Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 23 Issue 4

{ } = 1 ʋ(1 − ʋ) ∫ − ∞ 0 ( ʋ 1 − ʋ ) { } = 1 (1 − ʋ)̅ ( ʋ 1 − ʋ ) Assume that and are peicewise continuous function, or one of them is dirac’s generalized function [7]. The convolution of and is a function denoted ∗ by and given by the following expression ( ∗ )( ) = ∫ ( ) ( − ) … (2.4) 0 For every piecewise continuous function , ℎ the following properties hold. 1. Commutativity ∗ = ∗ 2. Associativity ∗ ( ∗ ℎ) = ( ∗ ) ∗ ℎ 3. Distributivity ∗ ( + ℎ) = ∗ + ∗ ℎ 4. Neutral element ∗ 0 = 0 5. Identity element ∗ 1 = If the function and have well defined transform { } and { } then Z { ∗ } = ʋ 2 { } { } … (2.5) { ∗ } = 1 ʋ ∫ −∞ 0 ( ∗ )(ʋ ) = 1 ʋ ∫ −∞ 0 [∫ ( ) (ʋ − ) ∞ 0 ] … (2.6) A New Integral Transform Called “Saxena & Gupta Transform” and Relation between New Transform and other Integral Transforms © 2023 Global Journals 1 Year 2023 12 Global Journal of Science Frontier Research Volume XXIII Issue ersion I V ( F ) The convolution of two function Definition: Theorem 1: Proof: R ef 7. Dirac, paul (1930), “The principal of Quantum Mechanics” (1st ed.), Oxford university press. IV iii)