scipy.special.expi#
- scipy.special.expi(x, out=None) = <ufunc 'expi'>#
- Exponential integral Ei. - For real \(x\), the exponential integral is defined as [1] \[Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt.\]- For \(x > 0\) the integral is understood as a Cauchy principal value. - It is extended to the complex plane by analytic continuation of the function on the interval \((0, \infty)\). The complex variant has a branch cut on the negative real axis. - Parameters:
- xarray_like
- Real or complex valued argument 
- outndarray, optional
- Optional output array for the function results 
 
- Returns:
- scalar or ndarray
- Values of the exponential integral 
 
 - Notes - The exponential integrals \(E_1\) and \(Ei\) satisfy the relation \[E_1(x) = -Ei(-x)\]- for \(x > 0\). - References [1]- Digital Library of Mathematical Functions, 6.2.5 https://dlmf.nist.gov/6.2#E5 - Examples - >>> import numpy as np >>> import scipy.special as sc - It is related to - exp1.- >>> x = np.array([1, 2, 3, 4]) >>> -sc.expi(-x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) >>> sc.exp1(x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) - The complex variant has a branch cut on the negative real axis. - >>> sc.expi(-1 + 1e-12j) (-0.21938393439552062+3.1415926535894254j) >>> sc.expi(-1 - 1e-12j) (-0.21938393439552062-3.1415926535894254j) - As the complex variant approaches the branch cut, the real parts approach the value of the real variant. - >>> sc.expi(-1) -0.21938393439552062 - The SciPy implementation returns the real variant for complex values on the branch cut. - >>> sc.expi(complex(-1, 0.0)) (-0.21938393439552062-0j) >>> sc.expi(complex(-1, -0.0)) (-0.21938393439552062-0j)