Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity :

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:

Current flow in a rectangular conducting sheet with voltage applied at a pair of opposite corners:

Plot the flow lines with bounds defined via EllipticK :

Construct lowpass elliptic filter for analog signal:

Compute filter ripple parameters and associate elliptic function parameter:

Use elliptic degree equation to find the ratio of the pass and the stop frequencies:

Compute corresponding stop frequency and elliptic parameters:

Compute ripple locations and poles and zeros of the transfer function:

Compute poles of the transfer function:

Assemble the transfer function:

Compare with the result of EllipticFilterModel :

This shows the branch cuts of the EllipticK function:

Numerically find a root of a transcendental equation:

Solve a differential equation:

EllipticK is a particular case of various mathematical functions:

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:

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 it returns to the origin:

Compare with the expected count at :

Riemann surface of TemplateBox[{m}, EllipticK]: