Jacobi Elliptic Functions

Author

Eric W. Weisstein
January 30, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/JacobiEllipticFunctions.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/JacobiEllipticFunctions.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Integral

Integrate[(1/(Sqrt[1-(k^2)((Sin[t])^2)])),{t,0,ϕ},Assumptions->-1<k<1&&0<ϕ<π/2]

Sin[JacobiAmplitude[EllipticF[ϕ,m],m]]

Cos[JacobiAmplitude[EllipticF[ϕ,m],m]]

FullSimplify[(Sqrt[1-m ((Sin[JacobiAmplitude[EllipticF[ϕ,m],m]])^2)]),0<m<1]

FullSimplify[(Sqrt[1-m ((Sin[JacobiAmplitude[EllipticF[ϕ,m],m]])^2)])-JacobiDN[EllipticF[ϕ,m],m],0<m<1]

Special values

JacobiSN[u,0]

JacobiCN[u,0]

JacobiDN[u,0]

Partity

JacobiSN[-u,m]

Periodicity

(JacobiSN[2 i (EllipticK[m])+2 i j (EllipticK[1-m])+u, m])==(((-1))^i) (JacobiSN[u, m])

JacobiSN[u+4EllipticK[m]+2 I EllipticK[1-m],m]

FullSimplify[JacobiSN[u+2i EllipticK[m]+2j I EllipticK[1-m],m],(i|j)∈Integers]

Refine[JacobiSN[u+2i EllipticK[m]+2j I EllipticK[1-m],m],(i|j)∈Integers]

Table[JacobiSN[u+2i EllipticK[m]+2j I EllipticK[1-m],m]-(((-1))^i)JacobiSN[u,m],{i,-5,5},{j,-5,5}]//Flatten//Union

In terms of Jacobi theta functions

(JacobiSN[u, m])==(((EllipticTheta[3, 0, q]) (EllipticTheta[1, (u/((EllipticTheta[3, 0, q])^2)), q]))/((EllipticTheta[2, 0, q]) (EllipticTheta[4, (u/((EllipticTheta[3, 0, q])^2)), q])))

FullSimplify[With[{m=(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))},{JacobiSN[u,m],((EllipticTheta[3,0,q])/(EllipticTheta[2,0,q]))((EllipticTheta[1,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))}],0<m<1&&u>0]

Subtract@@%/.{u->.2,q->.5}

(JacobiCN[u, m])==(((EllipticTheta[4, 0, q]) (EllipticTheta[2, (u/((EllipticTheta[3, 0, q])^2)), q]))/((EllipticTheta[2, 0, q]) (EllipticTheta[4, (u/((EllipticTheta[3, 0, q])^2)), q])))

With[{m=(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))},JacobiCN[u,m]-((EllipticTheta[4,0,q])/(EllipticTheta[2,0,q]))((EllipticTheta[2,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

%/.{u->.2,q->.5}

FullSimplify[((EllipticTheta[4,0,q])/(EllipticTheta[2,0,q]))((EllipticTheta[2,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

FullSimplify[JacobiCN[u,(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))]-((EllipticTheta[4,0,q])/(EllipticTheta[2,0,q]))((EllipticTheta[2,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

(JacobiDN[u, m])==(((EllipticTheta[4, 0, q]) (EllipticTheta[3, (u/((EllipticTheta[3, 0, q])^2)), q]))/((EllipticTheta[3, 0, q]) (EllipticTheta[4, (u/((EllipticTheta[3, 0, q])^2)), q])))

With[{m=(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))},JacobiDN[u,m]-((EllipticTheta[4,0,q])/(EllipticTheta[3,0,q]))((EllipticTheta[3,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

%/.{u->.2,q->.5}

FullSimplify[((EllipticTheta[4,0,q])/(EllipticTheta[3,0,q]))((EllipticTheta[3,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

