Complementary complete elliptic integral of the third kind
ellipticCPi(n,m)
ellipticCPi(
returns
the complementary
complete elliptic integral of the third kind.n
,m
)

Number, symbolic number, variable, expression, or function specifying the characteristic. This argument also can be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions. 

Number, symbolic number, variable, expression, or function specifying the parameter. This argument also can be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions. 
Compute the complementary complete elliptic integrals of the third kind for these numbers. Because these numbers are not symbolic objects, you get floatingpoint results.
s = [ellipticCPi(1, 1/3), ellipticCPi(0, 1/2),... ellipticCPi(9/10, 1), ellipticCPi(1, 0)]
s = 1.3703 1.8541 4.9673 Inf
Compute the complementary complete elliptic integrals of the
third kind for the same numbers converted to symbolic objects. For
most symbolic (exact) numbers, ellipticCPi
returns
unresolved symbolic calls.
s = [ellipticCPi(1, sym(1/3)), ellipticCPi(sym(0), 1/2),... ellipticCPi(sym(9/10), 1), ellipticCPi(1, sym(0))]
s = [ ellipticCPi(1, 1/3), ellipticCK(1/2), (pi*10^(1/2))/2, Inf]
Here, ellipticCK
represents the complementary
complete elliptic integrals of the first kind.
Use vpa
to approximate
this result with floatingpoint numbers:
vpa(s, 10)
ans = [ 1.370337322, 1.854074677, 4.967294133, Inf]
Differentiate these expressions involving the complementary complete elliptic integral of the third kind:
syms n m diff(ellipticCPi(n, m), n) diff(ellipticCPi(n, m), m)
ans = ellipticCK(m)/(2*n*(n  1)) ... ellipticCE(m)/(2*(n  1)*(m + n  1)) ... (ellipticCPi(n, m)*(n^2 + m  1))/(2*n*(n  1)*(m + n  1)) ans = ellipticCE(m)/(2*m*(m + n  1))  ellipticCPi(n, m)/(2*(m + n  1))
Here, ellipticCK
and ellipticCE
represent
the complementary complete elliptic integrals of the first and second
kinds.
[1] MilneThomson, 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.
ellipke
 ellipticCE
 ellipticCK
 ellipticE
 ellipticF
 ellipticK
 ellipticPi
 vpa