Definitions:
- 1) Euler: , ();
- 2) Gauss: , ()
- 3) Weierstrass: , where is the Euler-Mascheroni constant.
The Mittag-Leffler expansion:
It indicates that .
Properties:
- .
- , ().
- .
- The multiplication theorem: .
- The duplication formula: .
Hankel representations:
- ,
- ,
- .
Here the contour comes from plus infinity narrowly above the axis, circles once around origin and returns to plus infinity narrowly below the real axis.
Stirling formula:
Misc:
- , .
Source: “The Functions of Mathematical Physics” by Harry Hochstadt