RETRACTED GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM AND ITS APPLICATIONS ALIREZA ANSARI

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1 Kragujevac Journal of Mathematics Volume 36 Number 2 (22), Pages GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM AND ITS APPLICATIONS ALIREZA ANSARI Abstract In this paper, we introduce the generalized product theorem for the Mellin transform and we solve certain classes of singular integral equations with kernels coincided with conditions of this theorem Moreover, new inversion techniques for n-th iterate of the L 2 -transform are obtained A ver simple inversion formula for the Widder potential transform is also given Introduction and Preliminaries One of the classical integral transform is the Mellin transform () M{f(x); p} = F (p) = x p f(x)dx, c < <p <c 2 and its inversion formula is written in terms of the Bromwich s integral in the following form f(x) = Z c+i (2) F (p)x p dp, c <c<c 2 2ºi c i For convergence of the relation (2), function F (p) must belong to the K-class functions defined in Appendix This transform is used for expressing man problems in the applied sciences An application of this transform ma occur in problems leading to following singular integral equation (3) k(x, )g()d = f(x), x > Ke words and phrases Mellin transform, singular integral equation, n-th iterate of the L 2 - transform 2 Mathematics Subject Classification Primar: 33E, Secondar: 44A, 45E Received: August 8,

2 288 ALIREZA ANSARI It is of interest to have inversion techniques for formal solution of the above singular integral equation in terms of an improper integral as follows (4) g() = h(x, )f(x)dx For this purpose in this paper, in operational calculus of the Mellin transform we state the generalized product theorem for the Mellin transform and consider the following contributions The stud of the certain class of singular integral equation (3) which kernel is coincided with the conditions of the generalized product theorem Finding new inversion techniques for the Wright and the Mittag-Le er transforms arising in fractional calculus Obtaining the kernels of n-th iterates of the Laplace transform and the L 2 - transform and showing that the Mellin transform of kernels satisfies the conditions of the generalized product theorem Finding new inversion technique for n-th iterate of the L 2 -transform in terms of the inverse Mellin Transform For organization of our motivations, at first step we write some main properties of the Mellin transform and state the generalized product theorem In Section 2, we solve some singular integral equations with kernels in terms of the elementar functions In Section 3, we introduce new approaches for finding inversion formulas for the Wright and the Mittag-Le er integral transforms These integral transforms pla fundamental roles in fractional calculus and it is important to get inversion formulas for them In Section 4, we investigate on n-th iterates of the Laplace transform and the L 2 - transform and show that the kernels of n-th iterates of these transforms are coincided with conditions of the generalized product theorem B this result, new inversion formula for n-th iterate of the L 2 -transform is also given Finall, an Appendix is included for definition of K-class functions and some K-class functions used in this paper Main conclusions are also given First, we recall some fundamental properties of the Mellin transform which can be easil written with respect to the definition () For more details and properties of this transform, see [6-8], [], [3] Yurekli and Sadek [6] introduced the L 2 -transform f(x) = e x2 2 g()d and showed the Parseval-Goldstein theorems involving the L 2 -transform and the Laplace transform can be used to obtain identities involving several well-known integral transforms and infinite integrals of elementar and special functions Also, Aghili and Ansari applied this transform to solve some sstems of ODEs, PFDEs and singular integral equations with the special kernels [, 2]

3 GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM 289 i) The convolution theorem for the Mellin transform: ( (5) F (p)g(p) =M{f g; p} = M g(u)f( x ) u )du u ; p ii) The Mellin transform of ± x -derivatives: (6) M{± xf(x); n p} =( p) n F (p), ± x = x d dx iii) Change of scale propert of the Mellin transform : (7) M{f(x a ); p} = a F (p ), a> a iv) Translation propert of the Mellin transform: (8) M{x a f(x); p} = F (p + a), a > (9) () v) The Mellin transform of polar form of a function: M{=[f(re iµ )]; p} = F (p) sin(pµ) M{<[f(re iµ )]; p} = F (p)cos(pµ) Now, we state the generalized product theorem for the Mellin transform Theorem (The generalized product theorem) Let M{g(x); p} = G(p) 2 K and assume that (p) 2 K and 2 (p) is an analtic function such that, M{k(x, ); p} = (p) 2(p) Then the following relation holds for continuous function k(x, ) on the rectangular region a x b, c d, [a, b] [c, d] Ω (, ) (, ) Ω æ () M k(x, )g()d; p = (p)g( 2 (p)) Proof Using the definition of the Mellin transform and considering the condition of continuous function k(x, ) in order to change the order of integration, we get Ω æ M k(x, )g()d; p = x p k(x, )g()ddx = = (p) g() x p k(x, )dxd 2(p) g()d = (p)g( 2 (p)) With considering of the above theorem, in next section we find formal solutions of some singular integral equations with kernels in the K-class functions

