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Weierstrass Transforms JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 GENERALIZED PROPERTIES ON MELLIN - WEIERSTRASS TRANSFORMS Amarnath Kumar Thakur Department of Mathematics Dr.C.V.Raman University, Bilaspur, State, Chhattisgarh Email- [email protected] Gopi Sao Department of Mathematics Dr.C.V.Raman University, Bilaspur, State, Chhattisgarh Email- [email protected] Hetram Suryavanshi Department of Mathematics Dr.C.V.Raman University, Bilaspur, State, Chhattisgarh Email- [email protected] Abstract- In the present paper, we introduce definite integral transforms in operational calculus for the generalized properties on Mellin- Weierstrass associated transforms. These results are derived from the MW Properties These formulas may be considered as promising approaches in expressing in fractional calculus. Keywords – Laplace Transform , Convolution Transform Mellin Transforms , Weierstrass Transforms I. Introduction In mathematical terms the Fourier and Laplace transformations were introduced to solve physical problems. In fact, the first event of change is found by Reiman in which he used to study the famous zeta function. The Finnish mathematician Hjalmar Mellin (1854-1933), who was the first to give an orderly appearance change and its reversal,working in the theory of special functions, he developed applications towards a solution of hypergeometric differential equations and derivation of asymptotic expansions. Mellin's contribution gives a major place to the theory of analytic functions and essentially depends on the Cauchy theorem and the method of residuals. Actually, the Mellin transformation can also be placed in another framework, in which some cases is more closely related to Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 98 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 original ideas of Reimann's original.In addition to its use in mathematics, Mellin's transformation has been applied in many different ways and areas of Physics and Engineering. May be the most well-known application is the calculation of a solution to a potential problem in a the wedge-shaped region where the unknown function (e.g. Temperature or electrostatic potential) has to satisfy Laplace's equation with the given boundary conditions at the edges. Mellin's transformation has proved useful in resolving linear differential equations in x(d/dx) generation in electrical engineering by a process analogous to Laplace. Recently, traditional applications have grown and new one’s have surfaced.The most remarkable property of the Mellin transforms is the existence of direct mapping between asymptotic expansion of a function near zero and infinite , and the set of singularities of the transform in the complex plane . The Mellin transforms and its inverse as an important tool in applied mathematics , are closed releated to the Fourier and bi-laterl transforms . The Mellin transform has applications in various area , including digital data structure , probabilistic algorithem , asymptotes of gamma –related functions, coefficients of dirichlet series , asymptotic estimation of integral forms , asymptotic of algorithm and communication theory . The Mellin