ABSTRACT
Functions
3.1 The Family of Incomplete Gamma Functions
The exponential and logarithmic integral functions, sine and cosine integral
functions, error functions and Fresnel integrals and their generalizations be-
long to the family of the incomplete gamma functions
(; z) :=
Z
z
t
e
t
dt
= + i; > 0
; (3.1)
(; z) :=
Z
1e
i
z
t
e
t
dt = () (; z)
jj <
; > 0; jj =
; 0 < < 1
: (3.2)
It is to be noted that the integral in (3.2) exists without the restriction on
when z 6= 0 and jj < =2. In particular when = 0, we have ([183], p. 144)
(; z) :=
Z
z
t
e
t
dt
z 6= 0; z = 0; > 0
: (3.3)
We dene the family of the incomplete gamma functions and state their well-
known properties useful for applications and write their relationships with
the incomplete gamma functions. These relationships are found to be special
cases of the relations satised by the family of the generalized incomplete
gamma functions introduced in this chapter. For further details we refer to
The error functions dened by
erf(z) :=
p
Z
z
exp(t
)dt; (3.4)
erfc(z) :=
p
Z
z
exp(t
)dt = 1 erf(z); (3.5)
were originally introduced by Kramp [165]. Hartree [134] investigated the
repeated integrals
i
erfc(z) := erfc(z)
i
n
erfc(z) :=
Z
z
i
n1
erfc(t)dt =
p
Z
z
(t z)
n
n!
