New approach to the fractional derivatives
We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives of f, it is not sufficient to know the Taylor expansion of f, but we should also know the constants of all consecutive inte...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203206050 |
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| Summary: | We introduce a new approach to the fractional derivatives of the
analytical functions using the Taylor series of the functions. In
order to calculate the fractional derivatives of f, it is not
sufficient to know the Taylor expansion of f, but we should
also know the constants of all consecutive integrations of f.
For example, any fractional derivative of ex is ex only if
we assume that the nth consecutive integral of ex is ex
for each positive integer n. The method of calculating the
fractional derivatives very often requires a summation of
divergent series, and thus, in this note, we first introduce a
method of such summation of series via analytical continuation of
functions. |
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| ISSN: | 0161-1712 1687-0425 |