﻿Examination of a Special Function Defined by an Integral

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

H. Volkan Ersoy

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  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 .

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

1. 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
2. M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications, New York, 1972.