FURTHER RESULTS ON CONVOLUTIONS INVOLVING GAUSSIAN ERROR FUNCTION erf( x 1/2 )

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1 1 ADV MATH SCI JOURNAL Advances n Mathematcs: Scentc Journal 6 (217, no.1, 2938 ISSN prnted verson ISSN electronc verson UDC: : FURTHER RESULTS ON CONVOLUTIONS INVOLVING GAUSSIAN ERROR FUNCTION erf( x 1/2 BRIAN FISHER, FATMA AL-SIREHY AND EMIN ÖZÇAĞ 1 ABSTRACT. In the paper we consder some convolutons, whch are related to results of [7], of the Gassuan error functon and pseudo-functons. 1. INTRODUCTION Recall defnton of the error functon (also called Gaussan error functon erf(x [16] defned for x R by erf(x 2 x e u2 du 2 ( 1!(2 1 x21. The error functon erf(x s odd, convex on (, ], concave on [, and strctly ncreasng on R. The reader refer to [1, 2] for the other propertes of the error functon. The error functon s a specal functon of sgmod shape that occurs n probablty, statstcs and partal dfferental equatons descrbng dffuson. It plays an mportant roles n problems from mathematcal physcs, n analytc solutons for problems of thermomechancs and mass flow due to dffuson. Drschmd and Fscher defned the generalzed Gaussan error functon by erf (x 2 x u e u2 du, for N, [4] and t follows from the defnton that lm erf (x, x lm erf (x 2 x lm erf (x 2 x u e u2 du 1 Γ ( 1 1 Γ ( 1, 2 2 u e u2 du ( 11 Γ ( correspondng author 21 Mathematcs Subject Classfcaton. 33B2, 46F1. Key words and phrases. error functon, generalzed error functon, dstrbuton, neutrx, neutrx lmt, convoluton, neutrx convoluton. 29

2 3 B. FISHER, F. AL-SIREHY AND E. ÖZÇAĞ The locally summable functons erf(x and erf(x are defned by erf(x H(xerf(x, erf(x H( xerf(x, where H denotes Heavsde s functon. The functons erf( x 1/2, erf(x 1/2 and erf(x 1/2 are smlarly defned by erf( x 1/2 2 x 1/2 exp( u 2 du, erf(x 1/2 H(xerf( x 1/2, erf(x 1/2 H( xerf( x 1/2, and the functons erf 2 (x 1/2 and erf 2 (x 1/2 by erf 2 (x 1/2 H(xerf 2 ( x 1/2, erf 2 (x 1/2 H( xerf 2 ( x 1/2, for, 1, 2,.... It s easy to prove the followng equatons, whch we need for our results, by nducton: (1.1 and (1.2 erf 2 (x 2 x u 2 e u2 du 1 (2!( j! 2 2j!(2 2j! x2 2j 1 exp( x 2 (2! 2 2! erf(x j erf 21 (x 2 x u 21 e u2 du j! x 2 2j exp( x 2!, ( j! for, 1, 2,..., where the sum n (1.2 s empty when. In classcal analyss one often deals wth the convoluton of two functons f(x and g(x defned as follows: Defnton 1.1. Let f and g be functons. Then the convoluton f g s defned by (f g(x for all ponts x for whch the ntegral exst. f(tg(x t dt f(x tg(t dt It follows easly from the defnton that f f g exsts then g f exsts and f g g f; and f (f g and f g (or f g exsts, then (1.3 (f g f g (or f g. The followng result was proved n [1]: x r erf (x 1 ( ( 1 erf (xx r 1, for r, 1, 2,....

