# §5.9 Integral Representations

## §5.9(i) Gamma Function

, , and . (The fractional powers have their principal values.)

### ¶ Hankel’s Loop Integral

5.9.2

where the contour begins at , circles the origin once in the positive direction, and returns to . has its principal value where crosses the positive real axis, and is continuous. See Figure 5.9.1.

Figure 5.9.1: -plane. Contour for Hankel’s loop integral.

where the path is the real axis.

### ¶ Binet’s Formula

For additional representations see Whittaker and Watson (1927, §§12.31–12.32).

For ,

where and .