FullSimplify[JacobiDN[u,(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))]-((EllipticTheta[4,0,q])/(EllipticTheta[3,0,q]))((EllipticTheta[3,u ((EllipticTheta[3,0,q])^(-2)),q])/(EllipticTheta[4,u ((EllipticTheta[3,0,q])^(-2)),q]))]

Differential Equations

cn x

DSolve[y''[x]==-(1-2k^2)y[x]-2k^2y[x]^3,y[x],x]

Subtract@@{y''[x],-(1-2k^2)y[x]-2k^2y[x]^3}/.y->Function[u,JacobiCN[u,k^2]]//FullSimplify

dn x

DSolve[y''[x]==(2-k^2)y[x]-2y[x]^3,y[x],x]

Subtract@@{y''[x],(2-k^2)y[x]-2y[x]^3}/.y->Function[u,JacobiDN[u,k^2]]//FullSimplify

sn x

DSolve[y''[x]==-(1+k^2)y[x]+2k^2y[x]^3,y[x],x]

Subtract@@{y''[x],-(1+k^2)y[x]+2k^2y[x]^3}/.y->Function[u,JacobiSN[u,k^2]]//FullSimplify

Identities

((JacobiCN[u, m])^2)+((JacobiSN[u, m])^2)==1

FullSimplify[((JacobiSN[u,m])^2)+((JacobiCN[u,m])^2)]

((JacobiDN[u, m])^2)+m ((JacobiSN[u, m])^2)==1

FullSimplify[m ((JacobiSN[u,m])^2)+((JacobiDN[u,m])^2)]

FullSimplify[1-m JacobiSN[u,m]^2-((JacobiDN[u,m])^2)]

bug(102180):

FullSimplify[Sqrt[1-m JacobiSN[u,m]^2]-JacobiDN[u,m],0<u<1&&0<m<1]

ArrayPlot[Table[N[Sqrt[1-m JacobiSN[u,m]^2]-JacobiDN[u,m],10],{u,0,3,1/10},{m,0,3,1/10}]]

Table[{With[{m=RandomReal[],u=RandomReal[]},Sqrt[1-m JacobiSN[u,m]^2]-JacobiDN[u,m]]},{10^4}]//Chop//Union//Timing

Table[{With[{m=5RandomReal[],u=5RandomReal[]},Sqrt[1-m JacobiSN[u,m]^2]-JacobiDN[u,m]]},{10^3}]//Chop//Union//Timing

Table[{With[{m=5RandomComplex[],u=5RandomComplex[]},Sqrt[1-m JacobiSN[u,m]^2]-JacobiDN[u,m]]},{10^3}]//Chop//Union//Timing

m ((JacobiCN[u, m])^2)+(1-m)==((JacobiDN[u, m])^2)

FullSimplify[m ((JacobiCN[u,m])^2)+(1-m)-((JacobiDN[u,m])^2)]

bug(341503):

FullSimplify[m ((JacobiCN[u,m])^2)+(1-m)]

(JacobiCN[u, m])+(1-m) (JacobiSN[u, m])==((JacobiDN[u, m])^2)

FullSimplify[((JacobiCN[u,m])^2)+(1-m)((JacobiSN[u,m])^2)]

FullSimplify[((JacobiCN[u,(k^2)])^2)+(1-(k^2)) JacobiSN[u,(k^2)]^2]

ds(u|m) nd(u|m) sn(u|m)==1

JacobiSN[u,m]JacobiND[u,m]JacobiDS[u,m]//FullSimplify

Derivatives

((∂(JacobiCN[u, m]))/(∂u))==-(JacobiDN[u, m]) (JacobiSN[u, m])

D[JacobiCN[u,m],u]

TraditionalForm[JacobiAmplitude[u, m]]

Taylor Series

Series[JacobiCN[u,k^2],{u,0,6}]

Series[JacobiDN[u,k^2],{u,0,6}]

Series[JacobiSN[u,k^2],{u,0,6}]