Documentation

# ellipticPi

Complete and incomplete elliptic integrals of the third kind

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## Syntax

ellipticPi(n,<φ>,m)

## Description

ellipticPi(n,m) represents the complete elliptic integral of the third kind

$\Pi \left(n,m\right)=\Pi \left(n;\text{\hspace{0.17em}}\frac{\pi }{2}|m\right)=\underset{0}{\overset{\pi /2}{\int }}\frac{1}{\left(1-n{\mathrm{sin}}^{2}\theta \right)\sqrt{1-m{\mathrm{sin}}^{2}\theta }}d\theta$

ellipticPi(n,φ,m) represents the incomplete elliptic integral of the third kind

$\Pi \left(n,m\right)=\Pi \left(n;\text{\hspace{0.17em}}\phi |m\right)=\underset{0}{\overset{\phi }{\int }}\frac{1}{\left(1-n{\mathrm{sin}}^{2}\theta \right)\sqrt{1-m{\mathrm{sin}}^{2}\theta }}d\theta$

The elliptic integrals of the third kind are defined for complex arguments m, ϕ, and n.

If all arguments are numerical and at least one is a floating-point value, ellipticPi(n,<φ>,m) returns floating-point results. For most exact arguments, it returns unevaluated symbolic calls. You can approximate such results with floating-point numbers using the float function.

## Environment Interactions

When called with floating-point arguments, this function is sensitive to the environment variable DIGITS which determines the numerical working precision.

## Examples

### Example 1

Most calls with exact arguments return themselves unevaluated. To approximate such values with floating-point numbers, use float:

ellipticPi(PI/4, I);
float(ellipticPi(PI/4, I))

Alternatively, use a floating-point value as an argument:

ellipticPi(1/2, 1, 1/4);
ellipticPi(0.5, 1, 1/4)

Some special arguments return explicit symbolic representations:

ellipticPi(n, 0);
ellipticPi(0, m);
ellipticPi(0, p, m);
ellipticPi(1, p, m)

## Parameters

 m An arithmetical expression specifying the parameter. φ An arithmetical expression specifying the amplitude. The default is $\frac{\pi }{2}$. n An arithmetical expression specifying the characteristic.

## Return Values

Arithmetical expression.