Hankel complementary integral transformations of arbitrary order
Four selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx, ℋi,μ−1=ℋi,μ,where i=1,2,3,4; μ≥0; α1(x)=x1+2μ, ℊ1,μ(x)=x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x1−2μ, ℊ2,μ(x)=(−1)μx2μℊ1,μ(x); α3(x)=x−1−2μ,...
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Wiley
1992-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171292000401 |
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| author | M. Linares Linares J. M. R. Mendez Pérez |
| author_facet | M. Linares Linares J. M. R. Mendez Pérez |
| author_sort | M. Linares Linares |
| collection | DOAJ |
| description | Four selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx, ℋi,μ−1=ℋi,μ,where i=1,2,3,4; μ≥0; α1(x)=x1+2μ, ℊ1,μ(x)=x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x1−2μ, ℊ2,μ(x)=(−1)μx2μℊ1,μ(x); α3(x)=x−1−2μ, ℊ3,μ(x)=x1+2μℊ1,μ(x), and α4(x)=x−1+2μ, ℊ4,μ(x)=(−1)μxℊ1,μ(x). The simultaneous use of transformations ℋ1,μ, and ℋ2,μ, (which are denoted by ℋμ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ=D2+(1+2μ)x−1D, whereas the pair of transformations ℋ3,μ and ℋ4,μ, (which we express by ℋμ*) permits us to tackle those problems containing its adjoint operator Δμ*=D2−(1+2μ)x−1D+(1+2μ)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)dx=∫0∞(ℋμf)(y)(ℋμ*g)(y)dy,which is now valid for all real μ. |
| format | Article |
| id | doaj-art-0d26e4eec8ac482785ed57baed5b4e21 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1992-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0d26e4eec8ac482785ed57baed5b4e212025-08-20T03:54:15ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251992-01-0115232333210.1155/S0161171292000401Hankel complementary integral transformations of arbitrary orderM. Linares Linares0J. M. R. Mendez Pérez1Departamento de Informática y Sistemas, Universidad de Las Palmas, Canary Islands, Las Palmas de Gran Canaria, SpainDepartamento de Análisis Matemático, Facultad de Matemáticas, Universidad de La Laguna, Tenerife, Canary Islands, La Laguna, SpainFour selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx, ℋi,μ−1=ℋi,μ,where i=1,2,3,4; μ≥0; α1(x)=x1+2μ, ℊ1,μ(x)=x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order μ; α2(x)=x1−2μ, ℊ2,μ(x)=(−1)μx2μℊ1,μ(x); α3(x)=x−1−2μ, ℊ3,μ(x)=x1+2μℊ1,μ(x), and α4(x)=x−1+2μ, ℊ4,μ(x)=(−1)μxℊ1,μ(x). The simultaneous use of transformations ℋ1,μ, and ℋ2,μ, (which are denoted by ℋμ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ=D2+(1+2μ)x−1D, whereas the pair of transformations ℋ3,μ and ℋ4,μ, (which we express by ℋμ*) permits us to tackle those problems containing its adjoint operator Δμ*=D2−(1+2μ)x−1D+(1+2μ)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)dx=∫0∞(ℋμf)(y)(ℋμ*g)(y)dy,which is now valid for all real μ.http://dx.doi.org/10.1155/S0161171292000401complementary Hankel transformationsParseval equationgeneralized functions. |
| spellingShingle | M. Linares Linares J. M. R. Mendez Pérez Hankel complementary integral transformations of arbitrary order International Journal of Mathematics and Mathematical Sciences complementary Hankel transformations Parseval equation generalized functions. |
| title | Hankel complementary integral transformations of arbitrary order |
| title_full | Hankel complementary integral transformations of arbitrary order |
| title_fullStr | Hankel complementary integral transformations of arbitrary order |
| title_full_unstemmed | Hankel complementary integral transformations of arbitrary order |
| title_short | Hankel complementary integral transformations of arbitrary order |
| title_sort | hankel complementary integral transformations of arbitrary order |
| topic | complementary Hankel transformations Parseval equation generalized functions. |
| url | http://dx.doi.org/10.1155/S0161171292000401 |
| work_keys_str_mv | AT mlinareslinares hankelcomplementaryintegraltransformationsofarbitraryorder AT jmrmendezperez hankelcomplementaryintegraltransformationsofarbitraryorder |