This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

EllipticPi

 EllipticPigives the complete elliptic integral of the third kind . EllipticPigives the incomplete elliptic integral .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• .
• For certain special arguments, EllipticPi automatically evaluates to exact values.
• EllipticPi can be evaluated to arbitrary numerical precision.
Evaluate numerically:
Series expansions:
Evaluate numerically:
 Out[1]=

 Out[1]=

Series expansions:
 Out[1]=
 Out[2]=
 Scope   (7)
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 can be applied to power series:
 Applications   (4)
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 , -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:
New in 1