4 29 ALIREZA ANSARI 2 Singular Integral Equations with Kernels of Elementar Functions Problem 2 Solve the singular integral equation with the following logarithmic kernel (see [9]) (2) ln( x )g()d = f(x), x > B showing the above equation in the following form (22) ln( x )g()d = f(x) f(), x> and appling the Mellin transform on both sides of equation, we get ( Z M ln x! ) g()d; p = M{ (x), p}, where (x) is defined as (x) =f(x) f() Now, b using the generalized product theorem () and considering the relation A in Appendix 2 for the Mellin transform of ln x, we rewrite the above equation in the form Z º p cot(ºp) p g()d = (p) The above equation can be rewritten as the Mellin transform of function g(x) as (23) G(p +)= p º tan(ºp) (p) B implementation of the inverse Mellin transform and considering the translation propert (8) and the convolution propert (5), simultaneousl, we obtain (24) g() = Ω M p tan(ºp), æ (u) d º u u At this point, b reconsidering the following relation for M {p tan(ºp)} in view of the relation A7 in Appendix 2 ( M {p tan(ºp); } = M p 2 tan(ºp) ) ; = d! 2 ( ) tan(ºp) M ; p d p = d! 2 p! + (25) ln º d p and substituting in the relation (23), the solution of the singular integral equation (2) is written as (26) g() = d º 2 d d Z p p! + u ln p p (u) du d u u

5 GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM 29 Problem 22 Solve the singular integral equation with the following inverse trigonometric kernel q (27) sin ( 2 + x 2 )g()d = f(x), x> B appling the Mellin transform on both sides of equation Ω q æ M sin ( 2 + x 2 )g()d; p = M{f(x), p} and using the relation A2 in Appendix 2 for the Mellin transform of kernel sin ( p 2 + x 2 ), we get µ ºp p sin p g()d = F (p), 2 which implies that µ ºp (28) G(p +)=pcsc F (p) 2 B appling the inverse Mellin transform and using the translation and the convolution properties, we obtain µ ºp (29) g() = M Ωp csc, æ f(u) du 2 u u According to the relation A8 in Appendix 2 and the Mellin transform of delta derivatives (6), we finall get the solution of (27) as follows (2) g() = 2 d º d p p p f(u)du u( + u) Problem 23 Solve the singular integral equation with the following exponential kernel (2) e xæ g()d = f(x), Æ > In the same procedure as in the previous problem, after appling the Mellin transform on equation and using the relation A3 in Appendix 2, we get µ µ p p (22) G Æ Æ + = ÆF (p), or equivalentl (23) (p)g(p +)=ÆF (Æp) Also, b appling the inverse of Mellin transform and using the convolution propert we obtain (24) e u g(u)du = f( Æ )

6 292 ALIREZA ANSARI The above equation implies that the function g can be obtained in terms of the inverse Laplace transform as follows (25) g() = ( ) 2 L f(u Æ ); 3 Inversion Techniques for Some Integral Transforms Similar to the procedures in previous section for solving singular integral equations, in this section we find new inversion formulas for the Wright transform and Mittag- Le er transform These integral transforms have been recentl arisen in fractional calculus [], and it is necessar to have inversion techniques for them For the following Wright transform [] (3) 3 The Wright Transform W (, ; x )g()d = f(x), x>, < < where the Wright function is presented b the following relation X z k (32) W (Æ, Ø; z) =, Æ >, Ø 2 C, z2 C, k! (Æk + Ø) k= we show an inversion formula for the function g() At first, b appling the Laplace transform on both sides of equation with respect to x ( (33) L W (, ; ) x )g()d; s = L{f(x); s}, x> and using the fact that (see [2]) (34) e s = we get (35) e sx x W (, ; x )dx e s g()d = (s), where is the Laplace transform of the function (x) = f(x) x Now, b considering the singular integral equation with exponential kernel (2) and its solution we obtain the following relation for the inverse function g (36) g() = 2 L ( (s ); ) = ( ( ) f(x) 2 L L x ; s ; ), provided that the integrals involved converge absolutel

