,

sn(u,k)=(theta_3)/(theta_2)(theta_1(utheta_3^(-2)))/(theta_4(utheta_3^(-2)))
(12)
cn(u,k)=(theta_4)/(theta_2)(theta_2(utheta_3^(-2)))/(theta_4(utheta_3^(-2)))
(13)
dn(u,k)=(theta_4)/(theta_3)(theta_3(utheta_3^(-2)))/(theta_4(utheta_3^(-2)))
(14)

(Whittaker and Watson 1990, p. 492), wheretheta_i=theta_i(0) (Whittaker and Watson 1990, p. 464) and theelliptic modulus is given by

k=(theta_2^2(q))/(theta_3^2(q)).
(15)

Ratios of Jacobi elliptic functions are denoted by combining the first letter of thenumerator elliptic function with the first of thedenominator elliptic function. The multiplicative inverses of the elliptic functions are denoted by reversing the order of the two letters. These combinations give a total of 12 functions: cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. These functions are implemented in theWolfram Language asJacobiSN[z,m] and so on. Similarly, the inverse Jacobi functions are implemented asInverseJacobiSN[v,m] and so on.

TheJacobi amplitudephi is defined in terms ofsn(u,k) by

y=sinphi=sn(u,k).
(16)

Thek argument is often suppressed for brevity so, for example,sn(u,k) can be written assnu.

The Jacobi elliptic functions are periodic inK(k) andK^'(k) as

sn(u+2mK+2niK^',k)=(-1)^msn(u,k)
(17)
cn(u+2mK+2niK^',k)=(-1)^(m+n)cn(u,k)
(18)
dn(u+2mK+2niK^',k)=(-1)^ndn(u,k),
(19)

whereK(k) is thecomplete elliptic integral of the first kind,K^'(k)=K(k^'), andk^'=sqrt(1-k^2) (Whittaker and Watson 1990, p. 503).

Thecnx,dnx, andsnx functions may also be defined as solutions to the differential equations

(d^2y)/(dx^2)=(2-k^2)y-2y^3
(20)
(d^2y)/(dx^2)=-(1-2k^2)y-2k^2y^3
(21)
(d^2y)/(dx^2)=-(1+k^2)y+2k^2y^3,
(22)

respectively.

The standard Jacobi elliptic functions satisfy the identities

sn^2u+cn^2u=1
(23)
k^2sn^2u+dn^2u=1
(24)
k^2cn^2u+k^('2)=dn^2u
(25)
cn^2u+k^('2)sn^2u=dn^2u.
(26)

Special values include

cn(0,k)=cn(0)=1
(27)
cn(K(k),k)=cn(K(k))=0
(28)
dn(0,k)=dn(0)=1
(29)
dn(K(k),k)=dn(K(k))=k^'=sqrt(1-k^2),
(30)
sn(0,k)=sn(0)=0
(31)
sn(K(k),k)=sn(K(k))=1,
(32)

whereK=K(k) is acomplete elliptic integral of the first kind andk^'=sqrt(1-k^2) is the complementaryelliptic modulus (Whittaker and Watson 1990, pp. 498-499), and

cn(u,1)=sechu
(33)
dn(u,1)=sechu
(34)
sn(u,1)=tanhu.
(35)

In terms of integrals,

u=int_0^(sn(u,k))(1-t^2)^(-1/2)(1-k^2t^2)^(-1/2)dt
(36)
=int_(ns(u,k))^infty(t^2-1)^(-1/2)(t^2-l^2)^(-1/2)dt
(37)
=int_(cn(u,k))^1(1-t^2)^(-1/2)(k^('2)+k^2t^2)^(-1/2)dt
(38)
=int_1^(nc(u,k))(t^2-1)^(-1/2)(k^('2)t^2+k^2)^(-1/2)dt
(39)
=int_(dn(u,k))^1(1-t^2)^(-1/2)(t^2-k^('2))^(-1/2)dt
(40)
=int_1^(nd(u,k))(t^2-1)^(-1/2)(1-k^('2)t^2)^(-1/2)dt
(41)
=int_0^(sc(u,k))(1+t^2)^(-1/2)(1+k^('2)t^2)^(-1/2)dt
(42)
=int_(cs(u,k))^infty(t^2+1)^(-1/2)(t^2+k^('2))^(-1/2)dt
(43)
=int_0^(sd(u,k))(1-k^('2)t^2)^(-1/2)(1+k^2t^2)^(-1/2)dt
(44)
=int_(ds(u,k))^infty(t^2-k^('2))^(-1/2)(t^2+k^2)^(-1/2)dt
(45)
=int_1^(cd(u,k))(1-t^2)^(-1/2)(1-k^2t^2)^(-1/2)dt
(46)
=int_(dc(u,k))^1(t^2-1)^(-1/2)(t^2-k^2)^(-1/2)dt
(47)

(Whittaker and Watson 1990, p. 494).

