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 - 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 
- elliprd
- Symmetric elliptic integral of the second kind. 
- elliprg
- Completely-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