BUILT-IN MATHEMATICA SYMBOL

EllipticK

EllipticK[m]
gives the complete elliptic integral of the first kind

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

EllipticK is given in terms of the incomplete elliptic integral of the first kind by .

EllipticK[m] has a branch cut discontinuity in the complex m plane running from to .

For certain special arguments, EllipticK automatically evaluates to exact values.

EllipticK can be evaluated to arbitrary numerical precision.

EllipticK automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

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Scope (7)

Evaluate numerically for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Simple exact values are generated automatically:

EllipticK threads element-wise over lists:

Series expansions at branch points:

Find directional limits at branch cuts:

TraditionalForm formatting:

Generalizations & Extensions (1)

EllipticK can be applied to power series:

Applications (5)

Small angle approximation to the period of a pendulum:

Plot the period versus the initial angle:

Vector potential due to a circular current flow, in cylindrical coordinates:

The components of the magnetic field:

Plot the magnitude of the magnetic field:

Resistance between the origin and the point

in an infinite 3D lattice of unit resistors:

Energy for the Onsager solution of the Ising model:

Plot of the specific heat:

Find the critical temperature:

Calculate a singular value:

Properties & Relations (6)

This shows the branch cuts of the EllipticK function:

Numerically find a root of a transcendental equation:

Integrals:

Laplace transforms:

Solve a differential equation:

EllipticK is a particular case of various mathematical functions:

Possible Issues (3)

Machine-precision evaluation can result in numerically inaccurate answers near branch cuts:

The defining integral converges only under additional conditions:

Different argument conventions exist that result in modified results:

Neat Examples (1)

Probability that a random walker in a 3D cubic lattice returns to the origin:

Carry out a modeling run of 1000 walks and count how many return to the origin: