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Mathematics of Computation

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An integral analogue of Taylor's series and its use in computing Fourier transforms

Author: Thomas J. Osler
Journal: Math. Comp. 26 (1972), 449-460
MSC: Primary 44A15; Secondary 26A33, 65A05
MathSciNet review: 0306828
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Abstract: In this paper, an integral analogue of Taylor's series

$\displaystyle f(z) = \int_{ - \infty }^\infty {{D^\omega }f({z_0})} {(z - {z_0})^\omega }/\Gamma (\omega + 1)d\omega $

is discussed. $ {D^\omega }f(z)$ is a fractional derivative of order $ \omega $. Extensions of this integral are also given, one of which is an integral analogue of Lagrange's expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges.

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Keywords: Fractional derivative, Fourier integral theorem, Fourier transforms, Taylor series, Lagrange expansion, special functions
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society