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# ellipticCK

Complementary complete elliptic integral of the first kind

ellipticCK(m)

## Description

ellipticCK(m) returns the complementary 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 complementary complete elliptic integrals of the first kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

`s = [ellipticCK(1/2), ellipticCK(pi/4), ellipticCK(1), ellipticCK(inf)]`
```s =
1.8541    1.6671    1.5708       NaN```

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

```s = [ellipticCK(sym(1/2)), ellipticCK(sym(pi/4)),...
ellipticCK(sym(1)), ellipticCK(sym(inf))]```
```s =
[ ellipticCK(1/2), ellipticCK(pi/4), pi/2, ellipticCK(Inf)]```

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

`vpa(s, 10)`
```ans =
[ 1.854074677, 1.667061338, 1.570796327, NaN]```

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

```syms m
diff(ellipticCK(m))
diff(ellipticCK(m^2), m, 2)```
```ans =
ellipticCE(m)/(2*m*(m - 1)) - ellipticCK(m)/(2*m - 2)

ans =
(2*(ellipticCE(m^2)/(2*m^2 - 2) -...
ellipticCK(m^2)/(2*m^2 - 2)))/(m^2 - 1) -...
(2*ellipticCE(m^2))/(m^2 - 1)^2 -...
(2*ellipticCK(m^2))/(2*m^2 - 2) +...
(8*m^2*ellipticCK(m^2))/(2*m^2 - 2)^2 +...
(2*m*((2*m*ellipticCK(m^2))/(2*m^2 - 2) -...
ellipticCE(m^2)/(m*(m^2 - 1))))/(2*m^2 - 2) -...
ellipticCE(m^2)/(m^2*(m^2 - 1))```

Here, ellipticCE represents the complementary complete elliptic integral of the second kind.

Plot the complementary complete elliptic integral of the first kind:

```syms m
ezplot(ellipticCK(m), [0.1, 5])
title('Complementary complete elliptic integral of the first kind')
ylabel('ellipticCK(m)')
grid on
hold off
```

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

`ellipticCK(sym([pi/6 pi/4; pi/3 pi/2]))`
```ans =
[ ellipticCK(pi/6), ellipticCK(pi/4)]
[ ellipticCK(pi/3), ellipticCK(pi/2)]```

expand all

### Complementary Complete Elliptic Integral of the First Kind

The complementary complete elliptic integral of the first kind is defined as K'(m) = K(1–m), where K(m) is the complete elliptic integral of the first kind:

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

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, ellipticCK returns unresolved symbolic calls. You can approximate such results with floating-point numbers using the vpa function.

• If m is a vector or a matrix, then ellipticCK(m) returns the complementary 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.