content_image/306792.jpg content_image/306791.jpg content_image/306790.jpg content_image/306787.jpg content_image/306789.jpg content_image/306788.jpg
is 

<img src='content_image/306792.jpg'>

Note that there is no value of z for which Γ ( z ) = 0, hence [Γ ( z )] − 1 is an entire function. An integral representation is given by 

<img src='content_image/306791.jpg'>

Using (B.3), it can be shown that Γ ( 1 2 ) = √ π . The argument of the Gamma function can be simplified using the identity 

<img src='content_image/306790.jpg'>

In order to construct series expansions of the Gamma function, it is helpful to introduce the Digamma function 

and the closely related Polygamma function 

Numerical values of these functions at z = 1 are 

<img src='content_image/306787.jpg'>

<img src='content_image/306789.jpg'>

<img src='content_image/306788.jpg'>

where γ$_{E}$ ≃ 0 . 577 is the Euler-Mascheroni constant and ζ ( z ) is the Riemann Zeta function. 

