If , then the Cauchy principal value satisfies
Exact values of and for various special
values of are given in Byrd and Friedman (1971, 111.10 and 111.11) and
Cooper et al. (2006).
For the inverse Gudermannian function see
§4.23(viii). Compare also (19.10.2).
Circular and hyperbolic cases, including Cauchy principal values, are unified
by using . Let and
For the Cauchy principal value of when
, see §19.7(iii).