# ellipticE

Complete and incomplete elliptic integrals of the second kind

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```ellipticE(<`ϕ`>, `m`)
```

## Description

`ellipticE(m)` represents the complete elliptic integral of the second kind $E\left(m\right)$ which is defined as

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

`ellipticE(φ,m)` represents the incomplete elliptic integral of the second kind $E\left(\phi |m\right)$ which is defined as

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

The elliptic integrals of the second kind are defined for complex arguments ϕ and m.

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

```ellipticE(-PI/4), ellipticE(PI/4, I); float(ellipticE(-PI/4)), float(ellipticE(PI/4, I))```

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:

```ellipticE(1/2), ellipticE(1/4, I); ellipticE(0.5), ellipticE(0.25, I)```

Some special arguments return explicit symbolic representations:

```ellipticE(0), ellipticE(1), ellipticE(0, m), ellipticE(p, 0)```

## Parameters

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

## Return Values

Arithmetical expression.