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# ellipticF

Incomplete elliptic integral of the first kind

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
```

expand all

### 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.