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Mellin-Weierstrass Integral Transform of Order Partial Derivative of (, )

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Mellin-Weierstrass Integral Transform of ��� Order Partial Derivative of �(�, �) Journal of Mathematics and Computer Applications Research (JMCAR) ISSN(P): 2250-2408; ISSN (E): Applied Vol. 2, Issue 1, Jun 2015, 19-22 TJPRC Pvt. Ltd. MELLIN-WEIERSTRASS INTEGRAL TRANSFORM OF ORDER PARTIAL DERIVATIVE OF (, ) SHAHNAAZ ASSAD 1, PREETI TAMRAKAR 2 & AMARNATH KUMAR THAKUR 3 Dr. C. V. Raman University, Bilaspur, Chhattisgarh, India ABSTRACT An expression is obtained for Mellin-Weierstrass transform MW of the function in terms of their existence. It is shown, how the main result can be extended to hold for the MW transform of several functions and how the order partial derivative of the transform exists. KEYWORDS : Integral Transform, Mellin Transform, Weierstrass Transform, Mellin- Weierstrass Transform, Generalized Function, Existence Theorem, Multiplication Theorem INTRODUCTION In this page we have introduce the concept of MW transform which has use in several fields. The basic idea behind any transform is that the given problem can be solved more readily in the transform domain. The method is especially attractive in the linear mathematical models for physical system such as a spring/ mass system or a series of electrical circuit which involve discontinuous functions The Mellin transform is defined as: F(s)=M{f(y),s} ∞ ( ) = . Where a ≤ Re(s) ≤ b The conventional Weierstrass transform is defined by ∞ () () = () ∞ Where () a suitable restricted conventional is function on −∞ ∞ and ! is a complex variable. Our purpose in this work is to find Mellin-Weierstrass integral transform of order partial derivative of (!, ) The Mellin-Weierstrass integral transform is defined by "(#, !) = $% &(', )( (-.)/ ∞ ∞ = (', ), 0 ' *+ 1 1 Theorem: If $% &(', )( = "(#, !), then $% &2(', )( = (345#)2$%&(', )( www.tjprc.org [email protected] 20 Shahnaaz Assad, Preeti Tamrakar & Amarnath Kumar Thakur i'e; Proof:$% &(', )( = "(#, !) By definition of $% transform we have "(#, !) = $% &(', )( (-.)/ ∞ ∞ = (', ), 0 ' *+ 1 1 Now differentiating both sides with respect to x 6 "′(#, !) = &"(#, !)( 6 (-.)/ 6 ∞ ∞ = & (', ), 0 '( 6 1 1 (-.)/ ∞ ∞ = 345# (', ), 0 ' = 345# "(#, !) *+ 1 1 "′(#, !) = 345# "(#, !) (1) This shows the theorem is true for n=1 Next let the theorem is true for n=k , then we have $% &7(', )( = (345#)7$%&(', )( i'e; 68 "(#, !) = (345# )7"(#, !) 8 (-.)/ 6 ∞ ∞ i'e; "(#, !) = (345# )7 (', ), 0 ' 69 8 *+ 1 1 On differentiating it both sides w. r. t. x we get 8:; (-.)/ 6 6 ∞ ∞ "(#, !) = &(345# )7 (', ), 0 ' ( 6 8:; 6 *+ 1 1 (-.)/ ∞ ∞ = (345#)7. 345# (', ), 0 ' *+ 1 1 =(345# )7<"(#, !) Thus we get 68:; "(#, !) = (345# )7<"(#, !) 69 8:; Which implies theorem is true for = = + 1 also. Hence by principal of mathematical induction, the theorem is true for all positive integral values of n. www.tjprc.org [email protected] Mellin-Weierstrass Integral Transform of Order Partial Derivative of (, ) 21 REFERENCES 1. Chelo Ferreira. Jose L. Lopez, Approximation of the Poisson Transform for large and small values of the Transformation arameter, Ramanujan J.(2012) DOI 10,1007/11139-012-9438-y 2. Dr.P.A. Gulhaneand S.S Mathurkar, International journal of scientificEngineeringResearch,Volume4, issue 12,Decmber-2013 3. Saitoh S : The Weierstrass Transform and an Isometry in the Heat Equation Applicable Anal, (1983). 4. Pathak R S : A RepresAentation Theorem for a Class of Stieltjes Transformable Generalized Function (1974). 5. Srivastava H M : The Weierstrass-Laguerre Transform, National Academy of Sciences, Vol 68 , No 3, pp 554- 556,(1971) 6. Zemanian A H : Generalized Integral Transformation, Interscience Publishers, New York (1968). 7. Zemanian A H : Distribution Theory and Transform Analysis, McGraw- Hill,NewYork(1965). www.tjprc.org [email protected] .
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