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# `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.

#### Mathematical Modeling with Symbolic Math Toolbox

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