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Fractional Calculus

Published online by Cambridge University Press:  03 November 2016

W Fabian
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Extract

In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.

We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l by

where the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.

Type
Research Article
Information
The Mathematical Gazette , Volume 20 , Issue 240 , October 1936 , pp. 249 - 253
Copyright
Copyright © Mathematical Association 1936

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References

page no 249 note * Fabian, , Math. Gazette (1936), vol. xx, no. 1, pp. 88–92 CrossRefGoogle Scholar.

page no 249 note † Fabian, , Phil. Mag., ser. 7, vol. xx, pp. 781-78 (1935)CrossRefGoogle Scholar.

page no 251 note * Fabian, , Phil. Mag. (1936), ser. 7, vol. xxi, pp. 274–280 CrossRefGoogle Scholar.

page no 251 note † If f(z) has M cycles at p, f(z) is to be regarded as having M branch-points at p, and the theorem applies to each of the M branch-points at p separately.

page no 251 note ‡ Fabian, Phil. Mag. (1936).