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InverseJacobiAM

The inverse of the Jacobi amplitude function *am*

InverseJacobiSN, ..., InverseJacobiDC

The inverses of the Jacobi elliptic functions *sn*, ..., *dc*

 Calling Sequence InverseJacobiAM(z, k) InverseJacobiCD(z, k),    InverseJacobiCN(z, k),    InverseJacobiCS(z, k), InverseJacobiDC(z, k),    InverseJacobiDN(z, k),    InverseJacobiDS(z, k), InverseJacobiNC(z, k),    InverseJacobiND(z, k),    InverseJacobiNS(z, k), InverseJacobiSC(z, k),    InverseJacobiSD(z, k),    InverseJacobiSN(z, k)

Parameters

 z - algebraic expression k - algebraic expression, the modulus of the elliptic function

Description

 • The InverseJacobiAM and the twelve InverseJacobiPQ functions, where P and Q are any two of {C,D,N,S}, are all elliptic integrals related to the incomplete elliptic integral of the first kind EllipticF. The exact definition for each InverseJacobiPQ can be seen, for instance, via:
 $\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{{∫}}_{{0}}^{{\mathrm{φ}}}\frac{{1}}{\sqrt{{1}{-}{{k}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{_θ1}}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_θ1}}{,}{\mathrm{with no restrictions on}}{}\left({\mathrm{φ}}{,}{k}\right)\right]$ (1)
 $\left[{\mathrm{InverseJacobiSN}}{}\left({z}{,}{k}\right){=}{{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (2)
 From these definitions, InverseJacobiAM represents the trigonometric form of the incomplete Elliptic integral of the first kind (A&S 16.1.3, 17.2.6; G&R, 8.111, 8.141 - see References section) and InverseJacobiSN equals EllipticF,
 $\left[{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (3)
 EllipticF in turn is the Legendre normal form of the incomplete Elliptic integral of the first kind, (A&S 17.2.7; G&R 8.111). Note however that, unlike in those formulas, in the definition above the square root in the denominator of the integrand is split. The general relation between the Legendre and the trigonometric form of the incomplete Elliptic integral of the first kind, that is, between $\mathrm{InverseJacobiAM}\left(\mathrm{phi},k\right)$ and $\mathrm{EllipticF}\left(z,k\right)$, as well as the restrictions on the parameters such that the relations are valid are given by:
 $\left[{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{arcsin}}{}\left({z}\right){,}{k}\right){,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (4)
 $\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{ℜ}}{}\left({\mathrm{φ}}\right)}{{\mathrm{π}}}\right){+}{\mathrm{φ}}\right){,}{k}\right){-}{2}{}{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{ℜ}}{}\left({\mathrm{φ}}\right)}{{\mathrm{π}}}\right){}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{with no restrictions on}}{}\left({\mathrm{φ}}{,}{k}\right)\right]{,}\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({\mathrm{sin}}{}\left({\mathrm{φ}}\right){,}{k}\right){,}{-}\frac{{\mathrm{\pi }}}{{2}}{<}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right){<}\frac{{\mathrm{\pi }}}{{2}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{-}\frac{{\mathrm{\pi }}}{{2}}{=}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{\le }{\mathrm{\Im }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\frac{{\mathrm{\pi }}}{{2}}{=}{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\Im }}{}\left({\mathrm{\phi }}\right){\le }{0}\right]$ (5)
 • With some restrictions on the values of the function parameters, InverseJacobiAM is the inverse of the amplitude JacobiAM function:
 $\left[{z}{=}{\mathrm{JacobiAM}}{}\left({{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{1}{-}{{k}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}{,}{k}\right){,}{z}{::}\left({\mathrm{RealRange}}{}\left({-}\frac{{3}}{{2}}{,}\frac{{3}}{{2}}\right)\right)\right]$ (6)
 where in above the integral is equal to InverseJacobiAM(z,k). For all the other InverseJacobiPQ, where P and Q are any two of {C, D, N, S}, the InverseJacobiPQ function is the exact inverse of the corresponding JacobiPQ function for all values of the function parameters, so for instance,
 > JacobiSN( InverseJacobiSN(z,k), k);
 ${z}$ (7)
 > JacobiCN( InverseJacobiCN(z,k), k);
 ${z}$ (8)
 > JacobiDN( InverseJacobiDN(z,k), k);
 ${z}$ (9)
 • All InverseJacobiPQ (excluding InverseJacobiAM) satisfy $\mathrm{InverseJacobiPQ}\left(z,k\right)=\mathrm{InverseJacobiQP}\left(\frac{1}{z},k\right)$ For k = infinity, all the InverseJacobiPQ functions have the value zero, independent of the value of the first parameter z. For z = infinity, however, most of these functions have finite values (see below).

