Elliptic Functions
inverse_jacobi_cd (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm cd}(u,m).$ For $-1\le u \le 1,$ it can also be written (https://dlmf.nist.gov/22.15.E15DLMF 22.15.E15):
$${\rm inverse_jacobi_cd}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(1-mt^2)}}$$
$${\rm inverse_jacobi_cd}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(1-mt^2)}}$$
inverse_jacobi_cs (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm cs}(u,m).$ For all $u$ it can also be written (https://dlmf.nist.gov/22.15.E23DLMF 22.15.E23):
$${\rm inverse_jacobi_cs}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1+t^2)(t^2+(1-m))}}$$
$${\rm inverse_jacobi_cs}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1+t^2)(t^2+(1-m))}}$$
inverse_jacobi_dc (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm dc}(u,m).$ For $1 \le u,$ it can also be written (https://dlmf.nist.gov/22.15.E18DLMF 22.15.E18):
$${\rm inverse_jacobi_dc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(t^2-m)}}$$
$${\rm inverse_jacobi_dc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(t^2-m)}}$$
inverse_jacobi_dn (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm dn}(u,m).$ For $\sqrt{1-m}\le u \le 1,$ it can also be written (https://dlmf.nist.gov/22.15.E14DLMF 22.15.E14):
$${\rm inverse_jacobi_dn}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(t^2-(1-m))}}$$
$${\rm inverse_jacobi_dn}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(t^2-(1-m))}}$$
inverse_jacobi_ds (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm ds}(u,m).$ For $\sqrt{1-m}\le u,$ it can also be written (https://dlmf.nist.gov/22.15.E22DLMF 22.15.E22):
$${\rm inverse_jacobi_ds}(u, m) = \int_u^{\infty} {dt\over \sqrt{(t^2+m)(t^2-(1-m))}}$$
$${\rm inverse_jacobi_ds}(u, m) = \int_u^{\infty} {dt\over \sqrt{(t^2+m)(t^2-(1-m))}}$$
inverse_jacobi_nc (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm nc}(u,m).$ For $1\le u,$ it can also be written (https://dlmf.nist.gov/22.15.E19DLMF 22.15.E19):
$${\rm inverse_jacobi_nc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(m+(1-m)t^2)}}$$
$${\rm inverse_jacobi_nc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(m+(1-m)t^2)}}$$
inverse_jacobi_nd (u, m) — Function
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 (https://dlmf.nist.gov/22.15.E17DLMF 22.15.E17):
$${\rm inverse_jacobi_nd}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(1-(1-m)t^2)}}$$
$${\rm inverse_jacobi_nd}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(1-(1-m)t^2)}}$$
inverse_jacobi_ns (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm ns}(u,m).$ For $1 \le u,$ it can also be written (https://dlmf.nist.gov/22.15.E121DLMF 22.15.E121):
$${\rm inverse_jacobi_ns}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1-t^2)(t^2-m)}}$$
$${\rm inverse_jacobi_ns}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1-t^2)(t^2-m)}}$$
inverse_jacobi_sc (u, m) — Function
The inverse of the Jacobian elliptic function ${\rm sc}(u,m).$ For all $u$ it can also be written (https://dlmf.nist.gov/22.15.E20DLMF 22.15.E20):
$${\rm inverse_jacobi_sc}(u, m) = \int_0^u {dt\over \sqrt{(1+t^2)(1+(1-m)t^2)}}$$
$${\rm inverse_jacobi_sc}(u, m) = \int_0^u {dt\over \sqrt{(1+t^2)(1+(1-m)t^2)}}$$
inverse_jacobi_sd (u, m) — Function
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 (https://dlmf.nist.gov/22.15.E16DLMF 22.15.E16):
$${\rm inverse_jacobi_sd}(u, m) = \int_0^u {dt\over \sqrt{(1-(1-m)t^2)(1+mt^2)}}$$
$${\rm inverse_jacobi_sd}(u, m) = \int_0^u {dt\over \sqrt{(1-(1-m)t^2)(1+mt^2)}}$$
jacobi_am (u, m) — Function
The Jacobi amplitude function, jacobi_am, is defined implicitly by (see
http://functions.wolfram.com/09.24.02.0001.01)
$z = {\rm am}(w, m)$
where $w = F(z,m)$ where $F(z,m)$ is the incomplete elliptic
integral of the first kind (elliptic_005ff). It is defined for
all real and complex values of $z$ and $m$. In particular
for real $z$ and $m$ with $|m|<1$,
${\rm am}(z,m)$
maps the entire real line to the entire real line. For other values
of $z$ and $m$, the following relationship is used:
${\rm am}(z,m) = \sin^{-1}({\rm jacobi_sn}(z, m)).$
Some examples:
maxima
(%i1) jacobi_am(z,0);
(%o1) z
(%i2) jacobi_am(z,1);
z %pi
(%o2) 2 atan(%e ) - ---
2
(%i3) jacobi_am(0,m);
(%o3) 0
(%i4) jacobi_am(100, .5);
(%o4) 84.70311272411382
(%i5) jacobi_am(0.5, 1.5);
(%o5) 0.4707197897046991
(%i6) jacobi_am(1.5b0, 1.5b0+%i);
(%o6) 9.340542168700782b-1 - 3.723960452146071b-1 %i
maxima
(%i1) plot2d([jacobi_am(x,.4),jacobi_am(x,.7),jacobi_am(x,.99),jacobi_am(x,.999999)],[x,0,10*%pi]);
(%o1) false
Compare this plot with the plot from https://dlmf.nist.gov/22.16.ivDLMF 22.16.iv:
See also: jacobi_am, elliptic_f.
jacobi_cd (u, m) — Function
The Jacobian elliptic function ${\rm cd}(u,m) = {\rm cn}(u,m)/{\rm dn}(u,m).$
jacobi_cn (u, m) — Function
The Jacobian elliptic function ${\rm cn}(u,m).$
jacobi_cs (u, m) — Function
The Jacobian elliptic function ${\rm cs}(u,m) = {\rm cn}(u,m)/{\rm sn}(u,m).$
jacobi_dc (u, m) — Function
The Jacobian elliptic function ${\rm dc}(u,m) = {\rm dn}(u,m)/{\rm cn}(u,m).$
jacobi_dn (u, m) — Function
The Jacobian elliptic function ${\rm dn}(u,m).$
jacobi_ds (u, m) — Function
The Jacobian elliptic function ${\rm ds}(u,m) = {\rm dn}(u,m)/{\rm sn}(u,m).$
jacobi_nc (u, m) — Function
The Jacobian elliptic function ${\rm nc}(u,m) = 1/{\rm cn}(u,m).$
jacobi_nd (u, m) — Function
The Jacobian elliptic function ${\rm nd}(u,m) = 1/{\rm dn}(u,m).$
jacobi_ns (u, m) — Function
The Jacobian elliptic function ${\rm ns}(u,m) = 1/{\rm sn}(u,m).$
jacobi_sc (u, m) — Function
The Jacobian elliptic function ${\rm sc}(u,m) = {\rm sn}(u,m)/{\rm cn}(u,m).$
jacobi_sd (u, m) — Function
The Jacobian elliptic function ${\rm sd}(u,m) = {\rm sn}(u,m)/{\rm dn}(u,m).$
jacobi_sn (u, m) — Function
The Jacobian elliptic function ${\rm sn}(u,m).$