
1178 SEPARATION OF VARIABLES AND INTEGRAL TRANSFORM METHODS
where the integration path is parallel to the imaginary axis of the complex plane s and the
integral is understood in the sense of the C auchy principal value.
Formula (15.2.4.2) holds for continuous functions. If f (x) has a (finite) jump dis-
continuity at a point x = x
0
> 0, then the right-hand side of (15.2.4.2) evaluates to
1
2
f(x
0
− 0) + f (x
0
+ 0)
at this point (for x
0
= 0, the first term in the square brack-
ets must be omitted).
2
◦
. The main properties of the correspondence between the functions and their Mellin
transforms are gathered in Table 15.5.
TABLE 15.5
Main properties of the Mellin ...