Hankel complementary integral transformations of arbitrary order

M. Linares Linares and J. M. R. Mendez Pérez

International Journal of Mathematics and Mathematical Sciences, 1992, vol. 15, 1-10

Abstract:

Four selfreciprocal integral transformations of Hankel type are defined through ( ℋ i , μ f ) ( y ) = F i ( y ) = ∫ 0 ∞ α i ( x ) ℊ i , μ ( x y ) f ( x ) d x , ℋ i , μ − 1 = ℋ i , μ , where i = 1 , 2 , 3 , 4 ; μ ≥ 0 ; α 1 ( x ) = x 1 + 2 μ , ℊ 1 , μ ( x ) = x − μ J μ ( x ) , J μ ( x ) being the Bessel function of the first kind of order μ ; α 2 ( x ) = x 1 − 2 μ , ℊ 2 , μ ( x ) = ( − 1 ) μ x 2 μ ℊ 1 , μ ( x ) ; α 3 ( x ) = x − 1 − 2 μ , ℊ 3 , μ ( x ) = x 1 + 2 μ ℊ 1 , μ ( x ) , and α 4 ( x ) = x − 1 + 2 μ , ℊ 4 , μ ( x ) = ( − 1 ) μ x ℊ 1 , μ ( x ) . The simultaneous use of transformations ℋ 1 , μ , and ℋ 2 , μ , (which are denoted by ℋ μ ) allows us to solve many problems of Mathematical Physics involving the differential operator Δ μ = D 2 + ( 1 + 2 μ ) x − 1 D , whereas the pair of transformations ℋ 3 , μ and ℋ 4 , μ , (which we express by ℋ μ * ) permits us to tackle those problems containing its adjoint operator Δ μ * = D 2 − ( 1 + 2 μ ) x − 1 D + ( 1 + 2 μ ) x − 2 , no matter what the real value of μ be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation ∫ 0 ∞ f ( x ) g ( x ) d x = ∫ 0 ∞ ( ℋ μ f ) ( y ) ( ℋ μ * g ) ( y ) d y , which is now valid for all real μ .

Date: 1992
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:681761

DOI: 10.1155/S0161171292000401

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