JacobiCN

The Jacobi elliptic function $\operatorname{cn}(u,m)$, 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 LanguageJacobiCN!↗

JacobiCN(u, m)the Jacobi elliptic cosine, argument u, parameter m.

SageMathjacobi_cn(u, m)mpmathellipfun('cn', u, m=m)Juliaenumeratio_jacobi(:cn, 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 — not the elliptic modulus k itself.
• Numeric evaluation shares JacobiSN's descending Landen/arithmetic-geometric-mean kernel (Abramowitz & Stegun 16.4); see there for the m-outside-[0,1] and complex-u coverage this rests on.
• A genuinely complex m stays unevaluated.
• Exact at u = 0 (cn = 1, for any m), m = 0 (cn = Cos(u)), m = 1 (cn = 1/Cosh(u)), and the quarter period u = EllipticK(m) (cn = 0).
• 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.

Plot

Examples

23 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

live ↗

In$N(\mathrm{JacobiCN}(0.3, 0.5))$Out$0.955\,985\,861\,827\,787\,1$

Known value · mpmath 1.3.0 ellipfun

live ↗

In$\operatorname{Series}(\mathrm{JacobiCN}(z, m), z, 0, 7)$Out$1-\frac{z^2}{2}+\frac{1}{24}(4m+1)z^4+\frac{1}{720}(-16m^2-44m-1)z^6+O\left(z^8\right)$

live ↗

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

Scope #

m greater than 1, via the reciprocal-modulus transformation

live ↗

In$N(\mathrm{JacobiCN}(0.3, 1.5))$Out$0.957\,262\,699\,025\,984\,9$

Known value · mpmath 1.3.0 ellipfun

m less than 0, via the imaginary-modulus transformation

live ↗

In$N(\mathrm{JacobiCN}(0.3, -2))$Out$0.952\,664\,569\,791\,101\,9$

Known value · mpmath 1.3.0 ellipfun

A complex argument u

live ↗

In$N(\mathrm{JacobiCN}(0.3+0.4i, 0.5))$Out$1.027\,992\,872\,007\,962\,7-0.123\,775\,864\,136\,771\,64i$

Known value · mpmath 1.3.0 ellipfun

At $m=1$, cn is sech, and a matrix function gives $2$ at the diagonal

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{JacobiCN}((-(1/3)i)π, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack2, 0\bigr\rbrack, \bigl\lbrack0, 2\bigr\rbrack\bigr\rbrack$

live ↗

In$\bigl\lbrack\mathrm{JacobiCN}(z, 0), \mathrm{JacobiCN}(z, 1)\bigr\rbrack$Out$\bigl\lbrack\cos(z), \frac{1}{\cosh(z)}\bigr\rbrack$

live ↗

In$N(\mathrm{JacobiCN}(2, \frac{1}{3}), 50)$Out$-0.216\,378\,369\,067\,457\,741\,748\,259\,579\,609\,162\,442\,422\,719\,724\,244\,02$

live ↗

In$\bigl\lbrack\mathrm{JacobiCN}(0, m), \mathrm{JacobiCN}(\frac{\mathrm{EllipticK}(m)}{2}, m), \mathrm{JacobiCN}(\mathrm{EllipticK}(m), m)\bigr\rbrack$Out$\bigl\lbrack1, \mathrm{JacobiCN}(\frac{\mathrm{EllipticK}(m)}{2}, m), 0\bigr\rbrack$

live ↗

In$\operatorname{Series}(\mathrm{JacobiCN}(x, \frac{1}{3}), x, 0, 7)$Out$1-\frac{x^2}{2}+\frac{7x^4}{72}-\frac{157x^6}{6\,480}+O\left(x^8\right)$

live ↗

In$\operatorname{Series}(\mathrm{JacobiCN}(1, m), m, 0, 3)$Out$\cos(1)+m×(-\frac{\cos(1)}{16}+\frac{\cos(3)}{16}+\frac{\sin(1)}{4})+(-\frac{17\cos(1)}{256}+\frac{\cos(5)}{256}+\frac{\cos(3)}{32}+\frac{3\sin(3)}{64}+\frac{\sin(1)}{16})m^2+(-\frac{171\cos(1)}{4\,096}+\frac{\cos(7)}{4\,096}+\frac{5\cos(3)}{2\,048}+\frac{\cos(5)}{256}+\frac{5\sin(5)}{1\,024}+\frac{41\sin(1)}{1\,536}+\frac{39\sin(3)}{1\,024})m^3+O\left(m^4\right)$

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

live ↗

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

Properties #

u = 0 is 1, for any m

live ↗

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

m = 0 reduces to Cos(u)

live ↗

In$\mathrm{JacobiCN}(u, 0)$Out$\cos(u)$

The quarter period u = EllipticK(m) is 0

live ↗

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

cn(0) is 1 for any parameter

live ↗

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

Known value · Wolfram MathematicalFunctionData, JacobiCN

At $m=0$, $\mathrm{cn}(u)=\cos u$, so zero at $\pi/2$

live ↗

In$\mathrm{JacobiCN}(\frac{π}{2}, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, JacobiCN

At $m=1$, $\mathrm{cn}(0)=1$

live ↗

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

Known value · Wolfram MathematicalFunctionData, JacobiCN

At $m=1$ the quarter period $i\pi/2$ is a pole

live ↗

In$\mathrm{JacobiCN}(\frac{1}{2}i π, 1)$Out$\tilde∞$

Known value · Wolfram MathematicalFunctionData, JacobiCN

live ↗

In$N(\mathrm{JacobiCN}(\frac{2^{1/2}}{4}i, 2))$Out$1.068\,806\,917\,319\,438\,3$

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

live ↗

In$N(\mathrm{JacobiCN}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$0.659\,292\,746\,249\,578\,1$

Known value · Wolfram MathematicalFunctionData, JacobiCN, Gauss' transformation

live ↗

In$N(\mathrm{JacobiCN}(\sqrt{2}i, 2))$Out$-6.889\,282\,219\,354\,135$

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

live ↗

In$N(\mathrm{JacobiCN}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$-0.969\,150\,664\,490\,021\,4$

Known value · Wolfram MathematicalFunctionData, JacobiCN, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

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

About

Examples	48
Checked against	Wolfram Language 47 agree1 differSageMath 42 agree2 differ1 unevaluated2 errormpmath 35 agree1 unevaluatedJulia 35 agree1 error
Crosswalk	Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no Jacobi elliptic functions: sympy.jacobi is the Jacobi polynomial P_n^(a,b), not sn/cn/dn, and there is no ellipfun (mpmath, Sage and Nemo carry them).

See also: JacobiSN, JacobiDN, JacobiAmplitude, EllipticK