On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira
The classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex function F(z), to its derivative F(ν)(z) of complex order ν, understood as either a Liouville (1832) or a Rieman (1847) differintegration (Campos 1984, 1985); these results are distinc...
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| Language: | English |
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Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171290000941 |
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| author | L. M. B. C. Campos |
| author_facet | L. M. B. C. Campos |
| author_sort | L. M. B. C. Campos |
| collection | DOAJ |
| description | The classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex function F(z), to its derivative F(ν)(z) of complex order ν, understood as either a Liouville (1832) or a Rieman (1847) differintegration (Campos 1984, 1985); these results are distinct from, and alternative to, other extensions of Taylor's series using differintegrations (Osler 1972, Lavoie & Osler & Tremblay 1976). We consider a complex function F(z), which is analytic (has an isolated singularity) at ζ, and expand its derivative of complex order F(ν)(z), in an ascending (ascending-descending) series of powers of an auxiliary function f(z), yielding the generalized Teixeira (Lagrange) series, which includes, for f(z)=z−ζ, the generalized Taylor (Laurent) series. The generalized series involve non-integral powers and/or coefficients evaluated by fractional derivatives or integrals, except in the case ν=0, when the classical theorems of Taylor (1715), Lagrange (1770), Laurent (1843) and Teixeira (1900) are regained. As an application, these generalized series can be used to generate special functions with complex parameters (Campos 1986), e.g., the Hermite and Bessel types. |
| format | Article |
| id | doaj-art-1246e6d1ea8746afb6b28bbb211e7858 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1990-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-1246e6d1ea8746afb6b28bbb211e78582025-02-03T01:06:59ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251990-01-0113468770810.1155/S0161171290000941On generalizations of the series of Taylor, Lagrange, Laurent and TeixeiraL. M. B. C. Campos0Instituto Superior Técnico, Lisboa Codex 1096, PortugalThe classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex function F(z), to its derivative F(ν)(z) of complex order ν, understood as either a Liouville (1832) or a Rieman (1847) differintegration (Campos 1984, 1985); these results are distinct from, and alternative to, other extensions of Taylor's series using differintegrations (Osler 1972, Lavoie & Osler & Tremblay 1976). We consider a complex function F(z), which is analytic (has an isolated singularity) at ζ, and expand its derivative of complex order F(ν)(z), in an ascending (ascending-descending) series of powers of an auxiliary function f(z), yielding the generalized Teixeira (Lagrange) series, which includes, for f(z)=z−ζ, the generalized Taylor (Laurent) series. The generalized series involve non-integral powers and/or coefficients evaluated by fractional derivatives or integrals, except in the case ν=0, when the classical theorems of Taylor (1715), Lagrange (1770), Laurent (1843) and Teixeira (1900) are regained. As an application, these generalized series can be used to generate special functions with complex parameters (Campos 1986), e.g., the Hermite and Bessel types.http://dx.doi.org/10.1155/S0161171290000941fractional derivativesgeneralized Taylor and Laurent seriesspecial functionsand generalized Cauchy integral. |
| spellingShingle | L. M. B. C. Campos On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira International Journal of Mathematics and Mathematical Sciences fractional derivatives generalized Taylor and Laurent series special functions and generalized Cauchy integral. |
| title | On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira |
| title_full | On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira |
| title_fullStr | On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira |
| title_full_unstemmed | On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira |
| title_short | On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira |
| title_sort | on generalizations of the series of taylor lagrange laurent and teixeira |
| topic | fractional derivatives generalized Taylor and Laurent series special functions and generalized Cauchy integral. |
| url | http://dx.doi.org/10.1155/S0161171290000941 |
| work_keys_str_mv | AT lmbccampos ongeneralizationsoftheseriesoftaylorlagrangelaurentandteixeira |