Complete and incomplete elliptic integrals of the second kind
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ellipticE(<ϕ
>, m
)
ellipticE(m)
represents the complete elliptic
integral of the second kind $$E\left(m\right)$$ which
is defined as
$$E\left(m\right)=E\left(\frac{\pi}{2}m\right)={\displaystyle \underset{0}{\overset{\pi /2}{\int}}\sqrt{1m{\mathrm{sin}}^{2}\theta}\text{\hspace{0.17em}}d\theta}$$
ellipticE(φ,m)
represents the incomplete
elliptic integral of the second kind $$E\left(\phi m\right)$$ which
is defined as
$$E\left(\phi m\right)={\displaystyle \underset{0}{\overset{\phi}{\int}}\sqrt{1m{\mathrm{sin}}^{2}\theta}\text{\hspace{0.17em}}d\theta}$$
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 floatingpoint
value, ellipticE
returns floatingpoint results.
For most exact arguments, it returns unevaluated symbolic calls. You
can approximate such results with floatingpoint numbers using the float
function.
When called with floatingpoint 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 floatingpoint numbers, use float
:
ellipticE(PI/4), ellipticE(PI/4, I); float(ellipticE(PI/4)), float(ellipticE(PI/4, I))
Alternatively, use floatingpoint values as arguments. If one argument is a floatingpoint value and the others can be converted to a floatingpoint values, then a floatingpoint 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 $$\frac{\pi}{2}$$. 
Arithmetical expression.