The JacobiAM and the twelve JacobiPQ functions, where P and Q are any two of {C,D,N,S}, are inverses of elliptic integrals and doubly periodic elliptic functions. All JacobiPQ satisfy $\mathrm{JacobiPQ}=\frac{1}{\mathrm{JacobiQP}}=\frac{\mathrm{JacobiPR}}{\mathrm{JacobiQR}}$ where P, Q, R are any three of {C,D,N,S}. Hence, all JacobiPQ can be defined in terms of the three {JacobiSN, JacobiCN, JacobiDN }.

The three JacobiSN, JacobiCN and JacobiDN are in turn defined in terms of the amplitude function JacobiAM.

The JacobiDN function is also sometimes defined as a function of JacobiAM indirectly, through JacobiSN or JacobiCN, via:

JacobiDN(z,k)^2 = 1-k^2*JacobiSN(z,k)^2;

${\mathrm{JacobiDN}\left(z\,k\right)}^2=1-k^2{\mathrm{JacobiSN}\left(z\,k\right)}^2$

JacobiDN(k*z,(1/(k^2))^(1/2)) = JacobiCN(z,k);

$\mathrm{JacobiDN}\left(kz\,\sqrt{\frac{1}{k^2}}\right)=\mathrm{JacobiCN}\left(z\,k\right)$

For $-\frac{3}{2}<\mathrm{phi}<\frac{3}{2}$, the amplitude function JacobiAM(phi,k) is the inverse of the normal trigonometric form of the incomplete elliptic integral of the first kind, represented in Maple by the InverseJacobiAM function (see G&R, 8.111, 8.141, and A&S 16.1.3, 17.2.6):

F_trig := FunctionAdvisor( definition, InverseJacobiAM(phi,k))[1];

$\mathrm{F\_trig}≔\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)={\int }_0^{\mathrm{\phi }}\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\_\&theta;1}\right)}^2}}\&DifferentialD;\mathrm{\_\&theta;1}$

FunctionAdvisor( definition, JacobiAM(phi,k));

$\left[\mathrm{\phi }=\mathrm{JacobiAM}\left({\int }_0^{\mathrm{\phi }}\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\theta }\right)}^2}}\&DifferentialD;\mathrm{\theta }\&comma;k\right)\&comma;\mathrm{\phi }::\left[-\frac{3}{2}\&comma;\frac{3}{2}\right]\right]$

The JacobiPQ functions are well described in the "Table of Integrals Series and Products", Gradshteyn and Ryzhik (G&R) and in the popular "Handbook of Mathematical Functions" edited by Abramowitz and Stegun (A&S). In A&S, these functions are expressed in terms of a parameter m, representing the square of the modulus k entering the definition of these functions in Maple or G&R. So, for example, the formula ${\mathrm{JacobiDN}\left(z\&comma;k\right)}^2=1-k^2{\mathrm{JacobiSN}\left(z\&comma;k\right)}^2$ appears in A&S as ${\mathrm{dn}\left(z\&comma;m\right)}^2=1-m{\mathrm{sn}\left(z\&comma;m\right)}^2$.

It is worth noting here the difference between this normal trigonometric form $\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)$ and the Legendre normal form of this elliptic integral (see A&S 17.2.7), in Maple represented by the EllipticF function and with the square root in the denominator split.

EllipticF(z,k);

$\mathrm{EllipticF}\left(z\&comma;k\right)$

(5) = convert((5), Int);

$\mathrm{EllipticF}\left(z\&comma;k\right)={\int }_0^z\frac{1}{\sqrt{-{\mathrm{\_\&alpha;1}}^2+1}\sqrt{-k^2{\mathrm{\_\&alpha;1}}^2+1}}\&DifferentialD;\mathrm{\_\&alpha;1}$

For the general relationship between these two forms of the incomplete elliptic integral of the first kind see EllipticF and InverseJacobiAM.

