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.