EllipticPi

Evaluate numerically:

Evaluate for complex arguments:

Evaluate to high precision:

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

Simple exact values are generated automatically:

EllipticPi threads element-wise over lists:

TraditionalForm formatting:

EllipticPi can be applied to power series:

Carry out an elliptic integral:

Definition of the solid angle subtended by a disk (for instance a detector, a road sign) at the origin in the x,y-plane from a point :

Closed form result for the solid angle:

Numerical comparison:

Plot the solid angle as a function of horizontal distance and height:

This calculates the classical action for a relativistic 3D oscillator:

The action can be expressed using EllipticPi (for brevity, occurring roots are abbreviated):

A conformal map:

Visualize the image of lines of constant real and imaginary parts:

Expand special cases using assumptions:

This shows the branch cuts of the EllipticPi function:

Numerically find a root of a transcendental equation:

Integrals:

Limits at branch cuts can be wrong:

The defining integral converges only under additional conditions:

Different argument conventions exist that result in modified results: