Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Volume 2, Issue 8, August 2014, PP 736-741 ISSN 2347-307X (Pint) & ISSN 2347-3142 (Online) www.acjounals.og Results on the Commutative Neutix Convolution Poduct Involving the Logaithmic Integal li( s ) and Depatment of Mathematics, King Abdulaziz Univesity Jeddah, Saudi Aabia falseehi@kau.edu.sa Abstact: The logaithmic integal li(x and its associated functions x and li - (x whee s 1, 2,. ae defined as locally summable functions on the eal line. The commutative neutix convolution poduct of these functions and x ae evaluated fo 0, 1, 2, Futhe esults ae also given. Keywods and Phases: Logaithmic integal, distibution, neutix, neutix convolution, commutative neutix convolution. Mathematics Subject Classification 2000: 33B10, 46F10 1. INTRODUCTION The logaithmic integal li( ), see Abamowitz and Stegun [1] is defined by li( ) whee PV denotes the Cauchy pincipal value of the integal, we will use li( ) PV fo all values of The logaithmic integal li( ) was genealized to li( )PV and its associated functions ) and li_( ) ae defined by ) H(x) li( ), li - ( ) H(- ) li( ) whee H( ) denotes Heaviside's function. It follows that li( ) PV (1) see [6]. The distibution -1 ln -1 [li( )]' -1 ln -1 is then defined by and its associated distibutions ln -1 x + and ln -1 - ae defined by ARC Page 736
fo 1, 2,. ln -1 + H( ) ln -1 - H(- ) -1 ln -1 [ -1 ln -1 [li - ( The classical definition of the convolution of two functions f and g is as follows: Definition1. Let f and g be functions. Then the convolution f*g is defined by (f*g)( ) fo all points x fo which the integal exist. It follows fom Definition 1 that if f * g exists then g*f exists and f*g g*f. (2) Futhemoe, if (f*g )' and f*g' (o f'*g) exist, then (f*g )' f*g' (o f'*g) (3) Gel'fand and Shilov [9] extended Definition 1 to define the convolution f*g of two distibutions f and g in D', the space of infinitely diffeentiable functions with compact suppot. Definition2. Let f and g be distibutions in D'. Then the convolution f*g is defined by the equation fo evey in D, povided f and g satisfy eithe of the conditions (a) eithe f o g has bounded suppot, (b) the suppots of f and g ae bounded on the same side. Note that if f and g ae locally summable functions satisfying eithe of the above conditions and the classical convolution f*g exists, then it is in ageement with Definition 1.1. The commutative neutix convolution poduct is defined in [4] and it woks fo a lage class of pais of distibutions. In that definition, unit-sequences of functions in D ae used which allows one to appoximate a given distibution by a sequence of distibutions of bounded suppot. To ecall the definition of the commutative neutix convolution we fist let τ be a function in D, see [10], satisfying the the following popeties: i. τ( ) τ(- ), ii. 0 τ( ) 1, iii. τ( ) 1 fo, iv. τ( ) 0 fo 1. The function τ n is now defined by )]', )]', τ n ( ) fo n 1, 2,. We have the following definition of the commutative neutix convolution poduct. Definition3. Let f and g be distibutions in D' and let f n fτ n and gn gτ n fo n 1, 2,. Then the commutative neutix convolution poduct f g is defined as the neutix limit of the sequence {f n *g n } n N, povided the limit h exists in the sense that fo evey in D, whee N is the neutix, see van de Coput [2], having domain N' of positive integes and ange N'' the eal numbes, with negligible functions finite linea sums of the functions Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Page 737
Results on the Commutative Neutix Convolution Poduct Involving the Logaithmic Integal Li(X and X n λ ln -1 n, ln n: λ>0, 1, 2, and all functions which convege to zeo in the nomal sense as n tend to infinity. Note that in this definition, the convolution poduct f n * g n is in the sense of Definition 1.1, the distibutions f n and g n having bounded suppot since the suppot of τ n is contained in the inteval [-n n -n, n + n -n ]. This neutix convolution poduct is also commutative. It is obvious that any esults poved with the oiginal definition hold with the new definition. The following theoems, poved in [4] theefoe hold, the fist showing that the commutative neutix convolution poduct is a genealization of the convolution poduct. Theefoe the idea of a neutix lies in neglecting cetain numeical sequences diveging to ±, which makes a wide the class of pais of distibutions f and g fo which the poduct exists. It should be noted that, in geneal, the definition of a commutative neutix convolution poduct depends on the choice of the sequence τ n as well as the set of negligible sequences. Theoem1. Let f and g be distibutions in D', satisfying eithe condition (a) o condition (b) of Gel'fand and Shilov's definition. Then the commutative neutix convolution poduct f g exists and f g f*g. Note howeve that (f g)' is not necessaily equal to f' g, but we do have the following theoem poved in [5]. Theoem2. Let f and g be distibutions in D' and suppose that commutative neutix convolution poduct f g exists. If exists and equals (h, ) fo evey in D, then f' g exists and (f g)' f' g + h. In the following, we need to extend ou set of negligible functions to include finite linea sums of the functions n s li(n ) and n s ln - n, (n > 1) fo s 0, 1, 2,. and 1, 2,. 