InverseJacobiNC[v,m]
gives the inverse Jacobi elliptic function
.
InverseJacobiNC
InverseJacobiNC[v,m]
gives the inverse Jacobi elliptic function
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
gives the value of
for which
. - InverseJacobiNC has branch cut discontinuities in the complex
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, InverseJacobiNC automatically evaluates to exact values.
- InverseJacobiNC can be evaluated to arbitrary numerical precision.
- InverseJacobiNC automatically threads over lists.
Examples
open all close allBasic Examples (5)
InverseJacobiNC[2.5, 0.3]JacobiNC[%, 0.3]Plot the function over a subset of the reals:
Plot[InverseJacobiNC[x, 1 / 2], {x, 1, 3}]Plot over a subset of the complexes:
ComplexPlot3D[InverseJacobiCN[z, 1 / 3], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[InverseJacobiNC[z, m], {m, 0, 1}]Series expansion at Infinity:
Series[InverseJacobiNC[2, m], {m, ∞, 3}]//Normal//FullSimplifyScope (28)
Numerical Evaluation (5)
N[InverseJacobiNC[5 / 3, 3 / 5], 50]The precision of the output tracks the precision of the input:
InverseJacobiNC[1.6666666666666666666666666666666666666, 3 / 5]Evaluate for complex arguments:
InverseJacobiNC[0.2 - 0.3I, 0.1 + 0.5I]Evaluate InverseJacobiNC efficiently at high precision:
InverseJacobiNC[3, 0.5`500]//TimingInverseJacobiNC[3, 0.5`50000];//TimingCompute average-case statistical intervals using Around:
InverseJacobiNC[Around[2, 0.01], .2]Compute the elementwise values of an array:
InverseJacobiNC[{{Sqrt[2], 1}, {1, -Sqrt[2]}}, 0]Or compute the matrix InverseJacobiNC function using MatrixFunction:
MatrixFunction[InverseJacobiNC[#, 0]&, {{Sqrt[2], 1}, {1, -Sqrt[2]}}]//FullSimplifySpecific Values (3)
Simple exact values are generated automatically:
InverseJacobiNC[1, m]{InverseJacobiNC[ν, 0], InverseJacobiNC[ν, 1]}InverseJacobiNC[ν, Infinity]Find a real root of the equation
:
f[x_] := InverseJacobiNC[x, 1 / 2] - 1;
xzero = Solve[f[x] == 0 && 1.0 < x < 2.0, x][[1, 1, 2]]//QuietPlot[f[ν], {ν, 0, 2}, Epilog -> Style[Point[{xzero, f[xzero]}], PointSize[Large], Red]]Visualization (3)
Plot InverseJacobiNC for various values of the second parameter
:
Plot[{InverseJacobiNC[ν, -2], InverseJacobiNC[ν, 0], InverseJacobiNC[ν, 1 / 2], InverseJacobiNC[ν, 2]}, {ν, -8, 8}]Plot InverseJacobiNC as a function of its parameter
:
Plot[{InverseJacobiNC[6 / 5, m], InverseJacobiNC[2, m], InverseJacobiNC[3, m]}, {m, -3, 3}]ComplexContourPlot[Re[InverseJacobiNC[3, z]], {z, -3 - 3I, 3 + 3I}, IconizedObject[«PlotOptions»]]ComplexContourPlot[Im[InverseJacobiNC[3, z]], {z, -3 - 3I, 3 + 3I}, IconizedObject[«PlotOptions»]]Function Properties (6)
InverseJacobiNC is not an analytic function:
FunctionAnalytic[InverseJacobiNC[x, m], {x, m}]It has both singularities and discontinuities:
FunctionSingularities[InverseJacobiNC[x, 1], x]//QuietFunctionDiscontinuities[InverseJacobiNC[x, 1], x]//Quiet
is nondecreasing on its real domain:
FunctionMonotonicity[{InverseJacobiNC[x, 1], x > 1}, x]FunctionInjective[InverseJacobiNC[x, 1], x]Plot[{InverseJacobiNC[x, 1], .5}, {x, -3, 3}]FunctionSurjective[InverseJacobiNC[x, 1], x]Plot[{InverseJacobiNC[x, 1], -1}, {x, -5, 5}]
is non-negative on its real domain:
FunctionSign[{InverseJacobiNC[x, 1], x > 1}, x]
is concave on its real domain:
FunctionConvexity[{InverseJacobiNC[x, 1], x > 1}, x]Differentiation (4)
D[InverseJacobiNC[ν, m], ν]derivs = Table[D[InverseJacobiNC[ν, m], {ν, n}], {n, 1, 3}]//SimplifyPlot[Evaluate[derivs /. m -> 1 / 2], {ν, 0, 2}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Differentiate InverseJacobiNC with respect to the second argument
:
D[InverseJacobiNC[ν, m], m]derivs = Table[D[InverseJacobiNC[ν, m], {m, n}], {n, 1, 3}];
Plot[Evaluate[derivs /. ν -> 2], {m, 0, 1.5}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}, WorkingPrecision -> 35]Series Expansions (3)
Series[InverseJacobiNC[ν, m], {ν, 2, 3}]Plot the first three approximations for
around
:
terms = Normal@Table[Series[InverseJacobiNC[ν, 1 / 2], {ν, 2, n}], {n, 1, 3}];
Plot[{InverseJacobiNC[ν, 1 / 2], terms}, {ν, 1, 4}]Series[InverseJacobiNC[ν, m], {m, 0, 2}]//FullSimplifyPlot the first three approximations for
around
:
terms = Normal@Table[Series[InverseJacobiNC[2, m], {m, 0, n}], {n, 1, 3}];
Plot[{InverseJacobiNC[2, m], terms}, {m, -2, 2}]InverseJacobiNC can be applied to a power series:
InverseJacobiNC[ν, Log[1 + m] + O[m] ^ 3]//FullSimplifyFunction Identities and Simplifications (2)
InverseJacobiNC is the inverse function of JacobiNC:
Solve[ν == JacobiNC[u, m], u][[1, 1]]//QuietCompose with inverse function:
{InverseJacobiNC[JacobiNC[ν, m], m], JacobiNC[InverseJacobiNC[ν, m], m]}Use PowerExpand to disregard multivaluedness of the inverse function:
PowerExpand[%]Other Features (2)
InverseJacobiNC threads elementwise over lists:
InverseJacobiNC[{1.1, 1.2}, 0.5]TraditionalForm formatting:
InverseJacobiNC[ν, m]//TraditionalFormApplications (1)
Properties & Relations (1)
Obtain InverseJacobiNC from solving equations containing elliptic functions:
Solve[JacobiNC[x, m]^2 + 2JacobiNC[x, m] == a, x]Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0)
Text
Wolfram Research (1988), InverseJacobiNC, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseJacobiNC.html.
CMS
Wolfram Language. 1988. "InverseJacobiNC." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseJacobiNC.html.
APA
Wolfram Language. (1988). InverseJacobiNC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseJacobiNC.html
BibTeX
@misc{reference.wolfram_2026_inversejacobinc, author="Wolfram Research", title="{InverseJacobiNC}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/InverseJacobiNC.html}", note=[Accessed: 12-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_inversejacobinc, organization={Wolfram Research}, title={InverseJacobiNC}, year={1988}, url={https://reference.wolfram.com/language/ref/InverseJacobiNC.html}, note=[Accessed: 12-June-2026]}