Chebyshev approximations for the exponential integral 𝐸𝑖(𝑥)

Abstract

The computation of the exponential integral E i ( x ) Ei(x) , x > 0 x > 0 , using rational Chebyshev approximations is discussed. The necessary approximations are presented in well-conditioned forms for the intervals ( 0 , 6 ] (0,6] , [ 6 , 12 ] [6,12] , [ 12 , 24 ] [12,24] and [ 24 , ∞ ) [24,\infty ) . Maximal relative errors are as low as from 8 × 10 − 19 t o 2 × 10 − 21 8 \times {10^{ - 19}}to2 \times {10^{ - 21}} . In addition, the value of the zero of E i ( x ) Ei(x) is presented to 30 decimal places.