EllipticK

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:

EllipticK can be applied to power series:

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:

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:

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

The defining integral converges only under additional conditions:

Different argument conventions exist that result in modified results:

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: