JacobiCD

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

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

JacobiCD(u, m)the Jacobi elliptic function cd, argument u, parameter m.

SageMathjacobi_cd(u, m)mpmathellipfun('cd', u, m=m)Juliaenumeratio_jacobi(:cd, 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.
• $\operatorname{cd}(u,m) = \operatorname{cn}(u,m)/\operatorname{dn}(u,m)$ — JacobiCN 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).
• cd(0,m) = 1, 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$N(\mathrm{JacobiCD}(0.3, 0.5))$Out$0.977\,250\,336\,444\,249\,9$

Known value · mpmath 1.3.0 ellipfun

Scope #

m greater than 1, via the reciprocal-modulus transformation

live ↗

In$N(\mathrm{JacobiCD}(0.3, 1.5))$Out$1.023\,633\,106\,578\,735$

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

live ↗

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

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

live ↗

In$\operatorname{Series}(\mathrm{JacobiCD}(u, m), u, 0, 5)$Out$1+\frac{1}{2}(m-1)u^2+\frac{1}{24}(5m^2-6m+1)u^4+O\left(u^6\right)$

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

live ↗

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

Properties #

u = 0 is 1, for any m

live ↗

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

live ↗

In$N(\mathrm{JacobiCD}(\frac{2^{1/2}}{4}i, 2))$Out$0.942\,972\,425\,777\,385\,4$

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

live ↗

In$N(\mathrm{JacobiCD}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$0.664\,848\,107\,974\,773\,6$

Known value · Wolfram MathematicalFunctionData, JacobiCD, Gauss' transformation

live ↗

In$N(\mathrm{JacobiCD}(\sqrt{2}i, 2))$Out$0.710\,861\,047\,784\,087\,4$

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

live ↗

In$N(\mathrm{JacobiCD}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-0.970\,018\,357\,823\,858\,7$

Known value · Wolfram MathematicalFunctionData, JacobiCD, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

u = 10^16 spans far more than 10^6 periods of 4K(2/7): 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$N(\mathrm{JacobiCD}(10^{16}, \frac{2}{7}))$Out$\mathrm{JacobiCD}(10\,000\,000\,000\,000\,000.0, 0.285\,714\,285\,714\,285\,714\,286)$

The true value (mpmath, dps=30) — still not computed here. Argument reduction at bignum precision (reduce u mod 4K(m) before collapsing to a double) is deferred; this example only pins the gap, not a fix.

live ↗

Known value · mpmath 1.3.0 ellipfun

About

Examples	14
Checked against	Wolfram Language 12 agree1 differSageMath 10 agree1 differ1 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: JacobiCN, JacobiDN, JacobiDC