where is Euler’s constant (§5.2(ii)).
where the integration path is a simple loop contour, and is continuous on the path and takes its principal value at the intersection with the positive real axis.
where the integration path passes to the left of .
where is a positive constant and the integration path encloses the points .
where the square root has its principal value.
where is a positive constant. For the function see §10.25(ii).
where the path of integration separates the poles of from those of . See Paris and Kaminski (2001, p. 116) for related results.
For the function see §10.25(ii).
For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).