JacobiNC

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

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

JacobiNC(u, m)the Jacobi elliptic function nc, argument u, parameter m.

SageMathjacobi_nc(u, m)mpmathellipfun('nc', u, m=m)Juliaenumeratio_jacobi(:nc, 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.
• $\operatorname{nc}(u,m) = 1/\operatorname{cn}(u,m)$ — JacobiCN 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).
• nc(0,m) = 1, for any m; nc has a pole at the quarter period u = EllipticK(m) (cn = 0 there).
• 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$N(\mathrm{JacobiNC}(0.3, 1.5))$Out$1.044\,645\,321\,516\,758\,4$

Known value · mpmath 1.3.0 ellipfun

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In$\operatorname{Series}(\mathrm{JacobiNC}(z, m), z, 0, 7)$Out$1+\frac{z^2}{2}+\frac{1}{24}(5-4m)z^4+\frac{1}{720}(16m^2-76m+61)z^6+O\left(z^8\right)$

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

Scope #

m less than 0, via the imaginary-modulus transformation

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In$N(\mathrm{JacobiNC}(0.3, -2))$Out$1.049\,687\,404\,895\,594\,6$

K(0) = π/2, where cn is 0: the quarter-period pole holds at m = 0 too, not just cos(K(0)) left symbolic

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In$\mathrm{JacobiNC}(\mathrm{EllipticK}(0), 0)$Out$\tilde∞$

Known value · Wolfram MathematicalFunctionData, JacobiNC

At $m=1$, nc is cosh, giving $\frac12$ at $-i\pi/3$

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

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In$N(\mathrm{JacobiNC}(2, \frac{1}{3}), 50)$Out$-4.621\,534\,048\,480\,796\,847\,474\,129\,165\,488\,820\,572\,291\,436\,878\,365\,7$

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

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In$\bigl\lbrack\mathrm{JacobiNC}(\mathrm{EllipticK}(m), m), \mathrm{JacobiNC}(3\mathrm{EllipticK}(m), m), \mathrm{JacobiNC}(2i\mathrm{EllipticK}(1-m)+\mathrm{EllipticK}(m), m)\bigr\rbrack$Out$\bigl\lbrack\tilde∞, \mathrm{JacobiNC}(3\mathrm{EllipticK}(m), m), \mathrm{JacobiNC}(2i\mathrm{EllipticK}(1-m)+\mathrm{EllipticK}(m), m)\bigr\rbrack$

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In$\operatorname{Series}(\mathrm{JacobiNC}(x, \frac{1}{3}), x, 0, 7)$Out$1+\frac{x^2}{2}+\frac{11x^4}{72}+\frac{337x^6}{6\,480}+O\left(x^8\right)$

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In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{JacobiNC}(1, m), m, 0, 3))$Out$\frac{1}{\cos(1)}+\frac{m×(-4\sin(1)-\cos(3)+\cos(1))}{16\cos(1)^2}+\frac{(35-44\sin(2)-11\cos(4)-8\cos(2)-4\sin(4))m^2}{512\cos(1)^3}+\frac{(-2\,416\sin(1)-1\,840\sin(3)-633\cos(3)-387\cos(5)-192\sin(5)+3\cos(7)+2\,745\cos(1))m^3}{49\,152\cos(1)^4}+O\left(m^4\right)$

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

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

Properties #

u = 0 is 1, for any m

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

nc(0) is 1

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

Known value · Wolfram MathematicalFunctionData, JacobiNC

At $m=0$, $\mathrm{nc}=\sec$, a pole at $\pi/2$

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In$\mathrm{JacobiNC}(\frac{π}{2}, 0)$Out$\tilde∞$

Known value · Wolfram MathematicalFunctionData, JacobiNC

nc(0) is 1 at $m=1$ too

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

Known value · Wolfram MathematicalFunctionData, JacobiNC

At $m=1$ cosh vanishes at $i\pi/2$

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

Known value · Wolfram MathematicalFunctionData, JacobiNC

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In$N(\mathrm{JacobiNC}(\frac{2^{1/2}}{4}i, 2))$Out$0.935\,622\,687\,124\,812\,2$

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

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In$N(\mathrm{JacobiNC}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$1.516\,776\,888\,094\,937$

Known value · Wolfram MathematicalFunctionData, JacobiNC, Gauss' transformation

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In$N(\mathrm{JacobiNC}(\sqrt{2}i, 2))$Out$-0.145\,153\,002\,614\,799\,13$

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

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In$N(\mathrm{JacobiNC}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-1.031\,831\,310\,280\,545\,4$

Known value · Wolfram MathematicalFunctionData, JacobiNC, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

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

About

Examples	23
Checked against	Wolfram Language 21 agree2 differSageMath 17 agree3 differ2 errormpmath 10 agree2 unevaluatedJulia 9 agree3 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: JacobiCN, JacobiCD, JacobiDC