Examples

Reflection symmetry and special values for InverseJacobiAM and InverseJacobiSN:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{InverseJacobiAM}\right)$
 $\left[{\mathrm{InverseJacobiAM}}{}\left({-}{\mathrm{φ}}{,}{k}\right){=}{-}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){,}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{-}{k}\right){=}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){,}{\mathrm{InverseJacobiAM}}{}\left({0}{,}{k}\right){=}{0}{,}{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{0}\right){=}{\mathrm{φ}}{,}\left[{\mathrm{InverseJacobiAM}}{}\left(\frac{{1}}{{2}}{}{n}{}{\mathrm{π}}{,}{k}\right){=}{n}{}{\mathrm{EllipticK}}{}\left({k}\right){,}{n}{::}{\mathrm{integer}}\right]{,}\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{\mathrm{ln}}{}\left({\mathrm{sec}}{}\left({\mathrm{φ}}\right){+}{\mathrm{tan}}{}\left({\mathrm{φ}}\right)\right){,}\left|{\mathrm{\Re }}{}\left({\mathrm{\phi }}\right)\right|{<}\frac{{\mathrm{\pi }}}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{k}{\in }\left\{{-1}{,}{1}\right\}\right]{,}\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){=}{0}{,}{k}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left\{{\mathrm{∞}}{,}{-}{\mathrm{∞}}\right\}\right]{,}\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{∞}}{}{I}{,}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){-}\frac{{\mathrm{EllipticK}}{}\left(\frac{{1}}{{k}}\right)}{\sqrt{{k}}}{,}{{k}}^{{2}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{RealRange}}{}\left({\mathrm{Open}}{}\left({0}\right){,}{\mathrm{Open}}{}\left({1}\right)\right)\right]\right]$ (10)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{InverseJacobiSN}\right)$
 $\left[{\mathrm{InverseJacobiSN}}{}\left({0}{,}{k}\right){=}{0}{,}{\mathrm{InverseJacobiSN}}{}\left({1}{,}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{InverseJacobiSN}}{}\left({z}{,}{0}\right){=}{\mathrm{arcsin}}{}\left({z}\right){,}{\mathrm{InverseJacobiSN}}{}\left({z}{,}{1}\right){=}{\mathrm{arctanh}}{}\left({z}\right){,}{\mathrm{InverseJacobiSN}}{}\left({z}{,}{\mathrm{∞}}\right){=}{0}{,}\left[{\mathrm{InverseJacobiSN}}{}\left({\mathrm{∞}}{,}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){-}\frac{{\mathrm{EllipticK}}{}\left(\frac{{1}}{{k}}\right)}{{k}}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{1}{<}{k}\right]\right]$ (11)

Apart from InverseJacobiAM and InverseJacobiSN, none of the other InverseJacobiPQ functions have reflection symmetry with respect to z.

Branch points for InverseJacobiAM:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_points},\mathrm{InverseJacobiAM}\right)$
 $\left[{\mathrm{InverseJacobiAM}}{}\left({\mathrm{φ}}{,}{k}\right){,}{\mathrm{arcsin}}{}\left(\frac{{1}}{{k}}\right){+}{2}{}{\mathrm{\pi }}{}{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{integer}}{,}{\mathrm{φ}}{=}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}\right]$ (12)

Compositions between JacobiPQ and InverseJacobiPQ functions typically lead to elementary forms.

 > $\mathrm{JacobiSN}\left(\mathrm{InverseJacobiAM}\left(z,k\right),k\right)$
 ${\mathrm{sin}}{}\left({z}\right)$ (13)
 > $\mathrm{JacobiCN}\left(\mathrm{InverseJacobiAM}\left(z,k\right),k\right)$
 ${\mathrm{cos}}{}\left({z}\right)$ (14)
 > $\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(z,k\right),k\right)$
 $\sqrt{{-}{{z}}^{{2}}{+}{1}}$ (15)

In the case of (Inverse)JacobiCN and (Inverse)JacobiSN, the two possible compositions are equal.

 > $\mathrm{JacobiSN}\left(\mathrm{InverseJacobiCN}\left(z,k\right),k\right)$
 $\sqrt{{-}{{z}}^{{2}}{+}{1}}$ (16)
 > $\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(z,k\right),k\right)$
 $\sqrt{{-}{{z}}^{{2}}{+}{1}}$ (17)

Due to the large number of relationships between all the JacobiPQ functions, the InverseJacobiPQ functions also feature a large number of them. Typically, all of them can be expressed in terms of InverseJacobiAM and InverseJacobiSN, the two forms of the incomplete Elliptic integral of the first kind.

 > $\mathrm{InverseJacobiDN}\left(z,k\right)=\mathrm{convert}\left(\mathrm{InverseJacobiDN}\left(z,k\right),\mathrm{InverseJacobiAM}\right)$
 ${\mathrm{InverseJacobiDN}}{}\left({z}{,}{k}\right){=}\frac{{\mathrm{InverseJacobiAM}}{}\left({\mathrm{arccos}}{}\left({z}\right){,}\frac{{1}}{{k}}\right)}{{k}}$ (18)
 > $\mathrm{InverseJacobiCD}\left(z,k\right)=\mathrm{convert}\left(\mathrm{InverseJacobiCD}\left(z,k\right),\mathrm{InverseJacobiSN}\right)$
 ${\mathrm{InverseJacobiCD}}{}\left({z}{,}{k}\right){=}{\mathrm{EllipticK}}{}\left({k}\right){-}{\mathrm{InverseJacobiSN}}{}\left({z}{,}{k}\right)$ (19)
 > 

References

 [A&S] Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.
 [G&R] Gradshteyn, and Ryzhik. Table of Integrals Series and Products. 5th ed. Academic Press.