### 16.2 Functions and Variables for Elliptic Functions

See A&S Section 6.12 and DLMF 22.2 for more information.

Function: jacobi_sn (u, m)

  The Jacobian elliptic function \({\rm sn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_cn (u, m)

  The Jacobian elliptic function \({\rm cn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_dn (u, m)

  The Jacobian elliptic function \({\rm dn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_ns (u, m)

  The Jacobian elliptic function \({\rm ns}(u,m) = 1/{\rm sn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_sc (u, m)

  The Jacobian elliptic function \({\rm sc}(u,m) = {\rm sn}(u,m)/{\rm cn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_sd (u, m)

  The Jacobian elliptic function \({\rm sd}(u,m) = {\rm sn}(u,m)/{\rm dn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_nc (u, m)

  The Jacobian elliptic function \({\rm nc}(u,m) = 1/{\rm cn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_cs (u, m)

  The Jacobian elliptic function \({\rm cs}(u,m) = {\rm cn}(u,m)/{\rm sn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_cd (u, m)

  The Jacobian elliptic function \({\rm cd}(u,m) = {\rm cn}(u,m)/{\rm dn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_nd (u, m)

  The Jacobian elliptic function \({\rm nd}(u,m) = 1/{\rm dn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_ds (u, m)

  The Jacobian elliptic function \({\rm ds}(u,m) = {\rm dn}(u,m)/{\rm sn}(u,m).\)

  Categories:Elliptic functions ·

Function: jacobi_dc (u, m)

  The Jacobian elliptic function \({\rm dc}(u,m) = {\rm dn}(u,m)/{\rm cn}(u,m).\)

  Categories:Elliptic functions ·

Function: inverse_jacobi_sn (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): $$ {\rm inverse\_jacobi\_sn}(u, m) = \int_0^u {dt\over \sqrt{(1-t^2)(1-mt^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_cn (u, m)

  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): $$ {\rm inverse\_jacobi\_cn}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(1-m+mt^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_dn (u, m)

  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): $$ {\rm inverse\_jacobi\_dn}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(t^2-(1-m))}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_ns (u, m)

  The inverse of the Jacobian elliptic function \({\rm ns}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E121): $$ {\rm inverse\_jacobi\_ns}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1-t^2)(t^2-m)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_sc (u, m)

  The inverse of the Jacobian elliptic function \({\rm sc}(u,m).\) For all u it can also be written (DLMF 22.15.E20): $$ {\rm inverse\_jacobi\_sc}(u, m) = \int_0^u {dt\over \sqrt{(1+t^2)(1+(1-m)t^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_sd (u, m)

  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): $$ {\rm inverse\_jacobi\_sd}(u, m) = \int_0^u {dt\over \sqrt{(1-(1-m)t^2)(1+mt^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_nc (u, m)

  The inverse of the Jacobian elliptic function \({\rm nc}(u,m).\) For \(1\le u,\) it can also be written (DLMF 22.15.E19): $$ {\rm inverse\_jacobi\_nc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(m+(1-m)t^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_cs (u, m)

  The inverse of the Jacobian elliptic function \({\rm cs}(u,m).\) For all u it can also be written (DLMF 22.15.E23): $$ {\rm inverse\_jacobi\_cs}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1+t^2)(t^2+(1-m))}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_cd (u, m)

  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): $$ {\rm inverse\_jacobi\_cd}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(1-mt^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_nd (u, m)

  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): $$ {\rm inverse\_jacobi\_nd}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(1-(1-m)t^2)}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_ds (u, m)

  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): $$ {\rm inverse\_jacobi\_ds}(u, m) = \int_u^{\infty} {dt\over \sqrt{(t^2+m)(t^2-(1-m))}} $$

  Categories:Elliptic functions ·

Function: inverse_jacobi_dc (u, m)

  The inverse of the Jacobian elliptic function \({\rm dc}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E18): $$ {\rm inverse\_jacobi\_dc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(t^2-m)}} $$

  Categories:Elliptic functions ·