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