Abstract
There are many problems with the simplification of elementary functions, particularly over the complex plane, though not exclusively – see (20). Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have not yet reduced the simplification process to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.
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M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th printing (US Government Printing Office, 1972).
R.J. Bradford, Algebraic simplification of multiple-valued functions, in: Proc. DISCO'92, ed. J.P. Fitch, Lecture Notes in Computer Science, Vol. 721 (Springer, New York, 1993) pp. 13-21.
C. Carathéodory, Theory of Functions of a Complex Variable, 2nd ed., transl. F. Steinhardt (Chelsea, New York, 1958).
R.M. Corless, J.H. Davenport, D.J. Jeffrey, G. Litt and S.M. Watt, Reasoning about the elementary functions of complex analysis, in: Artificial Intelligence and Symbolic Computation, eds. J.A. Campbell and E. Roanes-Lozano, Lecture Notes in Artificial Intelligence, Vol. 1930 (Springer, New York, 2001) pp. 115-126.
R.M. Corless, J.H. Davenport, D.J. Jeffrey and S.M. Watt, According to Abramowitz and Stegun, SIGSAM Bulletin 34(2) (2000) 58-65.
R.M. Corless and D.J. Jeffrey, The unwinding number, SIGSAM Bulletin 30(2) (1996), 28-35.
R.M. Corless and D.J. Jeffrey, Graphing elementary Riemann surfaces, SIGSAM Bulletin 32(1) (1998), 11-17.
J.H. Davenport and H.-C. Fischer, Manipulation of expressions, in: Improving Floating-Point Programming, ed. P.J.L. Wallis (Wiley, New York, 1990) pp. 149-167.
Encyclopedia Britannica, 15th ed. (Encyclopedia Britannica Inc., Chicago, 1995).
P. Gianni, M. Seppälä, R. Silhol and B.M. Trager, Riemann surfaces, plane algebraic curves and their period matrices, J. Symbolic Comput. 26 (1998) 789-803.
N. Hur and J.H. Davenport, An exact real arithmetic with equality determination, in: Proc. ISSAC 2000, ed. C. Traverso (ACM, New York, 2000) pp. 169-174.
IEEE Standard Pascal Computer Programming Language (IEEE, 1983).
IEEE Standard 754 for Binary Floating-Point Arithmetic (IEEE, 1985) reprinted in SIGPLAN Notices 22 (1987) 9-25.
W. Kahan, Branch cuts for complex elementary functions, in: The State of Art in Numerical Analysis, eds. A. Iserles and M.J.D. Powell (Clarendon Press, Oxford, 1987) pp. 165-211.
G. Litt, Unwinding numbers for the logarithmic, inverse trigonometric and inverse hyperbolic functions, M.Sc. project, Department of Applied Mathematics, University of Western Ontario (December 1999).
R.H. Risch, Algebraic properties of the elementary functions of analysis, Amer. J. Math. 101 (1979) 743-759.
G.L. Steele, Jr., Common LISP: The Language, 2nd ed. (Digital Press, 1990).
M. Trott, Visualization of Riemann surfaces of algebraic functions, Mathematica in Education and Research 6(4) (1997) 15-36.
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Bradford, R., Corless, R.M., Davenport, J.H. et al. Reasoning about the Elementary Functions of Complex Analysis. Annals of Mathematics and Artificial Intelligence 36, 303–318 (2002). https://doi.org/10.1023/A:1016007415899
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DOI: https://doi.org/10.1023/A:1016007415899