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ellipticE

Complete and incomplete elliptic integrals of the second kind

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Syntax

ellipticE(<ϕ>, m)

Description

ellipticE(m) represents the complete elliptic integral of the second kind $E\left(m\right)$ which is defined as

$E\left(m\right)=E\left(\frac{\pi }{2}|m\right)=\underset{0}{\overset{\pi /2}{\int }}\sqrt{1-m{\mathrm{sin}}^{2}\theta }\text{\hspace{0.17em}}d\theta$

ellipticE(φ,m) represents the incomplete elliptic integral of the second kind $E\left(\phi |m\right)$ which is defined as

$E\left(\phi |m\right)=\underset{0}{\overset{\phi }{\int }}\sqrt{1-m{\mathrm{sin}}^{2}\theta }\text{\hspace{0.17em}}d\theta$

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 floating-point value, 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 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.

Examples

Example 1

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

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)

Parameters

 m An arithmetical expression specifying the parameter. φ An arithmetical expression specifying the amplitude. The default is $\frac{\pi }{2}$.

Return Values

Arithmetical expression.