JacobiND

The Jacobi elliptic function $\operatorname{nd}(u,m) = 1/\operatorname{dn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageJacobiND!↗

JacobiND(u, m)the Jacobi elliptic function nd, argument u, parameter m.

SageMathjacobi_nd(u, m)mpmathellipfun('nd', u, m=m)Juliaenumeratio_jacobi(:nd, u, m)

Typed signatures

Type	Package	Applies
(number, number) -> number	analytic	always

Domain: Special functions

Details

• $m = k^2$, the same convention EllipticK / EllipticF use.
• $\operatorname{nd}(u,m) = 1/\operatorname{dn}(u,m)$ — JacobiDN shares this head's numeric kernel and exact-value coverage (m outside [0,1] via the reciprocal- and imaginary-modulus transformations, complex u, a genuinely complex m declined).
• nd(0,m) = 1, for any m; nd(u,0) = 1, for any u (dn ≡ 1 at m = 0).
• Numeric only — a symbolic argument stays unevaluated; a floating-point argument (or N()) evaluates directly.
• N(x, d) past a double's digits runs the same AGM in BigDecimal for real $u$ and $0 < m < 1$; complex $u$ and other $m$ stay unevaluated there rather than print a double's digits as more.
• Derivatives of every order in $u$ and in $m$ are closed forms in sn, cn, dn and the Jacobi epsilon $\mathcal{E}(u,m)=Z(u,m)+u\,E(m)/K(m)$ (DLMF 22.13): D, Series in $u$ at $0$ or at a symbolic point, and Series in $m$ at $m=0$ all give concrete coefficients. The $m$-derivatives evaluate numerically for real $u$ and $m$ between 0 and 1.

Plot

Examples

Basic #

A floating-point argument evaluates directly to a decimal

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In$N(\mathrm{JacobiND}(0.3, -2))$Out$0.918\,684\,525\,634\,504\,4$

Known value · mpmath 1.3.0 ellipfun

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In$\operatorname{Series}(\mathrm{JacobiND}(z, m), z, 0, 6)$Out$1+0mz+\frac{mz^2}{2}+\frac{1}{24}(5m^2-4m)z^4+\frac{1}{720}(61m^3-76m^2+16m)z^6+O\left(z^8\right)$

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In$\operatorname{Series}(\mathrm{JacobiND}(z, m), m, 0, 2)$Out$1+m×(\frac{1}{4}-\frac{\cos(2z)}{4})+(-\frac{1}{8}z\sin(2z)-\frac{1}{16}×3\cos(2z)+\frac{\cos(4z)}{64}+\frac{11}{64})m^2+O\left(m^3\right)$

Scope #

m greater than 1, via the reciprocal-modulus transformation

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In$N(\mathrm{JacobiND}(0.3, 1.5))$Out$1.069\,333\,535\,737\,140\,7$

nd is 1 at $m=0$ and $\cosh$ at $m=1$

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In$\bigl\lbrack\mathrm{JacobiND}(z, 0), \mathrm{JacobiND}(z, 1)\bigr\rbrack$Out$\bigl\lbrack1, \cosh(z)\bigr\rbrack$

At $m=1$, nd is cosh, giving $\frac12$

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{JacobiND}((-(1/3)i)π, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack$

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In$\bigl\lbrack\mathrm{JacobiND}(0, m), \mathrm{JacobiND}(\frac{\mathrm{EllipticK}(m)}{2}, m), \mathrm{JacobiND}(\mathrm{EllipticK}(m), m)\bigr\rbrack$Out$\bigl\lbrack1, \mathrm{JacobiND}(\frac{\mathrm{EllipticK}(m)}{2}, m), \frac{1}{\sqrt{1-m}}\bigr\rbrack$

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In$\operatorname{Series}(\mathrm{JacobiND}(x, \frac{1}{3}), x, 0, 7)$Out$1+\frac{x^2}{6}-\frac{7x^4}{216}-\frac{23x^6}{19\,440}+O\left(x^8\right)$