Jacobi elliptic functions addition formulas include (where, for example,sn(u,k) is written assnu for conciseness),

sn(u+v)=(snucnvdnv+snvcnudnu)/(1-k^2sn^2usn^2v)
(48)
cn(u+v)=(cnucnv-snusnvdnudnv)/(1-k^2sn^2usn^2v)
(49)
dn(u+v)=(dnudnv-k^2snusnvcnucnv)/(1-k^2sn^2usn^2v).
(50)

Extended to integral periods,

sn(u+K)=(cnu)/(dnu)
(51)
cn(u+K)=-(k^'snu)/(dnu)
(52)
dn(u+K)=(k^')/(dnu)
(53)
sn(u+2K)=-snu
(54)
cn(u+2K)=-cnu
(55)
dn(u+2K)=dnu
(56)

Forcomplex arguments,

sn(u+iv)=(sn(u,k)dn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^'))+(icn(u,k)dn(u,k)sn(v,k^')cn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^'))
(57)
cn(u+iv)=(cn(u,k)cn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^'))-(isn(u,k)dn(u,k)sn(v,k^')dn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^'))
(58)
dn(u+iv)=(dn(u,k)cn(v,k^')dn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^'))-(ik^2sn(u,k)cn(u,k)sn(v,k^'))/(1-dn^2(u,k)sn^2(v,k^')).
(59)

Derivatives of the Jacobi elliptic functions include

(dsnu)/(du)=cnudnu
(60)
(dcnu)/(du)=-snudnu
(61)
(ddnu)/(du)=-k^2snucnu
(62)

(Hille 1969, p. 66; Zwillinger 1997, p. 136).

Double-period formulas involving the Jacobi elliptic functions include

sn(2u)=(2snucnudnu)/(1-k^2sn^4u)
(63)
cn(2u)=(1-2sn^2u+k^2sn^4u)/(1-k^2sn^4u)
(64)
dn(2u)=(1-2k^2sn^2u+k^2sn^4u)/(1-k^2sn^4u).
(65)

Half-period formulas involving the Jacobi elliptic functions include

sn(1/2K)=1/(sqrt(1+k^'))
(66)
cn(1/2K)=sqrt((k^')/(1+k^'))
(67)
dn(1/2K)=sqrt(k^').
(68)

Squared formulas include

sn^2u=(1-cn(2u))/(1+dn(2u))
(69)
cn^2u=(dn(2u)+cn(2u))/(1+dn(2u))
(70)
dn^2u=(dn(2u)+cn(2u))/(1+cn(2u)).
(71)

Taylor series of the Jacobi elliptic functions were considered by Hermite (1863), Schett (1977), and Dumont (1981),

cn(u,k)=1-1/2u^2+1/(24)(1+4k^2)u^4-1/(720)(1+44k^2+16k^4)u^6+...
(72)
dn(u,k)=1-1/2k^2u^2+1/(24)(4k^2+k^4)u^4-1/(720)(16k^2+44k^4+k^6)u^6+...
(73)
sn(u,k)=u-1/6(1+k^2)u^3+1/(120)(1+14k^2+k^4)u^5+...
(74)

(Abramowitz and Stegun 1972, eqn. 16.22).

See also

Elliptic Function,Jacobi Amplitude,Jacobi Differential Equation,Jacobi's Imaginary Transformation,Jacobi Function of the Second Kind,Jacobi Theta Functions,Weierstrass Elliptic Function

Related Wolfram sites

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Jacobian Elliptic Functions and Theta Functions." Ch. 16 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 567-581, 1972.Bellman, R. E.A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961.Briot, C. and Bouquet, C.Théorie des fonctions elliptiques, 2nd ed. Paris: Gauthier-Villars, 1875.Byrd, P. F. and Friedman, M. D.Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev. Berlin: Springer-Verlag, 1971.Dumont, D. "Une Approach combinatoire des fonctions elliptiques de Jacobi."Adv. Math.41, 1-39, 1981.Hermite, C. "Remarque sur le développement decosamx."Comptes Rendus de l'Académie des Sciences57, 613-618, 1863. Reprinted inJ. math. pures appliq.9, 289-295, 1864. Also reprinted inOeuvres de Charles Hermite, Vol. 2. Paris: Gauthier-Villars, pp. 264-270, 1908.Hille, E.Lectures on Ordinary Differential Equations. Reading, MA: Addison-Wesley, 1969.Morse, P. M. and Feshbach, H.Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 433, 1953.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Elliptic Integrals and Jacobi Elliptic Functions." §6.11 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254-263, 1992.Schett, A. "Recurrence Formula of the Taylor Series Expansion Coefficients of the Jacobi Elliptic Functions."Math. Comput.32, 1003-1005, 1977.Spanier, J. and Oldham, K. B. "The Jacobian Elliptic Functions." Ch. 63 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 635-652, 1987.Tölke, F. "Jacobische elliptische Funktionen und zugehörige logarithmische Ableitungen," "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gattung. Elliptische Amplitudenfunktionen sowie LegendrescheF- undE-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gattung. LegendreschePi-Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Chs. 5-7 inPraktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 1-144, 1967.Tölke, F.Praktische Funktionenlehre, vierter Band: Elliptische Integralgruppen und Jacobische elliptische Funktionen im Komplexen. Berlin: Springer-Verlag, 1967.Trott, M. "The Mathematica Guidebooks Additional Material: ODE for Jacobi Elliptic Function sn with Respect to the Modulus."http://www.mathematicaguidebooks.org/additions.shtml#S_3_04.Whittaker, E. T. and Watson, G. N.A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D.Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.

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Jacobi Elliptic Functions

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Weisstein, Eric W. "Jacobi Elliptic Functions."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/JacobiEllipticFunctions.html

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