InverseJacobiND[v,m]
gives the inverse Jacobi elliptic function
.
InverseJacobiND
InverseJacobiND[v,m]
gives the inverse Jacobi elliptic function
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of
for which
. - InverseJacobiND has branch cut discontinuities in the complex v plane with branch points at
and infinity, and in the complex m plane with branch points at
and infinity. - The inverse Jacobi elliptic functions are related to elliptic integrals.
- For certain special arguments, InverseJacobiND automatically evaluates to exact values.
- InverseJacobiND can be evaluated to arbitrary numerical precision.
- InverseJacobiND automatically threads over lists.
Examples
open all close allBasic Examples (5)
InverseJacobiND[0.5, 0.3]JacobiND[%, 0.3]Plot the function over a subset of the reals:
Plot[InverseJacobiND[x, 3], {x, 1, 5}]Plot the imaginary part of the function:
Plot[Im[InverseJacobiND[x, 1 / 3]], {x, -5, 5}]Plot over a subset of the complexes:
ComplexPlot3D[InverseJacobiND[z, 1 / 3], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansions at the origin:
Series[InverseJacobiND[z, m], {z, 0, 4}]Series[InverseJacobiND[z, m], {m, 0, 1}]Series expansion at Infinity:
Series[InverseJacobiND[2, m], {m, ∞, 3}]//Normal//FullSimplifyScope (28)
Numerical Evaluation (5)
N[InverseJacobiND[1 / 3, 1 / 2], 50]The precision of the input tracks the precision of the output:
InverseJacobiND[1 / 3, 0.50000000000000000000000000000000]Evaluate for complex arguments:
InverseJacobiND[0.2 + I, 0.1 - 0.3I]Evaluate InverseJacobiND efficiently at high precision:
InverseJacobiND[3, 1.3`500]//TimingInverseJacobiND[3, 1.3`10000];//TimingCompute average-case statistical intervals using Around:
InverseJacobiND[Around[3, 0.01], 2]Compute the elementwise values of an array:
InverseJacobiND[{{-1, 1}, {1, -1}}, 1]Or compute the matrix InverseJacobiND function using MatrixFunction:
MatrixFunction[InverseJacobiND[#, 1]&, {{-1, 1}, {1, 1}}]//FullSimplifySpecific Values (3)
Simple exact results are generated automatically:
InverseJacobiND[ν, 1]{InverseJacobiND[-1, m], InverseJacobiND[1, m]}InverseJacobiND[ν, Infinity]InverseJacobiND[Infinity, Infinity]Find a real root of the equation
:
f[x_] := InverseJacobiND[x, 2] - 1;
xzero = FindRoot[f[x] == 0, {x, 3.5}][[1, 2]]//ChopPlot[f[ν], {ν, 0, 4}, WorkingPrecision -> 10, Epilog -> Style[Point[{xzero, f[xzero]}], PointSize[Large], Red]]Visualization (3)
Plot InverseJacobiND for various values of the second parameter
:
Plot[{InverseJacobiND[ν, -2], InverseJacobiND[ν, 1], InverseJacobiND[ν, 1 / 2], InverseJacobiND[ν, 2]}, {ν, 0, 5}]Plot InverseJacobiND as a function of its parameter
:
Plot[{InverseJacobiND[7 / 10, m], InverseJacobiND[6 / 5, m], InverseJacobiND[3, m]}, {m, -10, 10}]ComplexContourPlot[Re[InverseJacobiND[z, 1 / 2]], {z, -5 - 5I, 5 + 5I}, IconizedObject[«PlotOptions»]]ComplexContourPlot[Im[InverseJacobiND[z, 1 / 2]], {z, -5 - 5I, 5 + 5I}, IconizedObject[«PlotOptions»]]Function Properties (6)
InverseJacobiND is not an analytic function:
FunctionAnalytic[InverseJacobiND[x, m], {x, m}]It has both singularities and discontinuities:
FunctionSingularities[InverseJacobiND[x, 1], x]//QuietFunctionDiscontinuities[InverseJacobiND[x, 1], x]//Quiet
is nondecreasing on its real domain:
FunctionMonotonicity[{InverseJacobiND[x, 1], x > 1}, x]FunctionInjective[InverseJacobiND[x, 1], x]Plot[{InverseJacobiND[x, 1], .3}, {x, -5, 5}]FunctionSurjective[InverseJacobiND[x, 1], x]Plot[{InverseJacobiND[x, 1], -1}, {x, -5, 5}]
is non-negative on its real domain:
FunctionSign[{InverseJacobiND[x, 1], x > 1}, x]
is concave on its real domain:
FunctionConvexity[{InverseJacobiND[x, 1], x > 1}, x]Differentiation (4)
D[InverseJacobiND[ν, m], ν]derivs = Table[D[InverseJacobiND[ν, m], {ν, n}], {n, 1, 3}]//SimplifyPlot[Evaluate[derivs /. m -> 2], {ν, 0, 2}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Differentiate InverseJacobiND with respect to the second argument
:
D[InverseJacobiND[ν, m], m]derivs = Table[D[InverseJacobiND[ν, m], {m, n}], {n, 1, 3}];
Plot[Evaluate[derivs /. ν -> 3], {m, 0, 2}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Series Expansions (3)
Series[InverseJacobiND[ν, m], {ν, 2, 3}]Plot the first three approximations for
around
:
terms = Normal@Table[Series[InverseJacobiND[ν, 2], {ν, 2, n}], {n, 1, 3}];
Plot[{InverseJacobiND[ν, 2], terms}, {ν, 1, 4}]Series[InverseJacobiND[ν, m], {m, 0, 2}]//SimplifyInverseJacobiND can be applied to a power series:
InverseJacobiND[x, Exp[m] + O[m] ^ 2]Function Identities and Simplifications (2)
InverseJacobiND is the inverse function of JacobiND:
Solve[ν == JacobiND[u, m], u][[1, 1]]//QuietCompose with inverse function:
{InverseJacobiND[JacobiND[ν, m], m], JacobiND[InverseJacobiND[ν, m], m]}Use PowerExpand to disregard multivaluedness of the inverse function:
PowerExpand[%]Other Features (2)
InverseJacobiND threads element‐wise over lists:
InverseJacobiND[{ν1, ν2}, m]TraditionalForm formatting:
InverseJacobiND[ν, m]//TraditionalFormApplications (1)
Properties & Relations (1)
Obtain InverseJacobiND from solving equations containing elliptic functions:
Solve[JacobiND[x, m]^2 + 2JacobiND[x, m] == a, x]See Also
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0)
Text
Wolfram Research (1988), InverseJacobiND, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseJacobiND.html.
CMS
Wolfram Language. 1988. "InverseJacobiND." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseJacobiND.html.
APA
Wolfram Language. (1988). InverseJacobiND. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseJacobiND.html
BibTeX
@misc{reference.wolfram_2026_inversejacobind, author="Wolfram Research", title="{InverseJacobiND}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiND.html}", note=[Accessed: 12-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_inversejacobind, organization={Wolfram Research}, title={InverseJacobiND}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiND.html}, note=[Accessed: 12-June-2026]}