firstly , I recommend you to read these Gamma function to have fundamental understanding of the Gamma function.

G.ID.1: Differentiation (integral representation)

differentiate gamma function in term of x variable

if x=1

G.ID.2: integral representation

proof

given that

variable substitution

(1)

(2)

from (1), lower and upper integrals boundaries change

(3)

(4)

(5)

G.ID.3: integral representation

proof

given that

variable substitution

G.ID.4: Weierstrass Identity

# = Euler–Mascheroni constant

Given

Proof

(1)

(2)

(3)

the cancels out, and based on (3) , therefore,

(4)

(5)

multiply 4 by 5 , therefore

(6)

(7)

(8)

# = Euler–Mascheroni constant

therefore;

(9)

G.ID.5: Reflection Formula

Given

(1)

(2)

Proof

multiplication of (1) and (2)

multiplication of (1) and (2)

(3)

sine identity

(4)

Recurrence identity of gamma function

(5)

based on (3) and (4) , (5)

(5)

(6)

replace x and (1-x) with the following identities

(7)

therefore

(8)

(9)

substituting (9) into (8) results in

G.ID.6: Recurrence Identity

proof

given

if s=s+1/2

integrating by parts results:

after n times of integration by parts:

similarly for s+1/3 and s+1/4

G.ID.7:

proof

(1)

(2)

(3)

(4)

(5)

(6)