ON THE LOGARITHMIC INTEGRAL

Transcription

1 Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x) ad ts assocated fuctos l +(x) ad l (x) are defed as locally summable fuctos o the real le. Some covolutos ad eutrx covolutos of these fuctos ad other fuctos are the foud. Keywords: Logarthmc tegral, Dstrbuto, Covoluto, Neutrx, Neutrx covoluto. 2 AMS Classfcato: 33 B 1, 46 F Itrocto The logarthmc tegral l(x), see Abramowtz ad Stegu [1], s defed by, for x < 1, l t l(x) = PV, for x > 1, l t PV l t, for x < 1, for x < 1, l t [ 1 ǫ x ] = lm ǫ + l t +, for x > 1, 1+ǫ l t [ 1+ǫ x ] lm ǫ + l t +, for x < 1 l t 1 ǫ where PV deotes the Cauchy prcpal value of the tegral. We wll therefore wrte l(x) = PV l t Departmet of Mathematcs, Uversty of Lecester, Lecester, LE1 7RH, Eglad. E-mal: fbr@le.ac.uk Correspodg Author Faculty of Electrcal Egeerg ad Iformatoal Techologes, Karpos II bb, Skopje, Republc of Macedoa. E-mal: bljaaj@fet.ukm.e.mk

2 394 B. Fsher, B. Jolevska-Tueska for all values of x. The assocated fuctos l +(x) ad l (x) are ow defed by l +(x) = H(x)l(x), l (x) = H( x)l(x), where H(x) deotes Heavsde s fucto. The dstrbuto l 1 x s defed by l 1 x = l (x) ad ts assocated dstrbutos l 1 x + ad l 1 x are defed by l 1 x + = H(x)l 1 x = l +(x), l 1 x = H( x)l 1 x = l (x). The classcal defto of the covoluto of two fuctos f ad g s as follows: 1.1. Defto. Let f ad g be fuctos. The the covoluto f g s defed by (f g)(x) = f(t)g(x t) for all pots x for whch the tegral exsts. It follows easly from the defto that f f g exsts the g f exsts, ad (1.1) f g = g f, ad f (f g) ad f g (or f g) exsts, the (1.2) (f g) = f g (or f g). Defto 1.1 ca be exteded to defe the covoluto f g of two dstrbutos f ad g D wth the followg defto, see Gel fad ad Shlov [6] Defto. Let f ad g be dstrbutos D. The the covoluto f g s defed by the equato (f g)(x),ϕ(x) = f(y), g(x),ϕ(x + y) for arbtrary ϕ D, provded f ad g satsfy ether of the coos (a) ether f or g has bouded support, or (b) the supports of f ad g are bouded o the same sde. It follows that f the covoluto f g exsts by ths defto the equatos (1.1) ad (1.2) are satsfed. The above defto of the covoluto s rather restrctve ad so a eutrx covoluto was defed [3]. I order to defe the eutrx covoluto, we frst of all let τ be a fucto D, see [7], satsfyg the followg propertes: () τ(x) = τ( x), () τ(x) 1, () τ(x) = 1 for x 1 2, (v) τ(x) = for x 1. The fucto τ s ow defed by 1, x, τ (x) = τ( x +1 ), x >, τ( x + +1 ), x <, for = 1,2,.... The followg defto of the o-commutatve eutrx covoluto was gve [3].

3 O the Logarthmc Itegral Defto. Let f ad g be dstrbutos D ad let f = fτ for = 1, 2,.... The the o-commutatve eutrx covoluto f g s defed as the eutrx lmt of the sequece {f g} N, provded the lmt h exsts the sese that N lm f g,ϕ = h, ϕ for all ϕ D, where N s the eutrx, see va der Corput [2], havg doma N the postve reals ad rage N the real umbers, wth eglgble fuctos fte lear sums of the fuctos λ l r 1, l r : λ >, r = 1, 2,..., ad all fuctos whch coverge to zero the ormal sese as teds to fty. It s easly see that ay results proved wth the orgal Deftos 1.1 ad 1.2 of the covoluto hold wth Defto 1.3 of the eutrx covoluto. The followg results proved [3] hold, frst showg that the eutrx covoluto s a geeralzato of the covoluto Theorem. Let f ad g be dstrbutos D, satsfyg ether coo (a) or coo (b) of Gel fad ad Shlov s defto. The the eutrx covoluto f g exsts ad f g = f g Theorem. Let f ad g be dstrbutos D, ad suppose that the eutrx covoluto f g exsts. The the eutrx covoluto f g exsts ad (f g) = f g. If N lm (fτ ) g, ϕ exsts ad equals h, ϕ for all ϕ D, the f g exsts ad (f g) = f g + h. I the followg, we eed to exted our set of eglgble fuctos to clude fte lear sums of the fuctos s l( r ) ad s l r, ( > 1) for s =, 1,2,... ad r = 1,2, Ma Results Before provg our ma results, we eed the followg lemmas Lemma. (2.1) l(x r ) = PV t r 1 l t. Proof. Makg the substtuto t = u r, we have l(x r ) = PV r provg Equato (2.1). l t = PV u r 1, l u 2.2. Lemma. (2.2) lm + for r = 1, 2,.... τ (t)l(t)(x t) r =

