Abstract
The theory introduced in previous chapters, especially the Fredholm Theory, was presented under the restrictive assumptions that the kernel was continuous on its domain of definition and that the interval of integration was finite. There is no guarantee that those results or similar ones will hold if the kernel has an infinite discontinuity or if the interval of integration is infinite.
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Notes
- 1.
See Oberhettinger, F., Tables of Mellin Transforms, Springer, New York, 1974.
- 2.
See Porter, D. and Stirling, D., Integral Equations: A Practical Treatment, from Spectral Theory to Applications, Cambridge University Press, Cambridge, 1990. For an alternate choice, see Andrews, L.C. and Shivamoggi, B.K., Integral Transforms for Engineers and Applied Mathematicians, MacMillan Publishing Company, New York, 1988.
- 3.
This example is discussed in greater detail in Widder, D.V., The Laplace Transform, Princeton University Press, Princeton, 1946, pp. 390–391.
- 4.
For a discussion of this problem, see Andrews, L.C. and Shivamoggi, B.K., Integral Transforms for Engineers and Applied Mathematicians, MacMillan Publishing Company, New York, 1988, pp. 240–242.
- 5.
For a proof of this theorem, see Privalov, I.I., Introduction to the Theory of Functions of a Complex Variable, 8th edn. State Technical Publishers, 1948.
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Zemyan, S.M. (2012). Singular Integral Equations. In: The Classical Theory of Integral Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8349-8_7
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DOI: https://doi.org/10.1007/978-0-8176-8349-8_7
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