2. MAIN RESULTS The following wee poved in [6] fo 0, 1, 2,., and s 1, 2,. * (4) (5) (6), (7) (8) Now we pove the following esults. Theoem 3 The neutix convolutions 0, (9) fo 0, 1, 2,, and s 1, 2,. Poof. Put [ ] n τ n ( ) and [ ] n compact suppot, the convolution poduct [ τ n ( ) fo n 1, 2,. Since these functions have ] n *[ ] n exists by definition 1 and so [ ] n * [ ] n I 1 +I 2. (10) If 0 n, then we have Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Page 738
I 1 PV PV PV. Using (7) and (8) we get, Next, if n I 1 whee 0, we have. (11) PV PV -. Using (7) and (8), we have. (12) Futhemoe by using (6), we get (13) We have fom equations (11), (12) and (13) that I 1 0. (14) Futhemoe, fo evey fixed x we have (15) Now equation (9) follows fom equations (10), (14) and (15), poving the theoem. Coolay1. The neutix convolution li_( li_( 0, (16) fo 0, 1, 2, and s 1, 2,. Poof. Equation (16) follows immediately on eplacing x by -x in equation(9). Coolay2. The neutix convolution li( li( 0, (17) fo 0, 1, 2,, and s 1, 2,. Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Page 739
Results on the Commutative Neutix Convolution Poduct Involving the Logaithmic Integal Li(X and X Poof. Equation (17) follows on adding equation (9) and (16). Coolay3. The neutix convolutions x - and li - ( + exist and - (18) + (19) fo 0, 1, 2,, and s 1, 2,. Poof. Equation (18) follows fom (4 ) and (9 ) by noting that + + (-1) s ). Equation (19) follows by eplacing x by -x in equation (18). Theoem4. The commutative neutix convolution s-1 + ln -1 + 0, (20) fo 0, 1, 2, and s 1, 2,. Poof. Diffeentiating equation (9) and applying Theoem 2 we get (21) whee, on integation by pats we have [ ]*(x ) n -li(n (x-n) τ n (x-t) - + + (22) Noting that τ n ( - n) is eithe 0 o 1 fo lage enough n, so (23) Also, it is clea that (24) (25) Now τ' n ( - t) 0 fo lage enough n and x 0, so If 0, then (26) This implies that (27) and now equation (20) follows fom the equations (22) to (27). Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Page 740
Coolay 4. The neutix convolution ( ) exists and s-1 ( ) fo 0, 1, 2,, and s 1, 2,. 0, (28) Poof. Equations (28) follows by eplacing by - in equations (20). Coolay5. The neutix convolution s-1 ln -1 s-1 ln -1 0, (29) fo 0, 1, 2,, and s 1, 2,. Poof. Equation (29) follows by adding equations (20) and (28). Coolay6. The neutix convolutions and exist and fo 0, 1, 2, and s 1, 2,. Poof. Since (30) (31), equation (30) follows fom (5) and (20). Equation (31) follows by eplacing by in equation (30). REFERENCES [1] M. Abamowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions with fomulas, Gaphs and Mathematical Tables, 9th pinting. New Yok: Dove, p. 879, 1972. [2] J.G. van de Coput, Intoduction to the neutix calculus, J. Analyse Math., 7(1959-60), 291-398. [3] B. Fishe, Neutices and the convolution of distibutions, Univ. u Novom Sadu Zb. Rad. Piod.-Mat. Fak. Se. Mat., 17(1987), 119-135. [4] B. Fishe and L. C. Kuan, A commutative neutix convolution poduct of distibutions, Zb, Rad. Piod.-Mat. Fak. Se. Mat. 23 (1993), no. 1, 13-27. [5] B. Fishe and E. Ozcag, The exponential integal and the commutative neutix convolution poduct, J. Anal., 7 (1999), 7-20. [6] B. Fishe, and B. Jolevska-Tuneska, On the logaithmic integal, Hacettepe Jounal of Mathematics and Statistics, 39(3)(2010), 393-401. [7] B. Fishe, B. Jolevska-Tuneska and A. Takaci, Futhe esults on the logaithmic integal, Saajev Jounal of Mathematics, Vol.8 (20) (2012), 91-100. [8] B. Jolevska-Tuneska and A. Takaci, Results on the commutative neutix con- volution of distibutions, Hacettepe Jounal of Mathematics and Statistics, 37(2)(2008), 135-141. [9] I. M. Gel'fand, and G. E. Shilov, Genealized functions, Vol. I, Academic Pess Chap. 1, 1964. [10] D.S. Jones, The convolution of genealized functions, Quat. J. Math. Oxfod (2), 24(1973), 145-163. Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Page 741