E1_imaginary {hawkesbow}

Exponential integral of imaginary argument

Description

Calculates the value of

E_1(ix) = \int_1^\infty \frac{e^{-ixt}}{t} \mathrm{d}t

using its relation to the trigonometric integrals (cf. https://en.wikipedia.org/wiki/Exponential_integral#Exponential_integral_of_imaginary_argument):

E_1(ix) = i \left[ -\frac{1}{2} \pi + Si(x) \right] - Ci(x)

and their Pad\'e approximants (cf. https://en.wikipedia.org/wiki/Trigonometric_integral#Efficient_evaluation)

Usage

E1_imaginary(x)

Arguments

x

A non-negative number

Value

The exponential integral of argument ix

Examples

E1_imaginary(1.0)