EllipticK[x]

Series[EllipticK[x],{x,0,3}]

phi[t_,p0_]:=2 ArcSin[Sin[p0/2] JacobiSN[t,Sin[p0/2]^2]];

FullSimplify[D[phi[t,p0],t]^2-2 (Cos[phi[t,p0]]-Cos[p0])]

p0=Pi/20;

Plot[phi[t,p0],{t,0,4 EllipticK[Sin[p0/2]^2]}]

JacobiSN[x,1]

Series[JacobiSN[x,m],{m,1,3}]

JacobiCN[x,1]
JacobiDN[x,1]

Series[JacobiCN[x,m],{m,1,3}]

FullSimplify[JacobiSN[x,m]^2+JacobiCN[x,m]^2]

Simplify[Pi+2 JacobiAmplitude[y,1]]

Series[EllipticK[m],{m,1,2}]

Series[Pi+2 JacobiAmplitude[k x-w t,m],{m,1,1}]

(1/2) (-t w Sech[t w-k x]+k x Sech[t w-k x]+Sinh[t w-k x])

psg[x_,t_,k_,w_]:=Pi+2 JacobiAmplitude[k x-w t,1/(k^2-w^2)];
epsilon=0.0001;
k1=1.1;
Plot[{psg[x,0,k1,Sqrt[k1^2-1-.0001]],psg[x,0,k1,Sqrt[k1^2-1-.001]],psg[x,0,k1,Sqrt[k1^2-1-.01]]},{x,0, 15}]

Plot3D[psg[x,t,k1,Sqrt[k1^2-1-.0001]],{x,0,15},{t,0,5}]
Animate[Plot[psg[x,t,k1,Sqrt[k1^2-1-.0001]],{x,0,15}],{t,0,5}]