3 CONVOLUTIONS OF ERROR FUNCTION 31 Further, the next two results were proved n [7]: x r erf(x 1/2 1 ( (1.4 ( 1 erf 2 ( x 1/2 x r 1 and (1.5 for r, 1, 2,.... x r [x 1/2 exp( x ] ( r ( 1 erf 2 (x 1/2 x r, 2. RESULTS ON CONVOLUTION Consder Posson s classcal formula n the theory of heat conducton (2.1 u(x, t 1 2 t exp ( It s well known that u(x, t satsfes the heat equaton (x ξ2 µ(ξ dξ. 4t u (2.2 t 2 u x 2, wth the ntal condton u(x, µ(x. Equaton (2.1 can be wrtten n the convoluton form 1 u(x, t µ(x 2 t exp( x2 4t, whch s a soluton of the heat equaton (2.2. As seen n generalzed functonal analyss, the convoluton s of great mportance. In order to defne the convoluton of two dstrbutons, we frst let D be the space of nfntely dfferentable functons wth compact support and D be the space of dstrbutons defned on D. Defnton 2.1. The convoluton f g of two dstrbutons f and g n D s defned by the equaton (f g(x, φ f(y, g(x, φ(x y for arbtrary φ n D, provded f and g satsfy ether of the condtons: (B1 ether f or g has bounded support, (B2 the supports of f and g are bounded on the same sde. See Gel fand and Shlov [11] (or [12]. Note that f f and g are locally summable functons satsfyng ether of the above condtons and the classcal convoluton f g exsts, then t s n agreement wth Defnton 1.1. Some attempts to defne the convoluton product of dstrbutons have been made. The convoluton product of dstrbutons may be defned n a more general way wthout any restrcton on the supports. The most well-known are gven by Vladmrov and Jones, see [17, 14]. However, there stll exst many pars of dstrbutons such that the convoluton products do not exst n the sense of these defntons. Usng the concepts of the neutrx and the neutrx lmt due to van der Corput [3], Fsher gave the general prncple for the dscardng of unwanted nfnte quanttes from asymptotc expansons and ths has been exploted n context of dstrbutons, partcularly

4 32 B. FISHER, F. AL-SIREHY AND E. ÖZÇAĞ n connecton wth convoluton product and dstrbutonal multplcaton. See [5, 6, 8, 13, 15]. To recall the defnton of neutrx convoluton product gven by Fsher, we frst of all let τ be a functon n D satsfyng the followng propertes: ( τ(x τ( x, ( τ(x 1, ( τ(x 1 for x 1 2, (v τ(x for x 1. The nfntely dfferentable functon τ n s now defned by 1, x n, τ n (x τ(n n x n n1, x > n, τ(n n x n n1, x < n, for n 1, 2,.... Defnton 2.2. Let f and g be dstrbutons n D and let f n fτ n for n 1, 2,.... Then the neutrx convoluton f g s defned as the neutrx lmt of the sequence {f n g}, provded that the lmt h exsts n the sense that N lm n f n g, φ h, φ, for all φ n D, where N s the neutrx (see van der Corput [3] havng doman N {1, 2,..., n,...} and range N the real numbers, wth neglgble functons fnte lnear sums of the functons n λ ln r 1 n, ln r n (λ >, r 1, 2,... and all functons whch converge to zero n the usual sense as n tends to nfnty. In ths defnton the convoluton product f n g exsts snce the dstrbuton f n has the bounded support and supp(τ n [ n n n, n n n ]. Note that because of the lack of symmetry n the defnton of f g, the neutrx convoluton s n general noncommutatve. Now let f and g be dstrbutons n D satsfyng ether condton (B1 or condton (B2 of Defnton 2.1. Then t was proved n [6] that the neutrx convoluton f g exsts and f g f g. Ths shows that the neutrx convoluton s a generalzaton of the convoluton. It was also proved that f f and g are dstrbutons n D and f g exsts, then the neutrx convoluton f g exsts and (f g f g. Note however that equaton (1.3 does not necessarly hold for the neutrx convoluton product and that (f g s not necessarly equal to f g. However we have the followng lemma whch was proved n [6]. Lemma 2.1. Let f and g be dstrbutons n D and suppose that f g exsts. If N lm n (fτ n g, φ exsts and equals h, φ for all φ D, then the neutrx convoluton f g exsts and (f g f g h.

5 CONVOLUTIONS OF ERROR FUNCTION 33 In order to prove our next results we need to extend our set of neglgble functons gven n Defnton 2.2 to also nclude fnte lnear sums of the functon n r erf[(x n 1/2 ], r 1, 2,.... The followng results were proved n [7]: x r erf(x 1/2 1 ( ( 1 (2! ( 2 2 x r 1,! x r [x 1/2 ( ( 1 (r 1(2! exp( x ] 2 2! for r, 1, 2,.... We now prove Theorem 2.1. The neutrx convoluton erf(x 1/2 x r exsts and erf(x 1/2 x r 1 ( ( 1 (2! ( ! for r, 1, 2,.... x r 1 Proof. We put [erf(x 1/2 ] n erf(x 1/2 τ n (x for n, 1, 2,.... Snce [erf(x 1/2 ] n has compact support, the convoluton [erf(x 1/2 ] n x r exsts and n 2 [erf(x1/2 ] n x r (x t r erf(t 1/2 dt (2.4 Now and so nn n n nn n n I 1 I 2. (x t r τ n (terf(t 1/2 dt (x t r τ n (terf(t 1/2 dt (x n r n n e n (2.5 lm I 2. n Further I 1 n n 1/2 (x t r t 1/2 exp( u 2 n 1/2 exp( u 2 du dt n u 2 (x t r dt du 1 [ (x u 2 r1 (x n r1] exp( u 2 du ( ( 1 x r 1 erf 2 (n 1/2 2( ( ( 1 x r 1 n erf(n 1/2. 2( x r

