Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas

458

A.2.3 Properties of the Mittag-Leffler functions: special values

z , E 1

E α , 1 ( z )= E α ( z ) , E 1 ( z )= E 1 , 1 ( z )= E t ( 0 , 1 )= e

α , β ( z )= E α , β ( z ) ,

1 1 − z

, | z | < 1 and E 1 ( z )= E 1 , 1 ( z )= E t ( 0 , 1 )= e z ,

E 0 ( z )=

sh ( z ) z

2 )= ch ( √ z ) and E

2 )=

2 )= E

E 2 ( z

2 , 1 ( z

2 , 2 ( z

,

1 2 " 1 2

3 √ z !# ,

√ 3 2

√ z

3 √ z

1 2

e 3

+ 2 e −

E 3 ( z )=

cos

cos 4 √ z + ch 4 √ z , and E 2 , 2 =

E 4 ( z )=

sh ( √ z ) √ z

e z − 1 z

E 1 , 2 ( z )=

.

A.2.4 Generalized exponential functions Let z , λ ∈ C , ℜ ( α ) and n ∈ N Then E α ( λ z α )= ∞ ∑ k = 0

λ k z k α Γ ( α k + 1 ) ,

(A.16)

∞ ∑ k = 0

λ k z ( k + 1 ) α − 1 Γ ( k α + α ) ,

e λ z

α − 1 E

α

α , α ( λ z

α = z

(A.17)

)=

e λ z

α − 1 E n + 1

α , ( n + 1 ) α ( λ z α

α , n = n ! z

(A.18)

) .

Generalized exponential functions - the following properties

c D α

a + E α ( λ ( z − a ) α

)( x )= λ E α ( λ ( x − a ) α ) ,

(A.19) (A.20)

E α ( λ ( z − a ) α

lim z → a +

)= 1 ,

lim z → a + h

α i =

1 Γ ( α )

α e λ ( z − a )

( z − a ) 1 −

(A.21)

,

∂

∂ z

n

α

λ z α

)]= z − n E

[ E α ( λ z

(A.22)

α , 1 − n (

) ,

∂

∂ z

n h

α i = z

e λ z

α − n − 1 E

λ z α

(A.23)

α , α − n (

) ,

∂

∂λ

n

α

α n E n + 1

α

[ E α ( λ z

α , α n + 1 ( λ z

)]= n ! z

(A.24)

) ,

∂

∂λ

n h

α i = n ! z

e λ z

α n + α − 1 E n + 1

α , ( n + 1 ) α ( λ z α

(A.25)

) ,

1 n !

∂ ∂λ

n h

e λ z α i , z̸ = 0 .

e λ z

(A.26)

α , n =

The generalized α -exponential functions do not have the index property, that is, in general

( λ + µ ) z α

E α ( λ z ) E α ( µ z )̸= E α (( λ + µ ) z ) ; e λ z α e µ z α̸ = e

(A.27)

.