transform f(p) of the function F(x) is defined by the equation, ( ) ∞ 푠−1 푓 푝 = ∫0 퐹(푥)푥 푑푥 ,where p > 0 , and its inverse transform is given by 1 훾+푖∞ 푓(푥) = ∫ 푒−푝푓(푝)푑푝 2휋푖 훾−푖∞ The WeierstrassTransform: Consider the known Laplace Transform 2 2 1 ∞ −푦 푒푐푠 = ∫ 푒−푠푦푒 ⁄4푐 dy 푐 > 0, − ∞ < 푠 < ∞ √4휋푐 −∞ Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 99 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 2 Then shows that if ∈ (푠) is the special function 푒−푐푠 , then its reciprocal is still be bilateral Laplace transform of a frequency function −1 −푥2 퐺(푥) = (4휋푐) ⁄2 푒 ⁄4푐 If we adopt this function as the kevel of a convolution transform, will be inverted by the operater 2 푒−퐶퐷 if properly interpreted. 2 In view of equation (30 it is natural to interpret 푒퐶퐷 ∅(푥) as 2 2 1 ∞ −푦 푒퐶퐷 ∅(푥) = ∫ 푒푦퐷∅(푥) 푒 ⁄4푐 dy √4휋푐 −∞ 2 1 ∞ −푦 = ∫ ∅(푥 − 푦)푒 ⁄4푐 dy, √4휋푐 −∞ 2 By above equation, we get 푒퐶퐷 ∅(푥) = 푓(푥).Then if 퐷 is treated as a number we obtain the symbolic 2 equation ∅(푥) = 푒−퐶퐷 푓(푥), The predicted invention formula. Observe that if 푥 is replaced by √푐 푥 and 푦 by √푐푦 in equation if becomes 2 1 ∞ −(푥−푦) 푓(√푐푥) = ∫ 푒 ⁄4 ∅(√푐푦) dy. √4휋 −∞ That is, 푓(√푐푠) is the convolution transform of ∅(√푐푦) having kevel equation with 푐 = 1 . The 2 resenting transform as the weierstraces transform, it will develop that 푒푡퐷 ∅(푥) is a solution of the heat 휕2푢 휕푢 equation = . Many resents interest in the meters will be proved about such salvation since the 휕푥2 휕푡 Laplace transform of equation (1) does not provide us with are interpretation of 푒−퐷2. However, the complex inversion of that transfom, Widder, suggesta an alternative prodedure. Replace 퐶 by (4푡)−1, 0 < 푡 ≤ 1 in above equation and inversion: 1 푎+푖∞ 푒−푦2푡 = ∫ 푘(푠, 푡)푒푠푦푑푦 , − ∞ < 푎 < ∞ 2휋 푎−푖∞ 1 휋 ⁄2 푠2 푘(푠, 푡) = ( ) 푒 ⁄4푡, 푡 Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 100 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 We are thus led to write 1 퐺+푖∞ 푒−푡퐷2푓(푥) = ∫ 푘(푠, 푡)푒푆퐷푓(푥)푑푥 2휋 푎−푖∞ 1 푎+푖∞ = ∫ 푘(푠, 푡) 푓(푥 + 푠)푑푠. 2휋 푎−푖∞ By Changing the variable the becomes 1 푑+푖∞ 푒−푡퐷2푓(푥) = ∫ 푘(푠 − 푥) 푡 푓(푠)푑푠, 2휋푖 푑−푖∞ Where the constant a is arbitrary and will be chosen so that the vertical line푟 = 푑1 will be one an which 푓(푠) is defined it will be chosen later that it is not appropriate to define 푒−퐷2 by printing 푡 = 1 is (1) for the integral would then diverge for certain Weierstron transform 푓(푥). 푒−퐷2푓(푥) = lim 푒−푡퐷2 푓(푥), 푛→1− And the operator will sence to convequent Weierstorn transform. If we set −1 −푥2 푘(푥, 푡) = (4휋푡) ⁄2푒 ⁄4푡 Statement becomes −퐷2 ∞ ( ) ( ) ( ) 푒 ∫−∞ 푘 푥 − 푦 , 휋 ∅ 푦 푑푦 = ∅ 푥 . Here, we now make our formula definition of the Weierstrass transform. Def. (a) −1 −푥2 퐾(푠, 푡) = (4휋푡) ⁄2 푒 ⁄4푡 , 0 < 푡 < ∞, −∞ < 푥 < ∞ This is the familiar source solution will be the kennel of the transform. Def. (b): The Weierstrass transform of a function ∅(푥) is the function Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 101 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 ( ) ∞ ( ) ( ) 푓 푥 = ∫−∞ 푘 푥 − 푦, 1 ∅ 푦 푑푦 ,whenever the integral converges. Def. (c) : The Weierstrass-stieltjes transform of a function 훼(푥) is the function ( ) ∞ ( ) ( ) 푓 푥 = ∫−∞ 퐾 푥 − 푦, 1 푑훼 푦 whenever the integral converges. Here is 훼(푦) of bounded variation in every