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:

$$\text{operatorName}Ei(x)=-{\int }_{-x}^{inf}\frac{e^{-t}dt}{t}$$

The exponential integral of order $n$ is defined as:

$${\text{E}}_n(x)={\int }_1^{inf}\frac{e^{-tx}dt}{t^n}$$

When $n=1$, we get the E1 function, which is related to the exponential integral by the relation

$$E_1(x)=-\text{operatorName}Ei(-x)$$

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

$$\operatorname{li}(x) = \operatorname{Ei}(\ln x).

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

$$\operatorname{Ei}(ix) = \operatorname{Ci}(x) + i\operatorname{Si}(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
ExpIntegralEi	The exponential integral Ei(x).
ExpIntegralE1	The exponential integral E1(x).
ExpIntegralE	The exponential integral of order n, En(x).
LogIntegral	The logarithmic integral li(x).
CosIntegral	The cosine integral Ci(x).
SinIntegral	The sine integral Si(x).