Mathematical Physics Vol 1

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

251

This function has poles of the first order and can further be expressed by the following formula

( − 1 ) k ( z + k ) k !

Γ ( z )=

[ 1 + O ( z + k )] , z → k , k ∈ N 0 ,

(5.160)

which is obtained by the substitutions z = 1 − n , n − 1 = k . The coefficient ( z + k ) − 1 in the vicinity of the pole z = − k is called the residue of the gamma function Re s Γ ( z )= ( − 1 ) k k ! . Here f ( z )= O ( g ( z )) z → a denotes a function, where for ε > 0 forwhich | z − a | < ε ⇒ f ( z ) g ( z ) < M , for some M < ∞ . The Euler Psi function is also defined as

Γ ′ ( z ) Γ ( z )

d d z

ψ ( z )=

ln Γ ( z )=

(5.161)

,

and Legendre formula as

2 2 z − 1 √ π

Γ ( 2 z )=

Γ ( z )( z + 1 / 2 )

(5.162)

as well as the generalized Gauss-Legendre multiplication formula

m − 1 ∏ k = 0

m mz − 1 / 2 2 π ( m − 1 ) / 2

Γ ( z + k / m ) , m = 2 , 3 ,...

Γ ( mz )=

(5.163)

Stirling formula

Γ ( 2 z )= √ 2 π z z − 1 / 2 e − z [ 1 + O ( 1 / z )] ,

| arg z | < π , | z |→ ∞

(5.164)

and its consequences

n ! = √ 2 π n

n e

n [ 1 + O ( 1 / n )] , n → ∞

(5.165)

√ 2 π

| y | x − 1 / 2 e − π | y | / 2

| Γ ( x + iy ) | =

[ 1 + O ( 1 / y )] , y → ∞ .

(5.166)

5.6.2 Beta function

Definition Beta function is defined by the following relation

1 Z 0

x m − 1 · ( 1 − x ) n − 1 d x

B ( m , n )=

(5.167)

for each m > 0 and n > 0. This condition is necessary for convergence of the integral.

The function (5.167) is also known as Euler integral of the first kind . Beta function can be related to Γ – function, starting from (5.167) and introducing the substitution x = cos 2 ϕ , which yields (according to (5.135)):

π / 2 Z 0 ( cos ϕ ) 2 m − 1 ( sin ϕ ) 2 n − 1 d ϕ =

Γ ( m ) · Γ ( n ) Γ ( m + n )

B ( m , n )= 2

= B ( m , n ) .

(5.168)

On this basis, the definition can now be broadened to the set of complex numbers.