JacobiNS

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

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

JacobiNS(u, m)the Jacobi elliptic function ns, argument u, parameter m.

SageMathjacobi_ns(u, m)mpmathellipfun('ns', u, m=m)Juliaenumeratio_jacobi(:ns, 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.
• $\operatorname{ns}(u,m) = 1/\operatorname{sn}(u,m)$ — JacobiSN 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).
• ns has a pole at u = 0 (sn(0,m) = 0, for any m).
• 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. At the pole $u=0$ the series in $u$ is the Laurent series, its coefficients polynomials in $m$.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

live ↗

In$N(\mathrm{JacobiNS}(0.3, 0.5))$Out$3.408\,168\,381\,792\,332\,3$

Known value · mpmath 1.3.0 ellipfun

Scope #

m less than 0, via the imaginary-modulus transformation

live ↗

In$N(\mathrm{JacobiNS}(0.3, -2))$Out$3.289\,220\,683\,752\,634\,8$

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

live ↗

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

The Laurent series at the pole u = 0, its coefficients polynomials in m

live ↗

In$\operatorname{Series}(\mathrm{JacobiNS}(u, m), u, 0, 4)$Out$\frac{1}{u}+u×(\frac{m}{6}+\frac{1}{6})+(\frac{7m^2}{360}-\frac{11m}{180}+\frac{7}{360})u^3+O\left(u^5\right)$

The series in the parameter m at m = 0, its coefficients trigonometric in z

live ↗

In$\operatorname{Series}(\mathrm{JacobiNS}(z, m), m, 0, 2)$Out$\frac{1}{\sin(z)}+\frac{m×(4z\cos(z)-\sin(z)-\sin(3z))}{16\sin(z)^2}+\frac{(24z^2+8\cos(2z)z^2-4z\sin(2z)+4z\sin(4z)+5\cos(4z)-5)m^2}{512\sin(z)^3}+O\left(m^3\right)$

Properties #

u = 0 is a pole, for any m

live ↗

In$\mathrm{JacobiNS}(0, m)$Out$\tilde∞$

live ↗

In$N(\mathrm{JacobiNS}(\frac{2^{1/2}}{4}i, -1))$Out$-2.832\,847\,693\,433\,667i$

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

live ↗

In$N(\mathrm{JacobiNS}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$1.329\,988\,245\,674\,117\,9$

Known value · Wolfram MathematicalFunctionData, JacobiNS, Gauss' transformation

live ↗

In$N(\mathrm{JacobiNS}(\sqrt{2}i, -1))$Out$-1.010\,704\,144\,622\,302\,2i$

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

live ↗

In$N(\mathrm{JacobiNS}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-4.057\,304\,661\,287\,208$

Known value · Wolfram MathematicalFunctionData, JacobiNS, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

Same gap as JacobiCD (the true value is tracked there, not repeated per head): u this far out has no argument reduction against its real period, so the guard in exceedsDoublePrecision's sibling (periodsExceedDouble, ce-patches/support/precise.ts) declines rather than return a wrong value.

live ↗

In$N(\mathrm{JacobiNS}(10^{16}, \frac{2}{3}))$Out$\mathrm{JacobiNS}(10\,000\,000\,000\,000\,000.0, 0.666\,666\,666\,666\,666\,666\,667)$

About

Examples	13
Checked against	Wolfram Language 12 agree1 differSageMath 8 agree2 differ2 unevaluated1 errormpmath 7 agree1 unevaluatedJulia 7 agree1 unevaluated
Crosswalk	Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: JacobiSN, JacobiSC, JacobiSD