# scipy.special.ellipe¶

scipy.special.ellipe(m) = <ufunc 'ellipe'>

Complete elliptic integral of the second kind

This function is defined as

$E(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{1/2} dt$
Parameters: m : array_like Defines the parameter of the elliptic integral. E : ndarray Value of the elliptic integral.

ellipkm1
Complete elliptic integral of the first kind, near m = 1
ellipk
Complete elliptic integral of the first kind
ellipkinc
Incomplete elliptic integral of the first kind
ellipeinc
Incomplete elliptic integral of the second kind

Notes

Wrapper for the Cephes  routine ellpe.

For m > 0 the computation uses the approximation,

$E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),$

where $$P$$ and $$Q$$ are tenth-order polynomials. For m < 0, the relation

$E(m) = E(m/(m - 1)) \sqrt(1-m)$

is used.

The parameterization in terms of $$m$$ follows that of section 17.2 in . Other parameterizations in terms of the complementary parameter $$1 - m$$, modular angle $$\sin^2(\alpha) = m$$, or modulus $$k^2 = m$$ are also used, so be careful that you choose the correct parameter.

References

  (1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html
  (1, 2) Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

#### Previous topic

scipy.special.ellipkinc

#### Next topic

scipy.special.ellipeinc