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 ![](https://www.scirp.org/html/8-1100067\efa99859-6369-418d-b5bf-bb99fb70b700.jpg)
The function
is defined as follows:
(1)
![](https://www.scirp.org/html/8-1100067\4af90d43-2551-4f27-a2e1-cc7e44f3b7eb.jpg)
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:
The function
that is a continuous function is an odd function of
, i.e.,
.
![](https://www.scirp.org/html/8-1100067\00b6f131-deac-48f6-beba-38a596a00696.jpg)
Figure 1. Variation of
with m (n = 0.1, r = 0, 1, 2, 3).
![](https://www.scirp.org/html/8-1100067\1a764394-33c8-459a-8a19-fdaedcd7a845.jpg)
Figure 2. Variation of
with m (n = 0.1, r = 4, 5, 6).
![](https://www.scirp.org/html/8-1100067\305aeb63-2f5d-471a-8f30-f84b7b3e83bb.jpg)
Figure 3. Variation of
with m (n = 0.5, r = 0, 1, 2, 3).
![](https://www.scirp.org/html/8-1100067\fe8f04a6-0384-4320-b4d6-865e8932e972.jpg)
Figure 4. Variation of
with m (n = 0.5, r = 4, 5, 6).
![](https://www.scirp.org/html/8-1100067\15750349-44f0-43f3-99ec-b65046a1225e.jpg)
Figure 5. Variation of
with m (n = 1, r = 0, 1, 2, 3).
![](https://www.scirp.org/html/8-1100067\50dc5a39-54fd-4480-ade1-a0e90dc70876.jpg)
Figure 6. Variation of
with m (n = 1, r = 4, 5, 6).
It becomes zero for
, i.e.,
.
The function
has the following relation for fixed
and
:
![](https://www.scirp.org/html/8-1100067\698e7567-6417-41d8-9c3e-2fedd3f6ea6a.jpg)
The function
increases with
for fixed
![](https://www.scirp.org/html/8-1100067\03dd74c0-9e71-477f-a30d-d0fdaeafd49b.jpg)
Table 7. Values of Ar(1, n) for r = 6 - 19.
n, i.e. ![](https://www.scirp.org/html/8-1100067\0ad99c55-5554-4afa-826c-830b9540decf.jpg)
The functions have maximum values for
. Moreover, they have the same values for
and any value of
. These values are as follows:
![](https://www.scirp.org/html/8-1100067\99ef2a14-ab96-4607-8106-499e823b747a.jpg)
or
![](https://www.scirp.org/html/8-1100067\5698e0aa-8f72-4581-bbf3-dd4f586f474c.jpg)
where
is the gamma function.
When
is larger, the function
is also larger for any fixed value of
, i.e.,
![](https://www.scirp.org/html/8-1100067\609f7371-a106-4ac4-bbea-46dccdb984bf.jpg)
For large values of
, it approximately varies linearly with
, i.e.,
.
is more linear than
.
For
, it is a linear function of
, i.e.,
.