The parameter k entering the definition of JacobiAM lines above is assumed to be real and such that $0<=k^2<1$. The JacobiAM function, and together with it all the JacobiPQ, however, can be evaluated at arbitrary complex values of k. For $k^2<0$ see A&S 16.10; for $1<k^2$ see A&S 16.11; for $\mathrm{\Im }\left(k^2\right)\ne 0$ see A&S 16.21.

The JacobiAM(z,k) function is an infinitely many valued function of z with period of $4\mathrm{K\text{'}}\left(k\right)I$ where $\mathrm{I}$ is the imaginary unit and $\mathrm{K\text{'}}$ is the complementary elliptic integral of the first kind (see also EllipticCK).

The inverse functions for the JacobiPQ are the InverseJacobiPQ functions, all of them (but for JacobiAM) satisfying $\mathrm{JacobiPQ}\left(\mathrm{InverseJacobiPQ}\left(z\&comma;k\right)\&comma;k\right)=z$ with no restrictions on the function parameters.

The JacobiPQ functions are frequently viewed as generalizations of the usual trigonometric functions in that they include all of them as particular cases for $k=0$; for instance,

map(Eval = eval, [JacobiSN(z,k), JacobiCN(z,k), JacobiSC(z,k)], k=0);

$\left[\genfrac{}{}{0ex}{}{\mathrm{JacobiSN}\left(z\&comma;k\right)}{\phantom{k=0}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiSN}\left(z\&comma;k\right)}}{k=0}=\sin \left(z\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{JacobiCN}\left(z\&comma;k\right)}{\phantom{k=0}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiCN}\left(z\&comma;k\right)}}{k=0}=\cos \left(z\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{JacobiSC}\left(z\&comma;k\right)}{\phantom{k=0}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiSC}\left(z\&comma;k\right)}}{k=0}=\tan \left(z\right)\right]$

From the above, JacobiSN, JacobiCN, JacobiSC are usually viewed as generalizations of sin, cos, and tan respectively.

$\mathrm{map}\left(\mathrm{Eval}=\mathrm{eval}\&comma;\left[\mathrm{JacobiSC}\&comma;\mathrm{JacobiNC}\&comma;\mathrm{JacobiSN}\right]\left(z\&comma;k\right)\&comma;k=1\right)$

$\left[\genfrac{}{}{0ex}{}{\mathrm{JacobiSC}\left(z\&comma;k\right)}{\phantom{k=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiSC}\left(z\&comma;k\right)}}{k=1}=\sinh \left(z\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{JacobiNC}\left(z\&comma;k\right)}{\phantom{k=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiNC}\left(z\&comma;k\right)}}{k=1}=\cosh \left(z\right)\&comma;\genfrac{}{}{0ex}{}{\mathrm{JacobiSN}\left(z\&comma;k\right)}{\phantom{k=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{JacobiSN}\left(z\&comma;k\right)}}{k=1}=\tanh \left(z\right)\right]$

${\mathrm{JacobiSN}\left(z\&comma;k\right)}^2+{\mathrm{JacobiCN}\left(z\&comma;k\right)}^2$

${\mathrm{JacobiSN}\left(z\&comma;k\right)}^2+{\mathrm{JacobiCN}\left(z\&comma;k\right)}^2$

$\mathrm{simplify}\left(\right)$

$1$

$\mathrm{JacobiSN}\left(\mathrm{InverseJacobiAM}\left(z\&comma;k\right)\&comma;k\right)$

$\sin \left(z\right)$

$\mathrm{JacobiCN}\left(\mathrm{InverseJacobiAM}\left(z\&comma;k\right)\&comma;k\right)$

$\cos \left(z\right)$

$\mathrm{JacobiDN}\left(\mathrm{InverseJacobiAM}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{1-k^2{\sin \left(z\right)}^2}$

$\mathrm{JacobiSN}\left(\mathrm{InverseJacobiCN}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{-z^2+1}$

$\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{-z^2+1}$