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See A&S Section 6.12 and DLMF 22.2 for more information.
The Jacobian elliptic function \({\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm ns}(u,m) = 1/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm sc}(u,m) = {\rm sn}(u,m)/{\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm sd}(u,m) = {\rm sn}(u,m)/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm nc}(u,m) = 1/{\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm cs}(u,m) = {\rm cn}(u,m)/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm cd}(u,m) = {\rm cn}(u,m)/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm nd}(u,m) = 1/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm ds}(u,m) = {\rm dn}(u,m)/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm dc}(u,m) = {\rm dn}(u,m)/{\rm cn}(u,m).\)
The inverse of the Jacobian elliptic function \({\rm sn}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E12):
The inverse of the Jacobian elliptic function \({\rm cn}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E13):
The inverse of the Jacobian elliptic function \({\rm dn}(u,m).\) For \(\sqrt{1-m}\le u \le 1,\) it can also be written (DLMF 22.15.E14):
The inverse of the Jacobian elliptic function \({\rm ns}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E121):
The inverse of the Jacobian elliptic function \({\rm sc}(u,m).\) For all \(u\) it can also be written (DLMF 22.15.E20):
The inverse of the Jacobian elliptic function \({\rm sd}(u,m).\) For \(-1/\sqrt{1-m}\le u \le 1/\sqrt{1-m},\) it can also be written (DLMF 22.15.E16):
The inverse of the Jacobian elliptic function \({\rm nc}(u,m).\) For \(1\le u,\) it can also be written (DLMF 22.15.E19):
The inverse of the Jacobian elliptic function \({\rm cs}(u,m).\) For all \(u\) it can also be written (DLMF 22.15.E23):
The inverse of the Jacobian elliptic function \({\rm cd}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E15):
The inverse of the Jacobian elliptic function \({\rm nd}(u,m).\) For \(1\le u \le 1/\sqrt{1-m},\) it can also be written (DLMF 22.15.E17):
The inverse of the Jacobian elliptic function \({\rm ds}(u,m).\) For \(\sqrt{1-m}\le u,\) it can also be written (DLMF 22.15.E22):
The inverse of the Jacobian elliptic function \({\rm dc}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E18):