scipy.special.ellipe#
- scipy.special.ellipe(m, out=None) = <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.
- outndarray, optional
Optional output array for the function values
- Returns:
- Escalar or ndarray
Value of the elliptic integral.
See also
ellipkm1Complete elliptic integral of the first kind, near m = 1
ellipkComplete elliptic integral of the first kind
ellipkincIncomplete elliptic integral of the first kind
ellipeincIncomplete elliptic integral of the second kind
elliprdSymmetric elliptic integral of the second kind.
elliprgCompletely-symmetric elliptic integral of the second kind.
Notes
Wrapper for the Cephes [1] routine ellpe.
For
m > 0the 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.
The Legendre E integral is related to Carlson’s symmetric R_D or R_G functions in multiple ways [3]. For example,
\[E(m) = 2 R_G(0, 1-k^2, 1) .\]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.
[3]NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i
Examples
This function is used in finding the circumference of an ellipse with semi-major axis a and semi-minor axis b.
>>> import numpy as np >>> 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