Beta Function

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

G.ID.1:  integral representation

 

G.ID.1: Recurrence

 

 

Proof

(1)

(2)

(3)

 

 (4)

(5)

integrating by parts of (1) 

 

 (6)

from (1) and (6) 

 (7)

from (5) and (7)  

 (8)

from (7) and (8),

 (9)

 (10)

 (11)

G.ID.2:  integral representation

given

proof

(1)

substitute, 

(2)

(3)

(4)

(5)

Zeta-Gamma Function

GZ.ID.1:  Fundamental Zeta-Gamma  function
the relation between Zeta function and Gamma function can be represented in the following form:

Proof

Given

(1)

(2)

 (3)

(4)

substituting (4) into (3)

(5)

(6)

(7)

(8)

Based on Geometric Series of Exponential (GS.ID.2),

(9)

(10)

(11) 

based on (10) and (11) , 

 

 (12) 

 

(13) 

(14) 

 

 (15) 

 (16) 

GZ.ID.2:   Zeta-Gamma  function (Trigonometric)

Proof

Given

(1)

 (2)

(3)

(4)

substituting (4) into (1) ,

(5)

(6)

(7)

substituting (2) into (7) :

(8)

from (16),  at (GZ.ID.1)  ,

(9)

(10)

based on (9) and (10) , then

 multiplying and dividing the denominator by   ,   and based on (3) therefore 

GZ.ID.3:   Zeta-Gamma  function

 

Given

(1)

 (2)

 (3)

substitute (3) into (2) , 

(4)

 (5)

(6)

differentiate (6) ,

(7)

substitute (7) into (5) ,

(8)