## Wednesday, May 4, 2011

### Gamma Function

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)

(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

given
proof

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)