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 fplot(ellipticF(pi/4, m)) hold on fplot(ellipticF(pi/3, m)) fplot(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.