finite interval. If, for example, 훼(푦) is constant except for a single unit jump at the origin, 푥2 1 − 푓(푥) = (4휋)− ⁄2 푒 4 The inversion operator : Let us the following definition : 2 1 푠 Definition (a) : 퐾(푠, 푡) = (휋/푡) ⁄2 푒4푡 = 2휋퐾(푖푠, 푡) 2 Definition (b) : The operator 푒−퐷 푓(푥) is defined to be 2 1 푑+푖 푒−퐷 푓(푥) = lim ∫ 퐾(푠 − 푥, 푡) 푓(푠)푑푠 푡→1 2휋푖 푑−푖∞ whenever the integral converges for 0 < t < 1 and the limit exists. At first sight this operator seems to depend on d, but for all functions 푓(푥) to which we shall apply it the result will be independent for d by virtue of Cauchy's integral theorem. Definition (c) : −퐷2 ( ) ∞ ( ) ( ) 푒 푓 푥 = ∫−∞ 퐾 푦 + 푖푥, 푡 푓 푖푦 푑푦 2 Let us introduce a notation for a class of functions to which 푒−퐷 will be applicable and to which all Weierstrass transform will belong. Definition(d) : Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 102 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 A function 푓(푥) belongs to class A in the integral a < x < b if and only if it can be extended analytically into the complex plane in such a way that (i) 푓(푥 + 푖푦) is analytic in the strip a < x < b 푦3 (ii) 푓(푥 + 푖푦) = 0 (|푦| 푒 3 ), |푦| → ∞,Uniformly in every closed subinterval of a < x < b. For inversion 2 operator 푒−퐷 inverts the Weierstrass transform in the form of special case in which 휙(푥) is bounded and continuous. For this if 휙(푦) is bounded and continuous in –∞ < 푦 < ∞, and if ( ) ∞ ( ) ( ) 푓 푥 = ∫−∞ 퐾 푥 − 푦, 1 휙 푦 푑푦 and the Weierstrass transform is defined by 1 ∞ −(푥−푦)2 퐹(푥) = ∫ 푓(푦) 푒 4 푑푦 ,Where 푓(푦) a suitable restricted conventional is function on −∞ < 푦 < ∞ and 푥 √4휋 −∞ is a complex variable. Thus , the Mellin-Weierstrass integral transform is defined by (푥−푦)2 1 ∞ ∞ − 퐹(푠, 푥) = 푀푊{푓(푡, 푦)} = ∫ ∫ 푓(푡, 푦)푦푠−1푒 4 푑푦푑푡 . √4휋 표 표 II. SOME THEOREMS ON MELLIN –WEIERSTRASS (MW) TRANSFORMS: Theorem-1 If f(x), f(y) and g(x), g(y) are four function a,b be two constants, then MW [ f(x) f(y) + b f(x) f(y) ;p] = a MW [f(x)f(y); p] + MW [g(x) g(y); p] Proof: We have 2 1 ∞ ∞ −(x−y) MW [ f(x) f(y) + b f(x) f(y) ;p]= ∫ ∫ xp−1{ a f(x) f(y) + b g(x) g(y) } e 4 √4π 0 0 2 2 a ∞ ∞ −(x−y) b ∞ ∞ −(x−y) = ∫ ∫ xp−1 e 4 f(x) f(y) dx dy + ∫ ∫ xp−1 e 4 f(x) f(y) dx dy √4π 0 0 √4π 0 0 Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 103 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 = a MW [f(x)f(y); p] + MW [g(x) g(y); p] Theorem-2 MW [f(ax) f(y):p]= a−p f ∗(p) Proof: 2 1 ∞ ∞ −(x−y) MW[f(ax) f(y):p]= ∫ ∫ xp−1 e 4 f(ax) f(y) dx dy √4π 0 0 Putting ax = u or , dx = du u −( −y)2 1 ∞ ∞ u a 1 = ∫ ∫ e 4 f(u) f(y) du dy √4π 0 0 a a u −( −y)2 a−p ∞ ∞ a = ∫ ∫ up−1 f(u) e 4 f(y) du dy √4π 0 0 2 a−p ∞ ∞ −(x−y) = ∫ ∫ xp−1 f(x) f(y) e 4 dx dy √4π 0 0 = 푎−푝 푓∗ (p) Theorem-3 푝 MW [ f(푥푎) f(y);p ] = 푎−1 푓∗ ( ) 푎 Proof 2 1 ∞ ∞ −(푥−푦) MW[ f(푥푎)f(y);p ] = ∫ ∫ 푥푝−1 f(푥푎) 푒 4 푓(푦) 푑푥 푑푦 √4휋 0 0 1 1 1 −1 Putting 푥푎 = u 푥= 푢푎 푑푥 = 푢푎 푑푢 푎 1 2 −푦 (푢푎 ) 1 1 1 ∞ ∞ 1 −1 = ∫ ∫ (푢푎 )푝 − 1 푓(푢) 푓(푦) 푢푎 푒 4 푑푢 푑푦 √4휋 0 0 푎 Volume XIII, Issue IX, SEPTEMBER 2020 Page No: 104 JAC : A JOURNAL OF COMPOSITION THEORY ISSN : 0731-6755 1 2 −푦 (푢푎 ) −1 푝 푎 ∞ ∞ −1 = ∫ ∫ 푢푎 푓(푢) 푓(푦) 푒 4 푑푢 푑푦 √4휋 0 0 −1 푝 (푥−푦)2 푎 ∞ ∞ −1 = ∫ ∫ 푥푎 푒 4 푓(푥) 푓(푦) 푑푥 푑푦= 푎−1 푓∗(p/a).
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