4 396 B. Fsher, B. Jolevska-Tueska Proof. We ote that l(t) > 1 whe t > e ad so l(t) = l(e) + t Thus, whe x < 1, we have + ad Equato (2.2) follows. e < l(e) + t e. l u τ (t) l(t)(x t) r < + l(t)(x t) r < (2 + r 1) r + l(t) < 4 r r [l(e) e ], 2.3. Lemma. (2.3) (2.4) for r = 1, 2,.... N lm l[(x + ) r ] =, N lm r l[(x + )] =, Proof. Wth x > 1, ad puttg f(x) = l(x r ), we have f (x) = rxr 1 l x. It follows that f (k) (x) s of the form (2.5) f (k) (x) = ad k j=1 α kj x k r l j x (2.6) f (r+1) ( + c) = O( 1 ), (c ). By Taylor s Theorem, we have x k f(x + ) = k! f(k) () + xr+1 ()! f(r+1) ( + ξx) = k= k k= j=1 α kj x k k r l j k! + O( 1 ) ad Equato (2.3) follows from Equatos (2.5) ad (2.6). Equato (2.4) follows smlarly. We ow prove a umber of results volvg the covoluto. Frst of all we have 2.4. Theorem. The covolutos l +(x) x r + ad l 1 x + x r + exst, ad l +(x) x r + = 1 r+1 (2.7) ( 1) r +1 x l +(x r +2 ), = l 1 x + x r r (2.8) + = ( 1) r x l +(x r +1 ) for r =, 1,2,.... =

5 O the Logarthmc Itegral 397 Proof. It s obvous that l +(x) x r + = f x <. Whe x >, we have l +(x) x r + = PV = PV (x t) r t 1 l u = PV 1 r+1 = 1 l u (x t) r u ( 1) r +1 x x u r +1 = l u r+1 ( 1) r +1 x l +(x r +2 ), = o usg Equato (2.1), ad Equato (2.7) s proved. Now, usg Equato (1.2) ad (2.7), we get l 1 x + x r + = r l +(x) x r 1 + r = ( 1) r x l +(x r +1 ), = provg Equato (2.8) Corollary. The covolutos l (x) x r ad l 1 x x r exst, ad (2.9) (2.1) l (x) x r = 1 r+1 ( 1) r +2 x l (x r +2 ), = l 1 x x r r = ( 1) r +1 x l (x r +1 ) for r =, 1,2,.... = Proof. Equatos (2.9) ad (2.1) are obtaed applyg a smlar procere as used obtag equatos (2.7) ad (2.8) Theorem. The eutrx covolutos l +(x) x r ad l 1 x + x r exst, ad (2.11) (2.12) l +(x) x r =, l 1 x + x r = for r =, 1,2,.... Proof. We put [l +(x)] = l +(x)τ (x). The the covoluto [l +(x)] x r exsts, ad (2.13) [l +(x)] x r = + l(t)(x t) r + τ (t)l(t)(x t) r,

6 398 B. Fsher, B. Jolevska-Tueska where l(t)(x t) r = PV = PV Thus from Lemma 2.3 we have (2.14) N lm (x t) r t 1 l u = PV 1 r+1 = 1 = l(t)(x t) r =. l u (x t) r u ( 1) r +1 x u r +1 r +1 = l u r+1 ( 1) r +1 x [ l( r +2 ) r +1 l() ]. Equato (2.11) ow follows usg Lemma 2.2, Equatos (2.13) ad (2.14). Dfferetatg Equato (2.11) ad usg Theorem 1.5 we get (2.15) l 1 x + x r = N lm[l +(x)τ (x)] x r, where o tegrato by parts we have (2.16) It s clear that (2.17) lm [l +(x)τ (x)] x r = + + = l()(x ) r τ (t)l(t)(x t) r l 1 (t)(x t) r τ (t) =, + l 1 (t)(x t) r τ (t) + + r l(t)(x t) r 1 τ (t). so Equato (2.12) follows from Lemma 2.2 ad Equatos (2.15), (2.16) ad (2.17) Corollary. The eutrx covolutos l (x) x r ad l 1 x x r exst, ad (2.18) (2.19) l (x) x r =, l 1 x x r = for r =, 1,2,.... Proof. Equatos (2.18) ad (2.19) are obtaed applyg a smlar procere as the case of Equatos (2.11) ad (2.12) Corollary. The eutrx covolutos l(x) x r ad l 1 x x r exst, ad (2.2) (2.21) l(x) x r =, l 1 x x r = for r =, 1,2,.... Proof. Equato (2.2) follows o addg Equatos (2.18) ad (2.11), ad Equato (2.21) follows o addg Equatos (2.12) ad (2.19).

