# ellipke

Complete elliptic integrals of the first and second kinds

## Syntax

``````[K,E] = ellipke(m)``````

## Description

``````[K,E] = ellipke(m)``` returns the complete elliptic integrals of the first and second kinds.```

## Examples

### Compute Complete Elliptic Integrals of First and Second Kind

Compute the complete elliptic integrals of the first and second kinds for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

```[K0, E0] = ellipke(0) [K05, E05] = ellipke(1/2)```
```K0 = 1.5708 E0 = 1.5708 K05 = 1.8541 E05 = 1.3506```

Compute the complete elliptic integrals for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, `ellipke` returns results using the `ellipticK` and `ellipticE` functions.

```[K0, E0] = ellipke(sym(0)) [K05, E05] = ellipke(sym(1/2))```
```K0 = pi/2 E0 = pi/2 K05 = ellipticK(1/2) E05 = ellipticE(1/2)```

Use `vpa` to approximate `K05` and `E05` with floating-point numbers:

`vpa([K05, E05], 10)`
```ans = [ 1.854074677, 1.350643881]```

### Compute Integrals When Input is Not Between `0` and `1`

If the argument does not belong to the range from 0 to 1, then convert that argument to a symbolic object before using `ellipke`:

`[K, E] = ellipke(sym(pi/2))`
```K = ellipticK(pi/2) E = ellipticE(pi/2)```

Alternatively, use `ellipticK` and `ellipticE` to compute the integrals of the first and the second kinds separately:

```K = ellipticK(sym(pi/2)) E = ellipticE(sym(pi/2))```
```K = ellipticK(pi/2) E = ellipticE(pi/2)```

### Compute Integrals for Matrix Input

Call `ellipke` for this symbolic matrix. When the input argument is a matrix, `ellipke` computes the complete elliptic integrals of the first and second kinds for each element.

`[K, E] = ellipke(sym([-1 0; 1/2 1]))`
```K = [ ellipticK(-1), pi/2] [ ellipticK(1/2), Inf] E = [ ellipticE(-1), pi/2] [ ellipticE(1/2), 1]```

## Input Arguments

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Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

## Output Arguments

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Complete elliptic integral of the first kind, returned as a symbolic expression.

Complete elliptic integral of the second kind, returned as a symbolic expression.

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### Complete Elliptic Integral of the First Kind

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

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

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.

### Complete Elliptic Integral of the Second Kind

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

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

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

• Calling `ellipke` for numbers that are not symbolic objects invokes the MATLAB® `ellipke` function. This function accepts only ```0 <= m <= 1```. To compute the complete elliptic integrals of the first and second kinds for the values out of this range, use `sym` to convert the numbers to symbolic objects, and then call `ellipke` for those symbolic objects. Alternatively, use the `ellipticK` and `ellipticE` functions to compute the integrals separately.

• For most symbolic (exact) numbers, `ellipke` returns results using the `ellipticK` and `ellipticE` functions. You can approximate such results with floating-point numbers using `vpa`.

• If `m` is a vector or a matrix, then `[K,E] = ellipke(m)` returns the complete elliptic integrals of the first and second kinds, evaluated for each element of `m`.

## Alternatives

You can use `ellipticK` and `ellipticE` to compute elliptic integrals of the first and second kinds separately.

## 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.