American Journal of Computational Mathematics
Vol.2 No.1(2012), Article ID:17969,4 pages DOI:10.4236/ajcm.2012.21008
Examination of a Special Function Defined by an Integral
Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey
Email: hversoy@yildiz.edu.tr
Received September 14, 2011; revised October 3, 2011; accepted October 15, 2011
Keywords: Special Function; Complementary Error Function; Gamma Function
ABSTRACT
The aim of this short note is to examine the properties of a special function defined by an integral which was appeared in a paper by Ersoy. It is revealed that the function 
for 
is expressed in terms of the gamma function and it varies linearly with 
for
. Its appropriate graphs are plotted and its pertinent values are tabulated.
1. Introduction
Ersoy [1] obtained a series solution which is rapidly convergent at small times while he investigated an expression for velocity field of an unsteady flow between eccentric rotating disks. In this short note, the properties of the special function used in the series solution are presented.
2. Definition of the Function 
The function 
is defined as follows:
(1)
where
(2)
and 
denote the complementary error function and the repeated integrals of the complementary error function, respectively [2].
3. Main Results
In order to acquire the properties of the function
, computer-assisted research is done. Furthermore, the illustrative graphs are shown in Figures 1-6 and the elucidative values are provided in Tables 1-7. The results are noted as follows:
Ÿ The function 
that is a continuous function is an odd function of
, i.e.,
.
Figure 1. Variation of 
with m (n = 0.1, r = 0, 1, 2, 3).
Figure 2. Variation of 
with m (n = 0.1, r = 4, 5, 6).
Figure 3. Variation of 
with m (n = 0.5, r = 0, 1, 2, 3).
Figure 4. Variation of 
with m (n = 0.5, r = 4, 5, 6).
Figure 5. Variation of 
with m (n = 1, r = 0, 1, 2, 3).
Figure 6. Variation of 
with m (n = 1, r = 4, 5, 6).
Table 1. Some pertinent values of
.
It becomes zero for
, i.e.,
.
Ÿ The function 
has the following relation for fixed 
and
:
Ÿ 
Ÿ The function 
increases with 
for fixed
Table 2. Some pertinent values of
.
Table 3. Some pertinent values of
.
Table 4. Some pertinent values of
.
Table 5. Some pertinent values of
.
Table 6. Some pertinent values of
.
Table 7. Values of Ar(1, n) for r = 6 - 19.
n, i.e.Ÿ 
Ÿ The functions have maximum values for
. Moreover, they have the same values for 
and any value of
. These values are as follows:
or
where 
is the gamma function.
Ÿ When 
is larger, the function 
is also larger for any fixed value of
, i.e.,
For large values of
, it approximately varies linearly with
, i.e.,
.
is more linear than
.
Ÿ For
, it is a linear function of
, i.e.,
.
REFERENCES
- H. V. Ersoy, “Unsteady Flow Due to a Sudden Pull of Eccentric Rotating Disks,” International Journal of Engineering Science, Vol. 39, No. 3, 2001, pp. 343-354. doi:10.1016/S0020-7225(00)00040-9
- M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications, New York, 1972.