Exponential Integrals

Exponential integrals arise in many engineering applications. At least three different definitions have been used in the literature. Here, we use the most common definition:

$$\Ei(x) = -\int_{-x}^{\inf}\frac{e^{-t}\,dt}{t}$$

The E1 function is related to the exponential integral by the relation

$$E_1(x) = -\Ei(-x)$$

The logarithmic integral is related to the exponential integral by the relation

li(x) = Ei(ln x).

The sine and cosine integrals are related to the exponential integral by the relation

Ei(ix) = Ci(x) + iSi(x).

The Special class provides static methods for evaluating the exponential integral and related functions for real arguments, as listed in the table below.

Method

Description

ExponentialIntegral

The exponential integral Ei(x).

E1

The exponential integral E1(x).

LogarithmicIntegral

The logarithmic integral li(x).

CosineIntegral

The cosine integral Ci(x).

SineIntegral

The sine integral Si(x).