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
- marray_like
Defines the parameter of the elliptic integral.
- Returns
- Endarray
Value of the elliptic integral.
See also
Notes
Wrapper for the Cephes [1] 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 [2]. 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
Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
- 2
Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
This function is used in finding the circumference of an ellipse with semi-major axis a and semi-minor axis b.
>>> from scipy import special
>>> a = 3.5 >>> b = 2.1 >>> e_sq = 1.0 - b**2/a**2 # eccentricity squared
Then the circumference is found using the following:
>>> C = 4*a*special.ellipe(e_sq) # circumference formula >>> C 17.868899204378693
When a and b are the same (meaning eccentricity is 0), this reduces to the circumference of a circle.
>>> 4*a*special.ellipe(0.0) # formula for ellipse with a = b 21.991148575128552 >>> 2*np.pi*a # formula for circle of radius a 21.991148575128552