7 O the Logarthmc Itegral Corollary. The eutrx covolutos l +(x) x r, l (x) x r +, l 1 x + x r ad l 1 x x r + exst, ad l +(x) x r = 1 r+1 (2.22) ( 1) x l +(x r +2 ), = l (x) x r + = 1 r+1 (2.23) ( 1) +1 x l (x r +2 ), = l 1 x + x r r (2.24) = ( 1) x l +(x r +1 ), = l 1 x x r r (2.25) + = ( 1) +1 x l (x r +1 ), for r =, 1,2,.... = Proof. Usg that x r = x r + +( 1) r x r, ad the fact that the eutrx covoluto proct s dstrbutve wth respect to ado, we have l +(x) x r = l +(x) x r + + ( 1) r l +(x) x r. Equato (2.22) follows from Equatos (2.7) ad (2.11). Equato (2.23) s obtaed by applyg a smlar procere as the case of Equato (2.22). Equato (2.24) follows from Equatos (2.8) ad (2.12), ad Equato (2.25) s obtaed by applyg a smlar procere as the case of Equato (2.24) Theorem. The eutrx covolutos x r l +(x) ad x r l 1 x + exst, ad (2.26) (2.27) x r l +(x) =, x r l 1 x + = for r =, 1,2,.... Proof. We put (x r ) = x r τ (x) for r =,1, 2,.... The the covoluto (x r ) l +(x) exsts, ad + ++ (2.28) (x r ) l +(x) = l(t)(x t) r + τ (x t) l(t)(x t) r, x+ where + l(t)(x t) r = PV = PV + + (x t) r t 1 l u = PV 1 r+1 = 1 + l u (x t) r u ( 1) r +1 x x+ = ( r+1 = ) PV ()r+1 u r +1 lu + ( 1) r +1 x l [ (x + ) r +2] ()r+1 l u l(x + ).

8 4 B. Fsher, B. Jolevska-Tueska Thus, o usg Lemma 2.3, we have (2.29) N lm + l(t)(x t) r =. Further, usg Lemma 2.2 t s easly see that (2.3) lm ++ x+ τ (x t)l(t)(x t) r =, ad Equato (2.26) follows from Equatos (2.28), (2.29) ad (2.3). Dfferetatg Equato (2.26) gves Equato (2.27) Corollary. The eutrx covolutos x r l (x) ad x r l 1 x exst, ad (2.31) (2.32) x r l (x) =, x r l 1 x = for r =, 1,2,.... Proof. Equatos (2.31) ad (2.32) are obtaed by applyg a smlar procere as for Equatos (2.26) ad (2.27) Corollary. The eutrx covolutos x r l(x) ad x r l 1 x exst, ad (2.33) (2.34) x r l(x) =, x r l 1 x = for r =, 1,2,.... Proof. Equato (2.33) follows o addg Equatos (2.31) ad (2.26), ad Equato (2.34) follows o addg Equatos (2.27) ad (2.32) Corollary. The eutrx covolutos x r l +(x), x r + l (x), x r l 1 x + ad x r + l 1 x exst, ad x r l +(x) = 1 r+1 (2.35) ( 1) x l +(x r +2 ), = x r + l (x) = 1 r+1 (2.36) ( 1) +1 x l (x r +2 ), = x r l 1 x + = 1 r (2.37) ( 1) x l +(x r +1 ), = x r + l 1 x = 1 r (2.38) ( 1) +1 x l (x r +1 ) for r =, 1,2,.... = Proof. Equato (2.35) follows from Equatos (2.7) ad (2.26) o otg that x r l +(x) = x r + l +(x) + ( 1) r x r l +(x). Equato (2.36) s obtaed by argug as the case of Equato (2.35). Equato (2.37) follows from Equatos (2.8) ad (2.27). Equato (2.38) s obtaed by argug as case of Equato (2.37).

9 O the Logarthmc Itegral 41 For further results volvg the covoluto the reader s referred to [4] ad [5]. Ackowledgemet. The authors would lke to thak the referee for hs help mprovg ths paper. Refereces [1] Abramowtz, M. ad Stegu, I. A. (Eds). Hadbook of Mathematcal Fuctos wth Formulas, Graphs ad Mathematcal Tables, 9th prtg (New York: Dover, p. 879, 1972). [2] va der Corput, J. G. Itrocto to the eutrx calculus, J. Aalyse Math. 7, , [3] Fsher, B. Neutrces ad the covoluto of dstrbutos, Uv. u Novom Sa Zb. Rad. Prrod.-Mat. Fak. Ser. Mat. 17, , [4] Fsher, B., Jolevska-Tueska, B. ad Taka c, A. O covolutos ad eutrx covolutos volvg the complete gamma fucto, Itegral Trasforms ad Specal Fuctos 15 (5), , 24. [5] Jolevska-Tueska, B. ad Taka c, A. Results o the commutatve eutrx covoluto proct of dstrbutos, Hacettepe J. Math. Stat. 37(2), , 28. [6] Gel fad, I. M. ad Shlov, G. E. Geeralzed fuctos, Vol. I (Academc Press Chap. 1, 1964). [7] Joes, D. S. The covoluto of geeralzed fuctos, Quart. J. Math. Oxford 24(2), , 1973.