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Expansions by means of the Fractional Calculus

Published online by Cambridge University Press:  03 November 2016

W Fabian
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Extract

We have shown how the theory of the Fractional Calculus enables us to expand functions in various forms. In the present paper we shall obtain some further expansions.

A λth integral of f(z) along a simple curve l is defined by

where γ is the least integer greater than or equal to zero such that R(λ) + γ>0, D stands for d/dz, and the integration and differentiation are along l.

Type
Research Article
Information
The Mathematical Gazette , Volume 21 , Issue 247 , December 1937 , pp. 396 - 400
Copyright
Copyright © Mathematical Association 1937 

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References

page no 396 note * Fabian, , Phil. Mag. (7) 20 (1935), 781-9CrossRefGoogle Scholar. Quarterly Journ. of Math. (1936).

page no 396 note † Fabian, , Phil. Mag. (7) 20 (1935), 781-9CrossRefGoogle Scholar.

page no 396 note ‡ f-λ-n(z 0) stands for Dλ+n(la)f(z0), and, generally, fλ(z will be written for D-λ (la)f(z), when no ambiguity can arise.

page no 396 note § Fabian, Quarterly Journ. of Math. (1936).

page no 397 note * Fabian, , Phil. Mag. (7) 20 (1935), 781-9CrossRefGoogle Scholar.

page no 398 note * Fabian, , Phil. Mag. (7) 20 (1935), 781-9CrossRefGoogle Scholar.