 The Geometry of ℂ n is Important for the Algebra of Elementary FunctionsJames H. Davenport ()
A chapter in Algebra, Geometry and Software Systems, 2003, pp 207-224 from Springer Abstract: Abstract On the one hand, we all “know” tha $$\sqrt {{z^2}} = z$$ , but on the other hand we know that this is false when z = −1. We all know that ln e x = x, and we all know that this is false when x = 2πi. How do we imbue a computer algebra system with this sort of “knowledge”? Why is it that $$\sqrt x \sqrt y = \sqrt {xy} $$ is false in general y =), but $$\sqrt {1 - z} \sqrt {1 + z} = \sqrt {1 - {z^2}} $$ is true everywhere? The root cause of this, of course, is that functions such as $$\sqrt {} $$ and log are intrinsically multi-valued from their algebraic definition. It is the contention of this paper that, only by considering the geometry of ℂ (or ℂ n if there are n variables) induced by the various branch cuts can we hope to answer these questions even semi-algorithmically (i.e. in a yes/no/fail way). This poses questions for geometry, and calls out for a more efficient formulation of cylindrical algebraic decomposition, as well as for cylindrical non-algebraic decomposition. It is an open question as to how far this problem can be rendered fully automatic. Keywords: Riemann Surface; Elementary Function; Multivalued Function; Computer Algebra System; Correct Equation (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-05148-1_11 Ordering information: This item can be ordered from DOI: 10.1007/978-3-662-05148-1_11 Access Statistics for this chapter More chapters in Springer Books from Springer |
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