JacobiCS

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

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

JacobiCS(u, m)the Jacobi elliptic function cs, argument u, parameter m.

SageMathjacobi_cs(u, m)mpmathellipfun('cs', u, m=m)Juliaenumeratio_jacobi(:cs, 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.
• $\operatorname{cs}(u,m) = \operatorname{cn}(u,m)/\operatorname{sn}(u,m)$ — JacobiCN and JacobiSN 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).
• cs has a pole at u = 0 (sn(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. At the pole $u=0$ the series in $u$ is the Laurent series, its coefficients polynomials in $m$.

Plot

Examples

Basic #

A floating-point argument evaluates directly to a decimal

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In$N(\mathrm{JacobiCS}(1.1, 0.9))$Out$0.722\,588\,061\,871\,721\,1$

Known value · mpmath 1.3.0 ellipfun

Scope #

m greater than 1, via the reciprocal-modulus transformation

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In$N(\mathrm{JacobiCS}(0.3, 1.5))$Out$3.309\,809\,750\,125\,873\,4$

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

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

The Laurent series at the pole u = 0, its coefficients polynomials in m

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In$\operatorname{Series}(\mathrm{JacobiCS}(u, m), u, 0, 4)$Out$\frac{1}{u}+u×(\frac{m}{6}-\frac{1}{3})+(\frac{7m^2}{360}+\frac{m}{45}-\frac{1}{45})u^3+O\left(u^5\right)$

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

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In$\operatorname{Series}(\mathrm{JacobiCS}(z, m), m, 0, 2)$Out$\frac{\cos(z)}{\sin(z)}+\frac{m×(2z-\sin(2z))}{8\sin(z)^2}+\frac{(32\cos(z)z^2+8z\sin(z)-12\cos(z)+13\cos(3z)-\cos(5z))m^2}{512\sin(z)^3}+O\left(m^3\right)$

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

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

Properties #

u = 0 is a pole, for any m

live ↗

In$\mathrm{JacobiCS}(0, m)$Out$\tilde∞$

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In$N(\mathrm{JacobiCS}(\frac{2^{1/2}}{4}i, -1))$Out$-3.004\,168\,113\,503\,711\,3i$

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

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In$N(\mathrm{JacobiCS}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$0.876\,851\,602\,970\,147\,7$

Known value · Wolfram MathematicalFunctionData, JacobiCS, Gauss' transformation

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In$N(\mathrm{JacobiCS}(\sqrt{2}i, -1))$Out$-1.421\,802\,682\,497\,364i$

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

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In$N(\mathrm{JacobiCS}(2(1+\frac{\sqrt{2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))$Out$3.932\,139\,508\,524\,959\,3$

Known value · Wolfram MathematicalFunctionData, JacobiCS, Gauss' transformation

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

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

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

Known value · mpmath 1.3.0 ellipfun

About

Examples	13
Checked against	Wolfram Language 13 agreeSageMath 9 agree2 differ1 unevaluated1 errormpmath 7 agreeJulia 6 agree1 error
Crosswalk	Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: JacobiCN, JacobiSN, JacobiSC