# ellipticF

Incomplete elliptic integral of the first kind

## Syntax

`ellipticF(phi,m)`

## Description

`ellipticF(phi,m)` returns the incomplete elliptic integral of the first kind.

## Input Arguments

 `m` Number, symbolic number, variable, expression, or function specifying the parameter. This argument also can be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions. `phi` Number, symbolic number, variable, expression, or function specifying the amplitude. This argument also can be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

## Examples

Compute the incomplete elliptic integrals of the first kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

```s = [ellipticF(pi/3, -10.5), ellipticF(pi/4, -pi),... ellipticF(1, -1), ellipticF(pi/2, 0)]```
```s = 0.6184 0.6486 0.8964 1.5708```

Compute the incomplete elliptic integrals of the first kind for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, `ellipticF` returns unresolved symbolic calls.

```s = [ellipticF(sym(pi/3), -10.5), ellipticF(sym(pi/4), -pi),... ellipticF(sym(1), -1), ellipticF(pi/6, sym(0))]```
```s = [ ellipticF(pi/3, -21/2), ellipticF(pi/4, -pi), ellipticF(1, -1), pi/6]```

Use `vpa` to approximate this result with floating-point numbers:

`vpa(s, 10)`
```ans = [ 0.6184459461, 0.6485970495, 0.8963937895, 0.5235987756]```

Differentiate this expression involving the incomplete elliptic integral of the first kind:

```syms m diff(ellipticF(pi/4, m))```
```ans = 1/(4*(1 - m/2)^(1/2)*(m - 1)) - ellipticF(pi/4, m)/(2*m) -... ellipticE(pi/4, m)/(2*m*(m - 1))```

Here, `ellipticE` represents the incomplete elliptic integral of the second kind.

Plot the incomplete elliptic integrals ```ellipticF(phi, m)``` for `phi = pi/4` and ```phi = pi/3```. Also plot the complete elliptic integral `ellipticK(m)`:

```syms m ezplot(ellipticF(pi/4, m)) hold on ezplot(ellipticF(pi/3, m)) ezplot(ellipticK(m)) title('Elliptic integrals of the first kind') ylabel('ellipticF(m)') legend('F(pi/4|m', 'F(pi/3|m)', 'K(m)', 'Location', 'Best') grid on hold off ```

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### Incomplete Elliptic Integral of the First Kind

The complete elliptic integral of the first kind is defined as follows:

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

Note that some definitions use the elliptical modulus k or the modular angle α instead of the parameter m. They are related as m = k2 = sin2α.

### Tips

• `ellipticF` returns floating-point results for numeric arguments that are not symbolic objects.

• For most symbolic (exact) numbers, `ellipticF` returns unresolved symbolic calls. You can approximate such results with floating-point numbers using `vpa`.

• At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, `ellipticF` expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

• `ellipticF(pi/2, m) = ellipticK(m)`.

## References

[1] Milne-Thomson, L. M. "Elliptic Integrals." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.