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In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{JacobiND}(1, m), m, 0, 3))$Out$1+\frac{1}{4}m×(1-\cos(2))+\frac{1}{64}(11-12\cos(2)-8\sin(2)+\cos(4))m^2+\frac{1}{1\,024}(140-136\sin(2)-127\cos(2)+16\sin(4)+20\cos(4)-\cos(6))m^3+O\left(m^4\right)$

Fifty digits of $\operatorname{nd}(3\mid 1/3)$ from the arbitrary-precision AGM

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In$N(\mathrm{JacobiND}(3, \frac{1}{3}), 50)$Out$1.034\,914\,869\,016\,700\,053\,096\,935\,478\,635\,939\,068\,417\,796\,527\,858\,7$

The derivative in the parameter m, in closed form through the Jacobi zeta

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In$\frac{\mathrm{d}\mathrm{JacobiND}(u, m)}{\mathrm{d}m}$Out$-\frac{(\frac{u\mathrm{EllipticE}(m)}{\mathrm{EllipticK}(m)}+\mathrm{JacobiZN}(u, m))\mathrm{JacobiSN}(u, m)\mathrm{JacobiCN}(u, m)}{2(1-m)\mathrm{JacobiDN}(u, m)^2}+\frac{u\mathrm{JacobiSN}(u, m)\mathrm{JacobiCN}(u, m)}{2\mathrm{JacobiDN}(u, m)^2}+\frac{\mathrm{JacobiSN}(u, m)^2}{2(1-m)\mathrm{JacobiDN}(u, m)}$

Properties #

u = 0 is 1, for any m

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In$\mathrm{JacobiND}(0, m)$Out$1$

nd(0) is 1

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In$\mathrm{JacobiND}(0, 1)$Out$1$

Known value · Wolfram MathematicalFunctionData, JacobiND

At $m=1$, $\cosh(i\pi/2)=0$

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In$\mathrm{JacobiND}(\frac{1}{2}i π, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, JacobiND

The same with a larger imaginary period

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In$\mathrm{JacobiND}(3i\mathrm{EllipticK}(1)+\mathrm{EllipticK}(0), 0)$Out$1$

Known value · Wolfram MathematicalFunctionData, JacobiND

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In$N(\mathrm{JacobiND}(\frac{2^{1/2}}{4}i, -1))$Out$1.068\,806\,917\,319\,438\,6$

Known value · Wolfram MathematicalFunctionData, JacobiND, Landen's transformation

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In$N(\mathrm{JacobiND}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$1.008\,426\,244\,269\,783\,6$

Known value · Wolfram MathematicalFunctionData, JacobiND, Gauss' transformation

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In$N(\mathrm{JacobiND}(\sqrt{2}i, -1))$Out$-6.889\,282\,219\,354\,136$

Known value · Wolfram MathematicalFunctionData, JacobiND, Landen's transformation

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In$N(\mathrm{JacobiND}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$1.000\,895\,313\,149\,575$

Known value · Wolfram MathematicalFunctionData, JacobiND, Gauss' transformation

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In$\mathrm{JacobiND}(i\mathrm{EllipticK}(1)+\mathrm{EllipticK}(0), 0)$Out$1$

Known value · Wolfram MathematicalFunctionData, JacobiND

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

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In$\mathrm{JacobiND}(u, m)$Out$\mathrm{JacobiND}(u, m)$

About

Examples	21
Checked against	Wolfram Language 20 agree1 differSageMath 16 agree2 differ2 errormpmath 9 agree2 unevaluatedJulia 10 agree1 error
Crosswalk	Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no Jacobi elliptic functions: sympy.jacobi is the Jacobi polynomial P_n^(a,b), not sn/cn/dn, and there is no ellipfun (mpmath, Sage and Nemo carry them).

See also: JacobiDN, JacobiDC, JacobiDS