Abstract
Although, formally, mathematics is clear that a function is a single-valued object, mathematical practice is looser, particularly with n-th roots and various inverse functions. In this paper, we point out some of the looseness, and ask what the implications are, both for Artificial Intelligence and Symbolic Computation, of these practices. In doing so, we look at the steps necessary to convert existing texts into
(a) rigorous statements
(b) rigorously proved statements.
In particular we ask whether there might be a constant “de Bruijn factor” [18] as we make these texts more formal, and conclude that the answer depends greatly on the interpretation being placed on the symbols.
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References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. US Government Printing Office (1964)
Apostol, T.M.: Calculus, 1st edn., vol. I. Blaisdell (1961)
Beaumont, J.C., Bradford, R.J., Davenport, J.H., Phisanbut, N.: Adherence is Better Than Adjacency. In: Kauers, M. (ed.) Proceedings ISSAC 2005, pp. 37–44 (2005)
Beaumont, J.C., Bradford, R.J., Davenport, J.H., Phisanbut, N.: Testing Elementary Function Identities Using CAD. AAECC 18, 513–543 (2007)
Bourbaki, N.: Théorie des Ensembles. In: Diffusion CCLS (1970)
Bradford, R.J., Corless, R.M., Davenport, J.H., Jeffrey, D.J., Watt, S.M.: Reasoning about the Elementary Functions of Complex Analysis. Annals of Mathematics and Artificial Intelligence 36, 303–318 (2002)
Carathéodory, C.: Theory of functions of a complex variable. Chelsea Publ., New York (1958)
Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison-Wesley, Reading (1973)
Corless, R.M., Davenport, J.H., Jeffrey, D.J., Watt, S.M.: According to Abramowitz and Stegun. SIGSAM Bulletin 34(2), 58–65 (2000)
Davenport, J.H.: Equality in computer algebra and beyond. J. Symbolic Comp. 34, 259–270 (2002)
Henrici, P.: Applied and Computational Complex Analysis I. Wiley, Chichester (1974)
Jeffreys, H., Jeffreys, B.: Methods of Mathematical Physics, 3rd edn. Cambridge University Press, Cambridge (1956)
Kahan, W.: Branch Cuts for Complex Elementary Functions. In: Iserles, A., Powell, M.J.D. (eds.) The State of Art in Numerical Analysis, pp. 165–211 (1987)
Lazard, D., Rioboo, R.: Integration of Rational Functions - Rational Computation of the Logarithmic Part. J. Symbolic Comp. 9, 113–115 (1990)
Mulders, T.: A note on subresultants and the Lazard/Rioboo/Trager formula in rational function integration. J. Symbolic Comp. 24, 45–50 (1997)
Risch, R.H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101, 743–759 (1979)
Ritt, J.F.: Differential Algebra. In: American Mathematical Society Colloquium Proceedings, vol. XXXIII (1950)
Wiedijk, F.: The De Bruijn Factor (2000), http://www.cs.kun.nl/~freek/notes/factor.pdf
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Davenport, J.H. (2010). The Challenges of Multivalued “Functions”. In: Autexier, S., et al. Intelligent Computer Mathematics. CICM 2010. Lecture Notes in Computer Science(), vol 6167. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14128-7_1
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DOI: https://doi.org/10.1007/978-3-642-14128-7_1
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