7 GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM The Mittag-Le er Transform If we consider the Mittag-Le er transform [] (37) E Æ ( s Æ )g()d = f(s), < Æ < where the Mittag-Le er function is given b the following relation X n (38) E Æ () = (Æn +), n= then b appling the inverse Laplace transform on both sides of equation with respect to s and using the fact that 2 (39) L {E Æ ( s Æ ); r} = r Æ sin(æº) º 2 + 2r Æ cos(æº)+r, 2Æ we get a transformed equation in the following form (3) º r Æ sin(æº) g()d = (r), < Æ <, 2 + 2r Æ cos(æº)+r2æ where (r) is the inverse Laplace transform of f(s) According to the relation A4 in Appendix 2, the integral equation (3) is coincided with the generalized product theorem Therefore, b appling the Mellin transform on the above equation we get µ µ p Æ G Æ + csc º p ( ) (r) sin(ºp) =M Æ r ; p, or equivalentl ( ) (r) G(p +) csc(ºp) sin(æºp) =ÆM r ; Æp At this point, b using the inverse Mellin transform and using the convolution and the change of scale properties, we get the function g as follows (3) g() = µ u u Æ du, u Æ + provided that the above integral converges absolutel Also, the function is given b ( ) sin(ºp) X µ (32) (x) =M sin(æºp) ; x = n=( ) n x næº n sin Æ 2 This relation can be obtained in view of the Titchmarsh theorem f(r) = º e sr ={F (se iº )}ds for inverses of the Laplace transform of functions which have branch cut on the real negative semiaxis, see [3]

8 294 ALIREZA ANSARI 4 Kernels of n-th Iterates of the Laplace transform and the L 2 -Transform In this section, b using the generalized product theorem for the Mellin transform, we find kernels for n-th iterates of the Laplace transform and the L 2 -transform Srivastava and Yurekli [2, 4] showed that the second iterates of the Laplace transform and the L 2 -transform are the Stieltjes and the Widder potential transforms respectivel, and Brown et al [4, 5] introduced that the third iterates of the Laplace transform and the L 2 -transform are as the exponential and the E 2, -transforms respectivel These integral transforms have been shown in the second and third rows of Table and Table 2 in terms of the exponential integral function E (x) defined as e u (4) E (x) = Ei( x) = x u du Now, for finding the n-th iterates, (n 4), of the Laplace transform and the L 2 - transform, we state the following theorem Theorem 4 The Mellin transform of the kernel of n-th iterates of the Laplace transform and the L 2 -transform satisfies the following relation (42) M{k n (x, ); p} = n, (p) n,2(p) where n, (p) belongs to K-class functions and n,2 (p) is an analtic function Proof B using the following relations which are easil obtained b definitions of the Laplace, Mellin and the L 2 -transform [5] (43) (44) M{L{k n (x, ); s}; p} = (p)m{k n (x, ); p}, M{L 2 {k n (x, ); s}; p} = 2 (p 2 )M{k n(x, ); 2 p}, and considering the relations A5, A6 in Appendix 2 as first iterates of the Laplace transform and the L 2 -transform we see that in each iterate the Mellin transform of kernels satisfies the relation (42) Table and Table 2 show the kernels of n-th iterates of the Laplace transform and the L 2 -transform and their Mellin transform The relations A7- A in Appendix 2 have been used to complete the third columns of Table and Table 2 Also, b using the generalized product theorem we can obtain new inversion techniques for n-th iterate of the L 2 -transform Since, the Mellin transforms of kernels k 2n (x, ) and k 2n+ (x, ) satisf the conditions of generalized product theorem (45) (46) M{k 2n (x, ); p} = º n csc n (ºp) p, M{k 2n+ (x, ); p} = º n csc n (ºp) (p) p, we can easil see that b implementation of relation (9) the inverse function g() is written as n-th iterate of imaginar part of the inverse Mellin transform These results have been shown in Table 3

