JacobiDC

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

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

JacobiDC(u, m)the Jacobi elliptic function dc, argument u, parameter m.

SageMathjacobi_dc(u, m)mpmathellipfun('dc', u, m=m)Juliaenumeratio_jacobi(:dc, 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.
• $\operatorname{dc}(u,m) = \operatorname{dn}(u,m)/\operatorname{cn}(u,m)$ — JacobiDN and JacobiCN 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).
• dc(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$N(\mathrm{JacobiDC}(0.6, 0.4))$Out$1.124\,780\,426\,911\,555$

Known value · mpmath 1.3.0 ellipfun

Scope #

m less than 0, via the imaginary-modulus transformation

live ↗

In$N(\mathrm{JacobiDC}(0.3, -0.7))$Out$1.079\,939\,770\,083\,225\,2$

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

live ↗

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

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

live ↗

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

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

live ↗

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

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

live ↗

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

Properties #

u = 0 is 1, for any m

live ↗

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

live ↗

In$N(\mathrm{JacobiDC}(\frac{2^{1/2}}{4}i, 2))$Out$1.060\,476\,396\,407\,456\,8$

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

live ↗

In$N(\mathrm{JacobiDC}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$1.504\,102\,949\,237\,757\,6$

Known value · Wolfram MathematicalFunctionData, JacobiDC, Gauss' transformation

live ↗

In$N(\mathrm{JacobiDC}(\sqrt{2}i, 2))$Out$1.406\,744\,683\,953\,669$

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

live ↗

In$N(\mathrm{JacobiDC}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-1.030\,908\,324\,501\,612\,8$

Known value · Wolfram MathematicalFunctionData, JacobiDC, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

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$N(\mathrm{JacobiDC}(10^{16}, \frac{5}{7}))$Out$\mathrm{JacobiDC}(10\,000\,000\,000\,000\,000.0, 0.714\,285\,714\,285\,714\,285\,714)$

About

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

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: JacobiDN, JacobiCN, JacobiCD