Abstract
The aim is to find conditions under which the equation
has a solution for f; F being the hypergeometric function. The method uses fractional differentiation to reduce the left side first to a generalized Stieltjes transform and then to a Stieltjes transform; the latter is then inverted by an adaptation of one of Widder's methods. The results obtained are: necessary conditions for a solution in a certain class, sufficient conditions for a solution in a comparable class, and an explicit formula for the solution, involving fractional derivatives.
Keywords
- Fractional Derivative
- Hypergeometric Function
- Fractional Integral
- Fractional Differentiation
- Absolute Convergence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
D.V. Widder: The Laplace Transform (Princeton, 1946).
A. Erdélyi and others: Higher Transcendental Functions, vol. 1 (McGraw-Hill, 1953).
A. Erdélyi and others: Tables of Integral Transforms, vol. 2 (McGraw-Hill, 1954).
R.G. Buschman: An inversion integral, Proc. American Math. Soc. 13 (1962) 675–677.
A. Erdélyi: An integral equation involving Legendre functions, J. Soc. Indust. App. Math. 12 (1964) 15–30.
T.P. Higgins: A hypergeometric function transform, J. Soc. Indust. App. Math. 12 (1964) 601–612.
J. Wimp: Two integral transform pairs involving hypergeometric functions, Proc. Glasgow Math. Assoc. 7 (1965) 42–44.
A. Erdélyi: Some integral equations involving finite parts of divergent integrals, Glasgow Math. J. 8 (1967) 50–54.
E.R. Love: Some integral equations involving hypergeometric functions, Proc. Edinburgh Math. Soc. 15 (1967) 169–198.
E.R. Love: Two more hypergeometric integral equations, Proc. Cambridge Philos. Soc. 63 (1967) 1055–1076.
A. Erdélyi: Fractional integrals of generalized functions, J. Australian Math. Soc. 14 (1972) 30–37.
E.R. Love: Two index laws for fractional integrals and derivatives, J. Australian Math. Soc. 14 (1972) 385–410.
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© 1975 Springer-Verlag
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Love, E.R. (1975). A hypergeometric integral equation. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067112
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DOI: https://doi.org/10.1007/BFb0067112
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07161-7
Online ISBN: 978-3-540-69975-0
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