Γ(x) · Γ(1 − x) = π/sin (xπ),
with the particular result
Γ(½)= √π.
The number
| − [ | d | { log Γ (1 + x) } ]x=0, or −Γ′(1), |
| dx |
is called “Euler’s constant,” and is equal to the limit
| lim.n=∞ [ ( 1 + ½ + 1⁄3 + ... + | 1 | ) − log n ]; |
| n |
its value to 15 decimal places is 0.577 215 664 901 532.
The function log Γ(1 + x) can be expanded in the series
| log Γ (1 + x) = ½ log ( | xπ | ) − ½ log | 1 + x | + { 1 + Γ′;(1) } x |
| sin xπ | 1 − x |