9 GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM 295 The n-th iterate The integral Transform The Mellin Transform of kernel R The first iterate e x g()d (p) p The second iterate The third iterate The fourth iterate The fifth iterate The sixth iterate The 2n-th iterate The 2n + -th iterate R º R csc(ºp)p e x E (x)g()d º (p) csc(ºp) p R g()d +x ln( x ) g()d x º2 csc 2 (ºp) p R k 5 (x, )g()d º 2 csc 2 (ºp) (p) p R k 6 (x, )g()d º 3 csc 3 (ºp) p R k 2n (x, )g()d º n csc n (ºp) p R k 2n+ (x, )g()d º n csc n (ºp) (p) p Table The n-th iterate of the Laplace transform The n-th iterate The integral Transform The Mellin Transform of kernel R The first iterate e x2 2 g()d 2 p ( p ) 2 R The second iterate º 2 g()d 2 +x 2 4 p csc( ºp ) 2 R The third iterate 4 e x2 2 E (x 2 2 º )g()d ( p 8 2 ) p csc( ºp ) 2 R 8 ln( x2 2 ) º The fourth iterate g()d 2 x csc2 ( ºp R The fifth iterate º k 5 (x, )g()d 2 ( p ) 32 2 csc2 ( ºp R The sixth iterate º k 6 (x, )g()d 3 64 csc3 ( ºp The 2n-th iterate R k 2n (x, )g()d ( º 4 )n csc n ( ºp 2 )p 2 ) p 2 )p 2 )p The 2n + -th iterate R k 2n+ (x, )g()d ( º 2 4 )n ( p ) 2 cscn ( ºp Table 2 The n-th iterate of the L 2 -transform 2 ) p

10 296 ALIREZA ANSARI The n-th iterate The integral Transform The inverse function The first iterate f(x) = R e x2 2 g()d g() = M { F (2 p) ( p 2 R The second iterate f(x) = g()d g() = 4 =[f(i)] 2 2 +x 2 º The third iterate f(x) = R 4 e x2 2 E(x 2 2 )g()d g() = 8 º =M { F (2 p) The fourth iterate f(x) = 8 R ln( x2 2 ) ( p 2 ); } ); i} g()d g() = 6 =M {=M {F (p); ir}; i} x 2 2 º 2 The fifth iterate f(x) = R k5(x, )g()d g() = 32 =M {=M { F (2 p) º 2 ( p 2 ); ir}; i} The sixth iterate f(x) = R k6(x, )g()d g() = 64 =M {=M {=M {F (p); ir}; iu}; i} º 3 The 2n-th iterate f(x) = R k2n(x, )g()d g() = ( 4)n (=M ) n {F (p); i} º n The 2n + -th iterate f(x) = R k2n+(x, )g()d g() = ( 2)2n+ (=M ) n { F (2 p) º n ( p 2 Table 3 The inverse function of n-th iterate of the L2-transform ); i}

