# Documentation

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# `ellipticCPi`

Complementary complete elliptic integral of the third kind

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## Syntax

```ellipticCPi(`n`,`m`)
```

## Description

`ellipticCPi(n,m)` represents the complementary complete elliptic integral of the third kind ${\Pi }^{\prime }\left(n|m\right)=\Pi \left(n|1-m\right)$, where $\Pi \left(n|m\right)$ is the complete elliptic integral of the third kind:

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

The complementary complete elliptic integral of the third kind is defined for complex arguments m and n.

If all arguments are numerical and at least one is a floating-point value, `ellipticCPi(n,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`:

```ellipticCPi(-1, PI/4); float(ellipticCPi(-1, PI/4))```

Alternatively, use floating-point values as arguments. If one argument is a floating-point value and the others can be converted to a floating-point values, then a floating-point result will be returned:

```ellipticCPi(1/2, 1/4); ellipticCPi(0.5, 1/4)```

Some special arguments return explicit symbolic representations:

```ellipticCPi(0, m); ellipticCPi(n, 1)```

## Parameters

 `m` An arithmetical expression specifying the parameter. `n` An arithmetical expression specifying the characteristic.

## Return Values

Arithmetical expression.