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)



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