# 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*) + *i*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 |
---|---|

The exponential integral Ei( | |

The exponential integral E1( | |

The logarithmic integral li( | |

The cosine integral Ci( | |

The sine integral Si( |