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`ellipticF`

Incomplete elliptic integral of the first kind

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Syntax

```ellipticF(`phi`, `m`)
```

Description

`ellipticF(phi,m)` represents the incomplete elliptic integral of the first kind $F\left(\phi |m\right)$ which is defined as

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

The incomplete elliptic integral of the first kind is defined for complex arguments ϕ and m.

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

```ellipticF(PI/4, I); float(ellipticF(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(1/4, I); ellipticE(0.25, I)```

Some special arguments return explicit symbolic representations:

```ellipticF(0, m), ellipticF(p, 0)```

Parameters

 `m` An arithmetical expression specifying the parameter. `phi` An arithmetical expression specifying the amplitude.

Return Values

Arithmetical expression.