Examples for

Elliptic Functions

Elliptic functions refer to some doubly periodic functions on the complex plane, and historically, they were discovered as the inverses of elliptic integrals. The Jacobi elliptic functions are the standard forms of these functions and can be defined using theta functions. Wolfram|Alpha can compute properties for elliptic functions as well as related elliptic integrals and theta functions.

Elliptic Integrals

Compute properties for different kinds of elliptic integrals and other related special functions.

Plot an elliptic integral:

plot EllipticE(t)

Compute a series expansion for nome:

series EllipticNomeQ(m)

Theta Functions

Compute properties for the Jacobi theta function and the four types of Neville theta functions.

Plot a Jacobi theta function:

plot theta2(x, 1/3)

Evaluate a Neville theta function numerically:

NevilleThetaC(2.5, 0.3)

Plot a Neville theta function:

plot NevilleThetaN[x, 0.7]

Jacobi Elliptic Functions

Compute properties for the three basic types of Jacobi elliptic functions.

Plot sn(z, m) for a fixed z:

plot sn(1/2, m)

Compute a series expansion for cn(z, m):

series cn(z, m) in z

Compute compositions of elliptic functions and integrals:

JacobiDN(EllipticK(k), k)

RELATED EXAMPLES

  • Calculus & Analysis
  • Complex Analysis
  • Conic Sections
  • Differential Equations
  • Trigonometry