Mathematical Physics Vol 1

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

245

It is easy to show that the above integral converges for all complex values z for which Re ( z ) > 0. Namely, Γ ( x + iy )= ∞ Z 0 e − t t x − 1 + iy d t = ∞ Z 0 e − t t x − 1 e iy ln t d t =

(5.118)

∞ Z 0 e − t t x − 1 [ cos ( y ln t )+ i sin ( y ln t )] d t

=

The expression in the square brackets is bounded for each t (as sin and cos are bounded functions), and thus the convergence, when t → ∞ , is provided by the term e − t , while for the convergence in zero it is necessary to ensure that x = Re ( z ) > 1. A more general definition of the gamma function can often be found in literature. Definition Gamma function Γ is a meromorphic function of the independent variable z (without the additional condition Re ( z ) > 0) so that its reciprocal value is 1 Γ ( z ) = ze γ z ∞ ∏ k = 1 1 + z k − z / k , (5.119) where γ = 0 , 57721566 ... is the Euler constant.

Definition Euler constant γ is determined by the following expression

ln n ! ≈ 0 , 5772156649 .

n ∑ k = 1

1 k −

γ = lim n → ∞

(5.120)

This function satisfies the reduction formula ,, that is, the following is valid

Γ ( z + 1 )= z Γ ( z ) , Re ( z ) > 0

(5.121)

which can be proved by partial integration

∞ Z 0

∞ Z 0

e − t t z d t = − e − t t z

t = ∞ t = 0

e − t t z − 1 d t = z Γ ( z ) .

Γ ( z + 1 )=

+ z

(5.122)

In the special case, when the variable z is a real number ( z = R ( z ) ), the Gamma function is determined by the expression

∞ Z 0

e − x · x n − 1 d x , n > 0 .

Γ ( n )=

(5.123)

This function is also known as the Euler integral of the second kind .