The Mehler-Fock transform of general order and arbitrary index and its inversion
An integral transform involving the associated Legendre function of zero order, P−12+iτ(x), x∈[1,∞), as the kernel (considered as a function of τ), is called Mehler-Fock transform. Some generalizations, involving the function P−12+iτμ(x), where the order μ is an arbitrary complex number, including t...
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| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117128400017X |
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| _version_ | 1832554485031895040 |
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| author | Cyril Nasim |
| author_facet | Cyril Nasim |
| author_sort | Cyril Nasim |
| collection | DOAJ |
| description | An integral transform involving the associated Legendre function of zero order, P−12+iτ(x), x∈[1,∞), as the kernel (considered as a function of τ), is called Mehler-Fock transform. Some generalizations, involving the function P−12+iτμ(x), where the order μ is an arbitrary complex number, including the case when μ=0,1,2,… have been known for some time. In this present note, we define a general Mehler-Fock transform involving, as the kernel, the Legendre function P−12+tμ(x), of general order μ and an arbitrary index −12+t, t=σ+iτ, −∞<τ<∞. Then we develop a symmetric inversion formulae for these transforms. Many well-known results are derived as special cases of this general form. These transforms are widely used for solving many axisymmetric potential problems. |
| format | Article |
| id | doaj-art-2f3b3fcdce4e43b8a1364e475bff62cd |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2f3b3fcdce4e43b8a1364e475bff62cd2025-02-03T05:51:17ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017117118010.1155/S016117128400017XThe Mehler-Fock transform of general order and arbitrary index and its inversionCyril Nasim0Department of Mathematics and Statistics, The University of Calgary, Alberta, Calgary, CanadaAn integral transform involving the associated Legendre function of zero order, P−12+iτ(x), x∈[1,∞), as the kernel (considered as a function of τ), is called Mehler-Fock transform. Some generalizations, involving the function P−12+iτμ(x), where the order μ is an arbitrary complex number, including the case when μ=0,1,2,… have been known for some time. In this present note, we define a general Mehler-Fock transform involving, as the kernel, the Legendre function P−12+tμ(x), of general order μ and an arbitrary index −12+t, t=σ+iτ, −∞<τ<∞. Then we develop a symmetric inversion formulae for these transforms. Many well-known results are derived as special cases of this general form. These transforms are widely used for solving many axisymmetric potential problems.http://dx.doi.org/10.1155/S016117128400017XMehler-Fock transformKontorovich-Lebedev transformsLegendre functions of first and second kindsMacdonald functionBessel functions. |
| spellingShingle | Cyril Nasim The Mehler-Fock transform of general order and arbitrary index and its inversion International Journal of Mathematics and Mathematical Sciences Mehler-Fock transform Kontorovich-Lebedev transforms Legendre functions of first and second kinds Macdonald function Bessel functions. |
| title | The Mehler-Fock transform of general order and arbitrary index and its inversion |
| title_full | The Mehler-Fock transform of general order and arbitrary index and its inversion |
| title_fullStr | The Mehler-Fock transform of general order and arbitrary index and its inversion |
| title_full_unstemmed | The Mehler-Fock transform of general order and arbitrary index and its inversion |
| title_short | The Mehler-Fock transform of general order and arbitrary index and its inversion |
| title_sort | mehler fock transform of general order and arbitrary index and its inversion |
| topic | Mehler-Fock transform Kontorovich-Lebedev transforms Legendre functions of first and second kinds Macdonald function Bessel functions. |
| url | http://dx.doi.org/10.1155/S016117128400017X |
| work_keys_str_mv | AT cyrilnasim themehlerfocktransformofgeneralorderandarbitraryindexanditsinversion AT cyrilnasim mehlerfocktransformofgeneralorderandarbitraryindexanditsinversion |