Complete and incomplete elliptic integrals of the third kind

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ellipticPi(n,m) represents the complete elliptic integral of the third kind


ellipticPi(n,φ,m) represents the incomplete elliptic integral of the third kind


The elliptic integrals of the third kind are defined for complex arguments m, ϕ, and n.

If all arguments are numerical and at least one is a floating-point value, ellipticPi(n,<φ>,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.


Example 1

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

ellipticPi(PI/4, I);
float(ellipticPi(PI/4, I))

Alternatively, use a floating-point value as an argument:

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

Some special arguments return explicit symbolic representations:

ellipticPi(n, 0);
ellipticPi(0, m);
ellipticPi(0, p, m);
ellipticPi(1, p, m)



An arithmetical expression specifying the parameter.


An arithmetical expression specifying the amplitude. The default is π2.


An arithmetical expression specifying the characteristic.

Return Values

Arithmetical expression.