Most properties of follow straightforwardly from those of . For an extensive treatment of see Chapter 6.
When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of , and unless indicated otherwise in the DLMF principal values are assumed.
with in both equations. For see §5.2(i).
again with in both equations. The right-hand sides are replaced by their limiting forms when .
For see §13.2(i).
See also Cuyt et al. (2008, pp. 277–285).
The general function is attained by extending the path in (8.19.2) across the negative real axis. Unless is a nonpositive integer, has a branch point at . For each branch of is an entire function of .
For and ,