# ellipticPi

Complete and incomplete elliptic integrals of the third kind

## Examples

### Compute the Incomplete Elliptic Integrals of Third Kind

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

s = [ellipticPi(-2.3, pi/4, 0), ellipticPi(1/3, pi/3, 1/2),...
ellipticPi(-1, 0, 1),  ellipticPi(2, pi/6, 2)]
s =
0.5877    1.2850         0    0.7507

Compute the incomplete elliptic integrals of the third kind for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, ellipticPi returns unresolved symbolic calls.

s = [ellipticPi(-2.3, sym(pi/4), 0), ellipticPi(sym(1/3), pi/3, 1/2),...
ellipticPi(-1, sym(0), 1),  ellipticPi(2, pi/6, sym(2))]
s =
[ ellipticPi(-23/10, pi/4, 0), ellipticPi(1/3, pi/3, 1/2),...
0, (2^(1/2)*3^(1/2))/2 - ellipticE(pi/6, 2)]

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

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

vpa(s, 10)
ans =
[ 0.5876852228, 1.285032276, 0, 0.7507322117]

### Differentiate Incomplete Elliptic Integrals of Third Kind

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

syms n m
diff(ellipticPi(n, m), n)
diff(ellipticPi(n, m), m)
ans =
ellipticK(m)/(2*n*(n - 1)) + ellipticE(m)/(2*(m - n)*(n - 1)) -...
(ellipticPi(n, m)*(- n^2 + m))/(2*n*(m - n)*(n - 1))

ans =
- ellipticPi(n, m)/(2*(m - n)) - ellipticE(m)/(2*(m - n)*(m - 1))

Here, ellipticK and ellipticE represent the complete elliptic integrals of the first and second kinds.

### Compute Integrals for Matrix Input

Call ellipticPi for the scalar and the matrix. When one input argument is a matrix, ellipticPi expands the scalar argument to a matrix of the same size with all its elements equal to the scalar.

ellipticPi(sym(0), sym([1/3 1; 1/2 0]))
ans =
[ ellipticK(1/3),  Inf]
[ ellipticK(1/2), pi/2]

Here, ellipticK represents the complete elliptic integral of the first kind.

## Input Arguments

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

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

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

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### Incomplete Elliptic Integral of the Third Kind

The incomplete elliptic integral of the third kind is defined as follows:

$\Pi \left(n;\phi |m\right)=\underset{0}{\overset{\phi }{\int }}\frac{1}{\left(1-n{\mathrm{sin}}^{2}\theta \right)\sqrt{1-m{\mathrm{sin}}^{2}\theta }}d\theta$

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 Third Kind

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

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

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

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

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

• All non-scalar arguments must have the same size. If one or two input arguments are non-scalar, then ellipticPi expands the scalars into vectors or matrices of the same size as the non-scalar arguments, with all elements equal to the corresponding scalar.

• ellipticPi(n, pi/2, m) = ellipticPi(n, 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.