# ellipticCPi

Complementary complete elliptic integral of the third kind

## Syntax

`ellipticCPi(n,m)`

## Description

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

## Input Arguments

 `n` 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. `m` 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.

## Examples

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.

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### Complementary Complete Elliptic Integral of the Third Kind

The complementary complete elliptic integral of the third kind is defined as Π'(m) = Π(n, 1–m), where Π(n,m) is the complete elliptic integral of the third kind:

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

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

• `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.

## 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.