where upper signs apply if and lower signs if . This dichotomy of signs (missing in several references) is due to Fettis (1970).
For two further transformations of this type see Erdélyi et al. (1953b, p. 316).
There are three relations connecting and , where is a rational function of . If and are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)).
The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of when (see (19.6.5) for the complete case). Let . Then
Since we have ; hence implies .
The second relation maps each hyperbolic region onto itself and each circular region onto the other:
The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: