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Complementary complete elliptic integral of the third kind

`ellipticCPi(`

returns
the complementary complete elliptic integral
of the third kind.`n`

,`m`

)

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

`ellipticCPi`

returns floating-point results for numeric arguments that are not symbolic objects.For most symbolic (exact) numbers,

`ellipticCPi`

returns unresolved symbolic calls. You can approximate such results with floating-point numbers using`vpa`

.At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then

`ellipticCPi`

expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

[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.

`ellipke`

| `ellipticCE`

| `ellipticCK`

| `ellipticE`

| `ellipticF`

| `ellipticK`

| `ellipticPi`

| `vpa`