Mathematical Physics Vol 1

A.2 Notation and Special Functions

457

Complementary error function

erfc ( z )= 1 − erf ( z ) , z ∈ C

(A.7)

Mittag-Leffler function, or one-parameter Mittag-Leffler function

∞ ∑ k = 0

z k Γ ( 1 + α k )

, ℜ ( α ) > 0 , z ∈ C .

E α ( z )=

Generalized Mittag-Leffler function with two-paramete function

∞ ∑ k = 0

z k Γ ( β + α k )

, ℜ ( α ) > 0 , ℜ ( β ) > 0 , z ∈ C .

E αβ ( z )=

(A.8)

Generalized Mittag-Leffler function with three-paramete function

∞ ∑ k = 0

( ρ ) k z k Γ ( β + α k ) k !

E ρ

, α , β , ρ ∈ C , ℜ ( α ) > 0 , z ∈ C .

αβ ( z )=

(A.9)

Miller-Ross function

+ ∞ ∑ k = 0

a k z k + ν Γ ( ν + k + 1 )

ν E

E z ( ν , a )=

= z

1 , ν + 1 ( az ) , z ∈ C .

(A.10)

Hypergeometric function

p F q ( a 1 ,..., a p ; b 1 ,..., b q ; z )= 1 + ∞ ∑ k = 1 " z k k !

. # (A.11)

k − 1 ∏ n = 0

( a 1 + n )( a 2 + n ) ··· ( a p + n ) ( b 1 + n )( b 2 + n ) ··· ( b q + n )

Bessel functions of the first kind

( − 1 ) m m ! Γ ( m + α + 1 )

x

2 m + α

∞ ∑ k = 0

1 2

J α ( x )=

(A.12)

.

Modified Bessel functions of the first kind

α J

I α ( x )= j −

α ( jx ) .

(A.13)

Bessel functions of the second kind

J α ( x ) cos ( απ ) − J

x )

− α (

Y α =

(A.14)

.

sin | ( απ )

Hermite polynomial

n d x n

e − x 2

x 2 d

H n ( x )= e

(A.15)

.