Mathematical Physics Vol 1

5.6 Special functions that are not a result of the Frobenius method

249

Figure 5.7: Euler Γ ( z ) function (blue) and its reciprocal function (red) in the interval [ − 4 , 4 ] .

Properties of Euler gamma function Euler gamma function has the following properties 1) for Re ( z ) ≤ 0 the integral ∞ R 0

e − t t z − 1 d t is equivalent to the following expression

1 Z 0

ln

1 t

z − 1

Γ ( z )=

d t ,

(5.146)

2) for z ∈ C \{ 0 , − 1 , − 2 ,... } the following reduction formula is valid Γ ( z + 1 )= z Γ ( z ) ,

(5.147)

3) for n ∈ N

Γ ( n )=( n − 1 ) ! ,

(5.148)

4) for z ∈ C \{ 0 , − 1 , − 2 ,... } the following formula is valid Γ ( 1 − z )= z Γ ( − z ) .

(5.149)

5) The gamma function can be expressed in terms of the limit value of the following expres sion Γ ( z )= lim n → ∞ n ! n z z ( z + 1 ) ··· ( z + n ) , Re ( z ) ≤ 0 . (5.150) The previous representation is equivalent to Euler’s infinite product

∞ ∏ n = 1

( 1 +( 1 / n )) z 1 +( z / n ) .

1 z

(5.151)

6) Let z ∈ C \{ 0 , − 1 , − 2 ,... } , then Euler gamma function can be defined by the following expression 1 Γ ( z ) = ze γ z ∞ ∏ k = 1 1 + z k − z / k , (5.152) where γ is the Euler constant. This expression is also known as the Weierstrass 12 definition of gamma function. 7) Euler gamma function is analytical for all z ∈ C \{ 0 , − 1 , − 2 ,... } . 8) Euler gamma function is always different from zero.