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
( , ) = ( , )
( ) ∞ ∞ = ( , ) √
Theorem: If ( , ) = ( , ), then
( , ) = ( ) ( , )
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20 Shahnaaz Assad, Preeti Tamrakar & Amarnath Kumar Thakur
i'e;
Proof: ( , ) = ( , )
By definition of transform we have
( , ) = ( , )
( ) ∞ ∞ = ( , ) √
Now differentiating both sides with respect to x
′( , ) = ( , )
( ) ∞ ∞ = ( , )
( ) ∞ ∞ = ( , ) = ( , ) √
′( , ) = ( , ) (1)
This shows the theorem is true for n=1
Next let the theorem is true for n=k , then we have
( , ) = ( ) ( , )
i'e;
( , ) = ( ) ( , )
( ) ∞ ∞ i'e; ( , ) = ( ) ( , ) √
On differentiating it both sides w. r. t. x we get
( ) ∞ ∞ ( , ) = ( ) ( , ) √
( ) ∞ ∞ = ( ) . ( , ) √
=( ) ( , )
Thus we get
( , ) = ( ) ( , )
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.
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REFERENCES
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