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# `ellipticK`

Complete elliptic integral of the first kind

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## Syntax

```ellipticK(`m`)
```

## Description

`ellipticK(m)` represents the complete elliptic integral of the first kind $K\left(m\right)$ which is defined as

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

The complete elliptic integral of the first kind is defined for a complex argument m.

For floating-point values `m`, `ellipticK(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`:

```ellipticK(PI/4); float(ellipticK(PI/4))```

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

```ellipticK(1/2); ellipticK(0.5)```

`ellipticK(0)` has a special value:

`ellipticK(0)`

## Parameters

 `m` An arithmetical expression specifying the parameter.

## Return Values

Arithmetical expression.

#### Mathematical Modeling with Symbolic Math Toolbox

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