6 34 B. FISHER, F. AL-SIREHY AND E. ÖZÇAĞ (2.6 It follows from equaton (1.1 and the n erf(n 1/2 beng neglgble functons that N lm n I 1 2( ( 1 x r 1 erf 2 ( ( 1 (2! 2 21 x r 1,! on notng that erf( 1. Equaton (2.3 now follows from equatons (2.4 to (2.6. Replacng x by x n equaton (2.3, we get Corollary 2.1. The neutrx convoluton erf(x 1/2 x r exsts and (2.7 for r, 1, 2,.... Notng that erf(x 1/2 x r 1 (2! 2 2! xr 1 erf( x 1/2 erf(x 1/2 erf(x 1/2 and usng equatons (2.3 and (2.7, we get Corollary 2.2. The neutrx convoluton erf( x 1/2 x r exsts and (2.8 for r, 1, 2,.... erf( x 1/2 x r 1 ( [( 1 1](2! 2 2! Corollary 2.3. The neutrx convoluton erf( x 1/2 x r exsts and erf( x 1/2 x r 1 (2.9 1 for r, 1, 2,.... Proof. Note that x r 1 ( 1 erf 2 ( x 1/2 [x r 1 ( 1 r x r 1 ] (2! 2 2! xr 1 erf( x 1/2 x r erf(x 1/2 x r erf(x 1/2 x r erf(x 1/2 x r erf(x 1/2 x r ( 1 r erf(x 1/2 x r. Then replacng x by x n equaton (2.8, we get (2.1 x r erf(x 1/2 1 ( 1 erf 2 ( x 1/2 x r 1

7 CONVOLUTIONS OF ERROR FUNCTION 35 and usng equatons (1.4, (2.7 and (2.1, we get erf( x 1/2 x r 1 Equaton (2.9 follows. 1 1 ( (2! 2 2! xr 1 ( 1 erf 2 ( x 1/2 x r 1 ( 1 r erf 2 ( x 1/2 x r 1. Replacng x by x n equaton (2.9, we get Corollary 2.4. The neutrx convoluton erf( x 1/2 x r exsts and erf( x 1/2 x r 1 for r, 1, 2, ( ( 1 r (2! 2 2! ( 1 erf 2 ( x 1/2 [x r 1 ( 1 r x r 1 ] x r 1 Theorem 2.2. The neutrx convoluton [x 1/2 exp( x ] x r exsts and (2.11 [x 1/2 r ( ( 1 exp( x ] x r (r 1(2! 2 2! for r 1, 2,... and (2.12 [x 1/2 exp( x ] 1. Proof. Dfferentatng the convoluton [erf(x 1/2 ] n x r, we get (2.13 for r 1, 2,.... Now and so 1 [x 1/2 exp( x τ n (x] x r [erf( x 1/2 τ n(x] x r r[erf(x 1/2 ] n x r 1, nn n [erf( x 1/2 τ n(x] x r (x t r erf( t 1/2 τ n (t dt n (x n r n n e n (2.14 lm n [erf( x 1/2 τ n(x] x r. x r

8 36 B. FISHER, F. AL-SIREHY AND E. ÖZÇAĞ It now follows from equatons (2.3, (2.13 and (2.14 that N lm n [x 1/2 exp( x τ n (x] x r rerf(x 1/2 x r 1 r ( ( 1 (r 1(2! 2 2 x r,! for r 1, 2,..., provng equaton (2.11. When r, equaton (2.13 s replaced by 1 [x 1/2 exp( x τ n (x] 1 [erf( x 1/2 τ n(x] 1, and t follows that 1 N lm n [x 1/2 exp( x τ n (x] 1 1 [x 1/2 exp( x ] 1, provng equaton (2.12. Replacng x by x n equatons (2.11 and (2.12, we get Corollary 2.5. The neutrx convoluton [x 1/2 exp( x ] x r exsts and (2.15 [x 1/2 r ( (r 1(2! exp( x ] x r 2 2! for r 1, 2,... and (2.16 [x 1/2 exp( x ] 1. Notng that x 1/2 exp( x x 1/2 exp( x x 1/2 exp( x and usng equatons (2.11, (2.12, (2.15 and (2.16, we get x r Corollary 2.6. The neutrx convoluton [ x 1/2 exp( x ] x r exsts and r ( r [1 ( 1 (2.17 [ x 1/2 exp( x ] x r ](r 1(2! 2 2! for r 1, 2,... and (2.18 [ x 1/2 exp( x ] 1. Corollary 2.7. The neutrx convoluton [ x 1/2 exp( x ] x r exsts and [ x 1/2 exp( x ] x r ( r (2! 2 2! xr ( r (2.19 ( 1 erf 2 ( x 1/2 x r ( r ( 1 r erf 2 ( x 1/2 x r for r 1, 2,... and (2.2 [ x 1/2 exp( x ] H(x. x r

