ellipticPi

Complete and incomplete elliptic integrals of the third kind

Syntax

ellipticPi(n,m)

ellipticPi(n,phi,m)

Description

ellipticPi(n,m) returns the complete elliptic integral of the third kind.

ellipticPi(n,phi,m) returns the incomplete elliptic integral 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

n — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

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

m — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

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

phi — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

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

More About

Incomplete Elliptic Integral of the Third Kind

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

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

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:

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

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.

Version History

Introduced in R2013a