# ellipticCE

Complementary complete elliptic integral of the second kind

ellipticCE(m)

## Description

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

s = [ellipticCE(0), ellipticCE(pi/4),...
ellipticCE(1), ellipticCE(pi/2)]
s =
1.0000    1.4828    1.5708    1.7753

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

s = [ellipticCE(sym(0)), ellipticCE(sym(pi/4)),...
ellipticCE(sym(1)), ellipticCE(sym(pi/2))]
s =
[ 1, ellipticCE(pi/4), pi/2, ellipticCE(pi/2)]

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

vpa(s, 10)
ans =
[ 1.0, 1.482786927, 1.570796327, 1.775344699]

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

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

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

Here, ellipticCK represents the complementary complete elliptic integral of the first kind.

Plot the complementary complete elliptic integral of the second kind:

syms m
ezplot(ellipticCE(m))
title('Complementary complete elliptic integral of the second kind')
ylabel('ellipticCE(m)')
grid on

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

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

collapse all

### Complementary Complete Elliptic Integral of the Second Kind

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

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

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

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

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

• If m is a vector or a matrix, then ellipticCE(m) returns the complementary complete elliptic integral of the second 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.