Fractional approximations for linear first-order differential equations with polynomial coefficients—application to E1(x)

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Abstract
A method is described to obtain fractional approximations for linear first-order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all of the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the higher and lower powers of the variable after the substitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Padé method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which gives three exact digits for most of the real values of the variable. Approximations of higher accuracy than those of other authors are also obtained.