# §10.9 Integral Representations

In particular,

## §10.9(ii) Contour Integrals

### ¶ Schläfli’s Integral

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.

### ¶ Hankel’s Integrals

In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing . Also, is continuous on the path, and takes its principal value at the intersection with the interval .

### ¶ Mellin–Barnes Type Integrals

where the integration path passes to the left of .

where is a positive constant and the integration path encloses the points .

For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116).

## §10.9(iii) Products

### ¶ Mellin–Barnes Type

where the path of integration separates the poles of from those of . See Paris and Kaminski (2001, p. 116) for related results.

### ¶ Nicholson’s Integral

10.9.30.

For the function see §10.25(ii).

## §10.9(iv) Compendia

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).