Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas

456

Heaviside function - H ( x )= ( 1 , if x ≥ x 0 0 , if x < x 0 . Pringsheim notation of continued fraction (which need not have an infinite number of terms)

= a 0 ;

, ··· = a 0 ,

b k a k

+ ∞

b 1

b 1 a 1

b 2 a 2

b 3 a 3

b 4 a 4

a 0 +

,

,

,

b 2

k = 1

a 1 +

b 3

a 2 +

b 4

a 3 +

b 5 a 5 + ···

a 4 +

Levi-Civita symbol

ε ℓ mn =  

+ 1 , if ( ℓ, m , n )=( 1 , 2 , 3 ) , ( 3 , 1 , 2 ) , ( 2 , 3 , 1 ) , − 1 , if ( ℓ, m , n )=( 1 , 3 , 2 ) , ( 3 , 2 , 1 ) , ( 2 , 1 , 3 ) , 0 , if ℓ = m ∨ ℓ = n , ∨ m = n .

A.2.2 Definitions of some Special Functions Euler’s gamma function

   + ∞ R 0

e − y y z − 1 d y , if ℜ ( z ) > 0 ,

Γ ( z )=

(A.1)

if ℜ ( z ) > − n , n ∈ N ∧ z / ∈ Z − 0 .

Γ ( z + n ) ( z ) n

,

Pochhammer function

( ρ ) 0 = 1 and ( ρ ) k = ρ ( ρ + 1 ) ··· ( ρ + k − 1 ) , k ∈ N .

(A.2)

Combinations of a things, b at a time

     0 ,

Γ ( a + 1 ) Γ ( b + 1 ) Γ ( a − b + 1 )

, if a , b , a − b / ∈ Z −

a

b

( − 1 ) b Γ ( b − a ) Γ ( b + 1 ) Γ ( − a ) ,

=

if a ∈ Z − ∧ b ∈ Z + 0

if [( b ∈ Z − ∨ b − a ∈ N ) ∧ a / ∈ Z − ] ∨ ( a , b ∈ Z − ∧| a | > | b | ) . (A.3)

Beta function

Γ ( x ) Γ ( y ) Γ ( x + y )

B ( x , y )= B ( y , x )=

(A.4)

.

Digamma function

dlog Γ ( x ) d x

d Γ ( x ) d x

1 Γ ( x )

ψ ( x )=

(A.5)

=

.

Error function

z Z 0

2 √ π

e − t 2 d t

erf ( z )=

z ∈ C .

(A.6)

,