Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

248

Based on (5.140), the Γ – function can be extended for n < 0, starting with the interval ( − 1 , 0 ) , followed by the interval ( − 2 , − 1 ) , etc. The function thus extended is depicted in Fig. 5.8.

Figure 5.6: Γ function.

Also, using the relation (5.119), the gamma function can also be extended to the left half plane, that is, for the values Re ( z ) ≤ 0

Γ ( z + n ) ( z ) n

Γ ( z )=

, Re ( z ) > − n , n ∈ N , z ∈ Z − 0 : = { 0 , − 1 , − 2 ,... }

(5.141)

where ( z ) n for z ∈ C , n ∈ N 0 is defined by

( z ) 0 = 1 , ( z ) n = z ( z + 1 ) ··· ( z + n − 1 ) ,

(5.142)

The previous relations yield the following identities

Γ ( n + 1 )=( 1 ) n = n ! ,

(5.143)

Γ ( z + n ) z ( z + 1 ) ··· ( z + n − 1 ) .

Γ ( z )=

(5.144)

Definition For z ∈ C \{ 0 , − 1 , − 2 , − 3 , ···} Euler Gamma function is defined by Γ ( z + 1 )=   ∞ R 0 e − t t z − 1 d t , if Re ( z ) > 0 Γ ( z + 1 ) / z , if Re ( z ) ≤ 0 , z̸ = 0 , − 1 , − 2 ,... that is, the function is defined in the entire complex plane except for zero and points having a negative integer value. (5.145)