9 CONVOLUTIONS OF ERROR FUNCTION 37 Proof. Note that [ x 1/2 exp( x ] x r [x 1/2 exp( x ] x r [x 1/2 exp( x ] x r [x 1/2 exp( x ] x r [x 1/2 exp( x ] x r ( 1 r [x 1/2 exp( x ] x r and replacng x by x n equaton (1.5, we get (2.21 x r [x 1/2 exp( x ] ( r Then ( 1 erf 2 (x 1/2 x r. [ x 1/2 exp( x ] x r ( r ( 1 erf 2 (x 1/2 x r r ( (r 1(2! 2 2! ( r ( 1 r erf 2 (x 1/2 x r, x r on usng equatons (1.5, (2.15 and (2.21. Equaton (2.19 follows for r 1, 2,.... Equaton (2.2 follows smlarly. In the partcular case r, we have [ x 1/2 exp( x ] H(x sgn(xerf( x 1/2, on usng equatons (2.17, (2.18 and (2.19. For further related results, see [9, 1]. REFERENCES [1] H. ALZER:A Kuczma-type functonal nequalty for error and complementary error functons, Aequat. Math. 89(215, [2] H. ALZER, M. N. KWONG:A subaddvty Property of the Error Functon, Proc. Amer. Math. Soc. 142(8(214, [3] J.G. VAN DER CORPUT:Introducton to the neutrx calculus, J. Analyse Math., 7 (1959 6, [4] H. DIRSCHMID, F. D. FISCHER:Generalzed Gausssan error functons and ther applcatons, Acta Mech., 226 (215, [5] B. FISHER:A non-commutatve neutrx product of dstrbutons Math. Nachr. 18 (1982, [6] B. FISHER:Neutrces and the convoluton of dstrbutons, Unv. u Novom Sadu Zb. Rad. Prrod.-Mat. Fak. Ser. Mat., 17 (1987, [7] B. FISHER, F. AL-SIREHY, E. ÖZÇAĞ:Results nvolvng Gaussan error functon erf( x 1/2 and the neutrx convoluton, Adv. Math. Sc. Journal, 5 (216, [8] B. FISHER, B. JOLEVSKA-TUNESKA, A. TAKACI:On convolutons and neutrx convolutons nvolvng the ncomplete gamma functon, Integral Transforms Spec. Funct. 15(5 (24, [9] B. FISHER, F. AL-SIREHY, M. TELCI :Convolutons nvolvng the error functon, Internat. J. Appl. Math., 13(4 (23, [1] B. FISHER, M. TELCI, F. AL-SIREHY:The error functon and the neutrx convoluton, Int. J. Appl. Math., 8 (22, [11] I. M. GEL FAND, G. E. SHILOV:Generalzed Functons Vol. I, Academc Press, [12] R. HOSKINS, J. S. PINTO:Dstrbutons Ultradstrbutons and Other Generalzed Functons, Ells Horward, [13] B. JOLEVSKA-TUNESKA, A. TAKACI, B. FISHER :On Some Neutrx Convoluton Product of Dstrbutons Integral Trans. Spec. Funct., 14(3 (23,

10 38 B. FISHER, F. AL-SIREHY AND E. ÖZÇAĞ [14] D. S. JONES:The convoluton of generalzed functons, Quart. J. Math., 24 (1973, [15] E. ÖZÇAĞ, H. KANSU:A result of neutrx convoluton of dstrbutons, Indan J. Pure Appl. Math., 28(1 (1997, [16] I. N. SNEDDON:Specal Functons of Mathematcal Physcs and Chemstry, Olver and Boyd, [17] V. S. VLADIMIROV: Equatons of Mathematcal Physcs, Nauka, Moskow, (1968; Englsh trans., Marcel Dekker, New York, DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE LEICESTER UNIVERSITY LEICESTER, LE1 7RH, ENGLAND Emal address: DEPARTMENT OF MATHEMATICS JEDDAH, KING ABDULAZIZ UNIVERSITY JEDDAH, SAUDI ARABIA Emal address: DEPARTMENT OF MATHEMATICS HACETTEPE UNIVERSITY ANKARA, TURKEY Emal address: