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# ellipticK

Complete elliptic integral of the first kind

ellipticK(m)

## Description

ellipticK(m) returns the complete elliptic integral of the first kind.

## Input Arguments

 m Number, symbolic number, variable, expression, or function. This argument also can be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

## Examples

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

`s = [ellipticK(1/2), ellipticK(pi/4), ellipticK(1),  ellipticK(-5.5)]`
```s =
1.8541    2.2253       Inf    0.9325```

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

```s = [ellipticK(sym(1/2)), ellipticK(sym(pi/4)),...
ellipticK(sym(1)),  ellipticK(sym(-5.5))]```
```s =
[ ellipticK(1/2), ellipticK(pi/4), Inf, ellipticK(-11/2)]```

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

`vpa(s, 10)`
```ans =
[ 1.854074677, 2.225253684, Inf, 0.9324665884]```

Differentiate these expressions involving the complete elliptic integral of the first kind:

```syms m
diff(ellipticK(m))
diff(ellipticK(m^2), m, 2)```
```ans =
- ellipticK(m)/(2*m) - ellipticE(m)/(2*m*(m - 1))

ans =
(2*ellipticE(m^2))/(m^2 - 1)^2 - (2*(ellipticE(m^2)/(2*m^2) -...
ellipticK(m^2)/(2*m^2)))/(m^2 - 1) + ellipticK(m^2)/m^2 +...
(ellipticK(m^2)/m + ellipticE(m^2)/(m*(m^2 - 1)))/m +...
ellipticE(m^2)/(m^2*(m^2 - 1))
```

Here, ellipticE represents the complete elliptic integral of the second kind.

Plot the complete elliptic integral of the first kind:

```syms m
ezplot(ellipticK(m))
title('Complete elliptic integral of the first kind')
ylabel('ellipticK(m)')
grid on
```

Call ellipticK for this symbolic matrix. When the input argument is a matrix, ellipticK computes the complete elliptic integral of the first kind for each element.

`ellipticK(sym([-2*pi -4; 0 1]))`
```ans =
[ ellipticK(-2*pi), ellipticK(-4)]
[             pi/2,           Inf]```

## Alternatives

You can use ellipke to compute elliptic integrals of the first and second kinds in one function call.

## More About

expand all

### 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.

### Tips

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

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

• If m is a vector or a matrix, then ellipticK(m) returns the complete elliptic integral of the first kind, evaluated for each element of 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.

## See Also

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