JacobiSD

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

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

JacobiSD(u, m)the Jacobi elliptic function sd, argument u, parameter m.

SageMathjacobi_sd(u, m)mpmathellipfun('sd', u, m=m)Juliaenumeratio_jacobi(:sd, 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.
• $\operatorname{sd}(u,m) = \operatorname{sn}(u,m)/\operatorname{dn}(u,m)$ — JacobiSN and JacobiDN share 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).
• sd(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.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

live ↗

In$N(\mathrm{JacobiSD}(0.3, -2))$Out$0.279\,301\,577\,474\,694\,6$

Known value · mpmath 1.3.0 ellipfun

Scope #

m close to its m = 1 boundary

live ↗

In$N(\mathrm{JacobiSD}(1.1, 0.9))$Out$1.267\,821\,766\,920\,661\,5$

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

live ↗

In$\frac{\mathrm{d}\mathrm{JacobiSD}(u, m)}{\mathrm{d}m}$Out$-\frac{(\frac{u\mathrm{EllipticE}(m)}{\mathrm{EllipticK}(m)}+\mathrm{JacobiZN}(u, m))\mathrm{JacobiCN}(u, m)}{2m×(1-m)}+\frac{u\mathrm{JacobiCN}(u, m)}{2m}-\frac{(\frac{u\mathrm{EllipticE}(m)}{\mathrm{EllipticK}(m)}+\mathrm{JacobiZN}(u, m))\mathrm{JacobiCN}(u, m)\mathrm{JacobiSN}(u, m)^2}{2(1-m)\mathrm{JacobiDN}(u, m)^2}+\frac{u\mathrm{JacobiCN}(u, m)\mathrm{JacobiSN}(u, m)^2}{2\mathrm{JacobiDN}(u, m)^2}+\frac{\mathrm{JacobiSN}(u, m)^3}{2(1-m)\mathrm{JacobiDN}(u, m)}+\frac{\mathrm{JacobiSN}(u, m)\mathrm{JacobiCN}(u, m)^2}{2(1-m)\mathrm{JacobiDN}(u, m)}$

The Taylor series at u = 0, its coefficients polynomials in m

live ↗

In$\operatorname{Series}(\mathrm{JacobiSD}(u, m), u, 0, 5)$Out$u+\frac{1}{6}(2m-1)u^3+\frac{1}{120}(16m^2-16m+1)u^5+O\left(u^7\right)$

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

live ↗

In$\operatorname{Series}(\mathrm{JacobiSD}(z, m), m, 0, 2)$Out$\sin(z)+m×(-\frac{z\cos(z)}{4}-\frac{\sin(3z)}{16}+\frac{7\sin(z)}{16})+(-\frac{1}{32}\sin(z)z^2-\frac{3z\cos(z)}{16}+\frac{1}{64}×3z\cos(3z)-\frac{\sin(3z)}{16}+\frac{\sin(5z)}{256}+\frac{79\sin(z)}{256})m^2+O\left(m^3\right)$

Properties #

u = 0 is 0, for any m

live ↗

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

live ↗

In$N(\mathrm{JacobiSD}(\frac{2^{1/2}}{4}i, -1))$Out$0.377\,290\,639\,308\,585i$

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

live ↗

In$N(\mathrm{JacobiSD}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$0.758\,221\,922\,298\,758\,8$

Known value · Wolfram MathematicalFunctionData, JacobiSD, Gauss' transformation

live ↗

In$N(\mathrm{JacobiSD}(\sqrt{2}i, -1))$Out$-6.816\,319\,351\,226\,808i$

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

live ↗

In$N(\mathrm{JacobiSD}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-0.246\,689\,710\,708\,594\,86$

Known value · Wolfram MathematicalFunctionData, JacobiSD, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

u = iπ/2 is a pole of sn AND cn individually (both ~oo at m = 1), but their ratio is finite — sharing one denominator in the m = 1 exact table (see jacobi-elliptic.ts) keeps this from an unresolved ComplexInfinity/ComplexInfinity.

live ↗

In$N(\mathrm{JacobiSD}(\frac{i π}{2}, 1))$Out$1.0i$

Known value · mpmath 1.3.0 ellipfun

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$N(\mathrm{JacobiSD}(10^{16}, \frac{1}{5}))$Out$\mathrm{JacobiSD}(10\,000\,000\,000\,000\,000.0, 0.2)$

About

Examples	14
Checked against	Wolfram Language 12 agree1 differSageMath 10 agree1 differ1 unevaluated1 errormpmath 7 agree1 unevaluated1 errorJulia 6 agree1 unevaluated2 error
Crosswalk	Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: JacobiSN, JacobiDN, JacobiDS