JacobiSN

The Jacobi elliptic function $\operatorname{sn}(u,m)$, one of the three primary Jacobi elliptic functions, in Wolfram/mpmath's $m=k^2$ parameter convention.

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

JacobiSN(u, m)the Jacobi elliptic sine, argument u, parameter m.

Typed signatures

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

Domain: Special functions

Details

• $m = k^2$, the same convention EllipticK / EllipticF use — not the elliptic modulus k itself.
• Numeric evaluation is the descending Landen/arithmetic-geometric-mean method (Abramowitz & Stegun 16.4) for real u, real m between 0 and 1, extended to m outside that range by the reciprocal-modulus (m > 1) and imaginary-modulus (m < 0) transformations, and to complex u via DLMF 22.8's real addition formulas (real-u/real-(1−m) evaluations combined algebraically — verified against mpmath to ~1e-13 relative across all four quadrants, large Im(u), and close to sn's poles).
• A genuinely complex m stays unevaluated — neither the AGM recursion nor either parameter transform has been verified there.
• Exact at u = 0 (sn = 0, for any m), m = 0 (sn = Sin(u)), m = 1 (sn = Tanh(u)), and the quarter period u = EllipticK(m) (sn = 1).
• $\operatorname{sn}^2(u,m) + \operatorname{cn}^2(u,m) = 1$ and $\operatorname{dn}^2(u,m) + m\operatorname{sn}^2(u,m) = 1$, checked at complex u as part of this head's tests.
• Numeric only — a symbolic argument (outside the exact table above) 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.

Examples

23 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

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In$N(\mathrm{JacobiSN}(0.3, 0.5))$Out$0.293\,412\,733\,168\,455\,4$

Known value · mpmath 1.3.0 ellipfun

Scope #

m greater than 1, via the reciprocal-modulus transformation

live ↗

In$N(\mathrm{JacobiSN}(0.3, 1.5))$Out$0.289\,219\,855\,911\,530\,8$

Known value · mpmath 1.3.0 ellipfun

m less than 0, via the imaginary-modulus transformation

live ↗

In$N(\mathrm{JacobiSN}(0.3, -2))$Out$0.304\,023\,383\,088\,759\,87$

Known value · mpmath 1.3.0 ellipfun

A complex argument u

live ↗

In$N(\mathrm{JacobiSN}(0.3+0.4i, 0.5))$Out$0.328\,927\,004\,182\,561\,6+0.386\,835\,694\,367\,635\,2i$

Known value · mpmath 1.3.0 ellipfun

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

live ↗

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

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

live ↗

In$\operatorname{Series}(\mathrm{JacobiSN}(u, m), u, 0, 5)$Out$u+\frac{1}{6}((-m-1)u^3)+\frac{1}{120}((m^2+14m+1)u^5)+O\left(u^7\right)$

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

live ↗

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

The series about a symbolic point u0, its coefficients the Jacobi functions at u0

live ↗

In$\operatorname{Series}(\mathrm{JacobiSN}(u, m), u, \mathrm{u0}, 2)$Out$\mathrm{JacobiSN}(\mathrm{u0}, m)+(u-\mathrm{u0})\mathrm{JacobiDN}(\mathrm{u0}, m)\mathrm{JacobiCN}(\mathrm{u0}, m)+\frac{1}{2}((-(m\mathrm{JacobiSN}(\mathrm{u0}, m)\mathrm{JacobiCN}(\mathrm{u0}, m)^2)-\mathrm{JacobiSN}(\mathrm{u0}, m)\mathrm{JacobiDN}(\mathrm{u0}, m)^2)(u-\mathrm{u0})^2)+O\left((u-\mathrm{u0})^3\right)$

Properties #

u = 0 is 0, for any m

live ↗

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

m = 0 reduces to Sin(u)

live ↗

In$\mathrm{JacobiSN}(u, 0)$Out$\sin(u)$

m = 1 reduces to Tanh(u)

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In$\mathrm{JacobiSN}(u, 1)$Out$\tanh(u)$

live ↗

In$N(\mathrm{JacobiSN}(\frac{2^{1/2}}{4}i, -1))$Out$0.353\,001\,681\,776\,936\,5i$

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

live ↗

In$N(\mathrm{JacobiSN}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$0.751\,886\,344\,298\,584\,4$

Known value · Wolfram MathematicalFunctionData, JacobiSN, Gauss' transformation

live ↗

In$N(\mathrm{JacobiSN}(\sqrt{2}i, -1))$Out$0.989\,409\,220\,611\,930\,6i$

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

live ↗

In$N(\mathrm{JacobiSN}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-0.246\,469\,043\,732\,777\,75$

Known value · Wolfram MathematicalFunctionData, JacobiSN, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

A genuinely complex m is declined — neither the AGM recursion nor its parameter transforms are verified there

live ↗

In$N(\mathrm{JacobiSN}(0.3, 0.5+0.2i))$Out$\mathrm{JacobiSN}(0.3, 0.5+0.2i)$

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$N(\mathrm{JacobiSN}(10^{16}, \frac{1}{3}))$Out$\mathrm{JacobiSN}(10\,000\,000\,000\,000\,000.0, 0.\overline{3})$

About

Examples	41
Checked against	Wolfram Language 39 agree2 differ
Crosswalk	Wolfram Language

See also: JacobiCN, JacobiDN, JacobiAmplitude, EllipticK