Connections within the group of exponential integrals and with other function groups

Representations through more general functions

The exponential integrals , , , , , , and are the particular cases of the more general hypergeometric and Meijer G functions.

For example, they can be represented through hypergeometric functions or the Tricomi confluent hypergeometric function :

Representations of the exponential integrals and , the sine and cosine integrals and , and the hyperbolic sine and cosine integrals and through classical Meijer G functions are rather simple:

Here is the Euler gamma constant and the complicated‐looking expression containing the two logarithm simplifies piecewise:

But the last four formulas that contain the Meijer G function can be simplified further by changing the classical Meijer functions to the generalized one. These formulas do not include factors and terms :

The corresponding representations of the logarithmic integral through the classical Meijer G function is more complicated and includes composition of the G function and a logarithmic function:

All six exponential integrals of one variable are the particular cases of the incomplete gamma function:

Representations through related equivalent functions

The exponential integral can be represented through the incomplete gamma function or the regularized incomplete gamma function:

Relations to inverse functions

The exponential integral is connected with the inverse of the regularized incomplete gamma function by the following formula:

Representations through other exponential integrals

The exponential integrals , , , , , , and are interconnected through the following formulas: