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)

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)