Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

252

Definition Beta function B ( z , ω ) with respect to two variables z , ω ∈ C is defined by B ( z , ω )=

Γ ( z ) Γ ( ω ) Γ ( z + ω ) .

(5.169)

Figure 5.8: Beta function.

Properties of Beta function Beta function has the following properties. a) For Re ( z ) > 0 , Re ( ω ) > 0 the equation (5.169) is equivalent to

∞ Z 0

1 Z 0

t z − 1 ( 1 + t ) z + ω

ω − 1 d t

t z − 1 ( 1 − t )

B ( z , ω )=

d t =

=

(5.170)

π / 2 Z 0

ω − 1 d t

( sin t ) 2 z − 1 ( cos t ) 2

= 2

.

b) B ( z + 1 , ω + 1 ) is a solution of the Beta integral

1 Z 0

ω d t

t z ( 1 − t )

= B ( z + 1 , ω + 1 ) .

(5.171)

c) The following identities stand

B ( z , ω )= B ( ω , z ) , B ( z , ω )= B ( z + 1 , ω )+ B ( z , ω + 1 ) ,

(5.172)

ω z

ω z + ω

B ( z , ω + 1 )=

B ( z + 1 , ω )=

B ( z , ω ) .