Incomplete elliptic integral of the first kind

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ellipticF(phi, m)


ellipticF(phi,m) represents the incomplete elliptic integral of the first kind F(φ|m) which is defined as


The incomplete elliptic integral of the first kind is defined for complex arguments ϕ and m.

If all arguments are numerical and at least one is a floating-point value, ellipticF 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:

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

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:

ellipticE(1/4, I);
ellipticE(0.25, I)

Some special arguments return explicit symbolic representations:

ellipticF(0, m),
ellipticF(p, 0)



An arithmetical expression specifying the parameter.


An arithmetical expression specifying the amplitude.

Return Values

Arithmetical expression.