11 GENERALIZED PRODUCT THEOREM FOR THE MELLIN TRANSFORM 297 Remark 4 The second row of the Table 3 implies that for the Widder potential transform (47) P{g(); p} = f(x) = 2 + x g()d 2 a ver simple inversion formula can be written as the imaginar part of function 2 f(i) º 5 Conclusions In this paper we provided new results in operational calculus for the Mellin transform These results are derived from the generalized product theorem New inversion formulas for the Wright and the Mittag-Le er transforms were obtained These formulas ma be considered as promising approaches in expressing the Wright and the Mittag-Le er functions in fractional calculus Also, new inversion technique for n-th iterate of the L 2 -transform was written and a simple inversion formula was derived for the Widder potential transform as second iterate of the L 2 -transform Appendix Appendix The definition of the K-class functions A complex variable function F (p) is said to belong to the K-class functions if it is regular in the infinite strip S = {s = æ + iø : c < æ <c 2 } and for an arbitrar >, F (p) tends to zero uniforml as ø! in the strip c + æ c 2 + Also, the integral R F (æ + iø)dø is absolutel convergent for each value of æ in the open internal (c,c 2 ) Appendix 2 The Mellin transform of the K-class functions coincides with the generalized product theorem [5] A): M n ln x ; po = º p p cot(ºp), < <p <, A2): M n sin ( p 2 + x 2 ); p o = p p sin ºp 2, < <p <, A3): M n e xæ ; p o = p Æ p Æ Æ, <p >, ( ) x sin(æº) A4): M 2 + 2x cos(æº)+x ; p = º p csc(ºp) sin(æºp), < <p <, 2 A5): M{e x ; p} = (p) p, < <p, A6): M{e 2 x 2 ; p} = 2 p p 2, <p >, ( ) A7): M + x ; p = º p csc(ºp), < <p <, ( ) A8): M 2 + x ; p = º 2 2 p csc ºp 2, < <p < 2, A9): M{e x E (x); p} = º (p) p csc(ºp), < <p <,

12 298 ALIREZA ANSARI A): M{e x2 2 E (x 2 2 ); p} = º p 2 2 p csc ºp 2, < <p < 2 The Mellin transform of other functions used in this note A): M n ln( +p x p ); x po = º tan(ºp), < <p <, p A2): M n 2 º + p ; x po = csc( ºp ), < <p < 2 Acknowledgement: The author would like to express his sincere appreciation to the Shahrekord Universit and the center of excellence for mathematics for financial supports References [] A Aghili and A Ansari, A new approach to solving SIEs and sstem of PFDEs using the L 2 -transform, DiÆ Equat Cont Proces, 3, (2) [2] A Aghili and A Ansari, Solution to sstem of partial fractional diæerential equation using the L 2 -transform, Analsis and Applications, World Scientific Publishing, Vol 9, No (2) -9 [3] AV Boblev and C Cercignani, The inverse laplace transform of some analtic functions with an application to the eternal solutions of the boltzmann equation, Appl Math Lett, 5 (22), [4] D Brown, N Dernek and O Yurekli, Identities for the E 2, -Transform and their applications, Appl Math Comput, 82 (26), [5] D Brown, N Dernek and O Yurekli, Identities for the exponential integral and the complementar error transforms, Appl Math Comput, 82 (26), [6] B Davies, Integral transforms and their applications, 3rd Ed, Springer-Verlag, New York, 2 [7] L Debnath and D Bhatta, Integral transforms and their applications, 2nd edition, Chapman & Hall/CRC, New York, 26 [8] VA Ditkin and A P Prudnikov, Integral transforms and Operational Calculus, Pergamon Press, Oxford, 965 [9] R Estrada and R P Kanwal, Singular Integral Equations, Springer-Verlag New York, 2 [] A A Kilbas, HM Srivastava and JJ Trujillo, Theor and Applications of Fractional DiÆerential Equations, North-Holland Mathematics Studies, 24, Elsevier Science Publishers, Amsterdam, Heidelberg and New York, 26 [] I N Sneddon, The Use of Integral Transforms, McGraw-Hill Book Compan, New York, 972 [2] HM Srivastava and O Yurekli, A theorem on Widder s potential transform and its applications, J Math Anal Appl, 54 (99), [3] HM Srivastava, RG Buschman, Theor and Applications of Convolution Integral Equations, Kluwer Academic Publishers, 992 [4] HM Srivastava and O Yurekli, A theorem on a Stieltjes-tpe integral transform and its applications, Complex Var Theo Appl, 28 (995), [5] F Oberhettinger, Tables of Mellin transforms, Springer-Verlag, Berlin, 974 [6] O Yurekli and I Sadek, A Parseval-Goldstein tpe theorem on the Widder potential transform and its applications, Int J Math Math Sci, 4 (99), Department of Applied Mathematics, Facult of Mathematical Sciences, Shahrekord Universit, P O Box 5, Shahrekord, Iran address: alireza 38@ahoocom