Complete and incomplete elliptic integrals of the second kind
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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.
If all arguments are numerical and at least one is a floating-point
ellipticE returns floating-point results.
For most exact arguments, it returns unevaluated symbolic calls. You
can approximate such results with floating-point numbers using the
When called with floating-point arguments, this function is
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
Most calls with exact arguments return themselves unevaluated.
To approximate such values with floating-point numbers, use
ellipticE(-PI/4), ellipticE(PI/4, I); float(ellipticE(-PI/4)), float(ellipticE(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(1/2), ellipticE(1/4, I); ellipticE(0.5), ellipticE(0.25, I)
Some special arguments return explicit symbolic representations:
ellipticE(0), ellipticE(1), ellipticE(0, m), ellipticE(p, 0)
An arithmetical expression specifying the parameter.
An arithmetical expression specifying the amplitude. The default is .