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Elliptic integral of the second kind

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.


 (<ϕ>, m)


ellipticE(m) represents the complete elliptic integral of the second kind which is defined as


ellipticE( φ ,m) represents the incomplete elliptic integral of the second kind which is defined as


The elliptic integrals of the second kind are defined for complex arguments ϕ and m.

A floating-point value is computed if all arguments are numerical and at least one is a floating-point value. Unevaluated symbolic calls are returned for most exact arguments. For some special cases explicit symbolic representations are returned.

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:

ellipticK(1/2); ellipticF(PI/4, I);

Some special arguments return explicit symbolic representations:

ellipticF(PI/2, 1/2); ellipticF(1, 1);

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:

ellipticPi(0.5, PI/3, 1);



An arithmetical expression specifying the parameter.


An arithmetical expression specifying the amplitude. In case of ellipticE and ellipticPi, the default is .

Return Values

Arithmetical expression.

See Also

MuPAD Functions

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