ellipticCPi

Complementary complete elliptic integral of the third kind

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

ellipticCPi(n,m)

Description

ellipticCPi(n,m) represents the complementary complete elliptic integral of the third kind ${\Pi }^{\prime }\left(n|m\right)=\Pi \left(n|1-m\right)$, where $\Pi \left(n|m\right)$ is the complete elliptic integral of the third kind:

$\Pi \left(n,m\right)=\Pi \left(n;\text{\hspace{0.17em}}\frac{\pi }{2}|m\right)=\underset{0}{\overset{\pi /2}{\int }}\frac{1}{\left(1-n{\mathrm{sin}}^{2}\theta \right)\sqrt{1-m{\mathrm{sin}}^{2}\theta }}d\theta$

The complementary complete elliptic integral of the third kind is defined for complex arguments m and n.

If all arguments are numerical and at least one is a floating-point value, ellipticCPi(n,m) returns floating-point results. For most exact arguments, it returns unevaluated symbolic calls. You can approximate such results with floating-point numbers using the float function.

Environment Interactions

When called with floating-point arguments, this function is sensitive to the environment variable DIGITS which determines the numerical working precision.

Examples

Example 1

Most calls with exact arguments return themselves unevaluated. To approximate such values with floating-point numbers, use float:

ellipticCPi(-1, PI/4);
float(ellipticCPi(-1, PI/4))

Alternatively, use floating-point values as arguments. If one argument is a floating-point value and the others can be converted to a floating-point values, then a floating-point result will be returned:

ellipticCPi(1/2, 1/4);
ellipticCPi(0.5, 1/4)

Some special arguments return explicit symbolic representations:

ellipticCPi(0, m);
ellipticCPi(n, 1)

Parameters

 m An arithmetical expression specifying the parameter. n An arithmetical expression specifying the characteristic.

Return Values

Arithmetical expression.