The principal value of the exponential integral is defined by
where the path does not cross the negative real axis or pass through the origin. As in the case of the logarithm (§4.2(i)) there is a cut along the interval and the principal value is two-valued on .
Unless indicated otherwise, it is assumed throughout the DLMF that assumes its principal value. This is also true of the functions and defined in §6.2(ii).
is sometimes called the complementary exponential integral. It is entire.
In the next three equations .
( is undefined when , or when is not real.)
The logarithmic integral is defined by
The generalized exponential integral , , is treated in Chapter 8.
is an odd entire function.
where the path does not cross the negative real axis or pass through the origin. This is the principal value; compare (6.2.1).
is an even entire function.