With careful standardization of argument conventions, the Wolfram Language provides full coverage of all standard types of elliptic functions, with arbitrary-precision numerical evaluation for complex values of all parameters, as well as extensive symbolic transformations and simplifications.

Jacobi Elliptic Functions

JacobiSN   ▪ JacobiCN   ▪ JacobiDN   ▪ JacobiCD   ▪ JacobiCS   ▪ JacobiDC   ▪ JacobiDS   ▪ JacobiNC   ▪ JacobiND   ▪ JacobiNS   ▪ JacobiSC   ▪ JacobiSD   ▪ JacobiEpsilon   ▪ JacobiZN

Inverse Jacobi Elliptic Functions

InverseJacobiSN   ▪ InverseJacobiCN   ▪ InverseJacobiDN   ▪ InverseJacobiCD   ▪ InverseJacobiCS   ▪ InverseJacobiDC   ▪ InverseJacobiDS   ▪ InverseJacobiNC   ▪ InverseJacobiND   ▪ InverseJacobiNS   ▪ InverseJacobiSC   ▪ InverseJacobiSD

Weierstrass Elliptic Functions

WeierstrassP   ▪ WeierstrassPPrime   ▪ WeierstrassSigma   ▪ WeierstrassZeta

WeierstrassHalfPeriodW1   ▪ WeierstrassHalfPeriodW2   ▪ WeierstrassHalfPeriodW3   ▪ WeierstrassE1   ▪ WeierstrassE2   ▪ WeierstrassE3   ▪ WeierstrassEta1   ▪ WeierstrassEta2   ▪ WeierstrassEta3   ▪ WeierstrassInvariantG2   ▪ WeierstrassInvariantG3

Inverse Weierstrass Elliptic Functions

InverseWeierstrassP

Theta Functions

EllipticTheta   ▪ EllipticThetaPrime   ▪ SiegelTheta

NevilleThetaC   ▪ NevilleThetaD   ▪ NevilleThetaN   ▪ NevilleThetaS

Elliptic Exponential Functions

EllipticExp   ▪ EllipticExpPrime   ▪ EllipticLog

JacobiAmplitude — convert from argument and parameter to amplitude

EllipticNomeQ — convert from parameter to nome

InverseEllipticNomeQ — convert from nome to parameter

WeierstrassInvariants — convert from half-periods to invariants

WeierstrassHalfPeriods — convert from invariants to half-periods

Related Tech Notes

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• Elliptic Integrals and Elliptic Functions

Related Guides

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• Elliptic Integrals
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• Special Functions

Related Links

• Demonstrations related to Elliptic Functions (Wolfram Demonstrations Project)
• Wolfram Core Areas: Mathematical Functions