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:
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).
It becomes zero for, i.e.,
.
The function has the following relation for fixed and:
The function increases with for fixed
Table 7. Values of Ar(1, n) for r = 6 - 19.
n, i.e.
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.
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.
For, it is a linear function of, i.e.,
.