Abstract
There is a one-to-one correspondence between formal power series (FPS) \(\sum\limits_{k = 0}^\infty {a_k x^k }\) with positive radius of convergence and corresponding analytic functions. Since a goal of Computer Algebra is to work with formal objects and preserve such symbolic information, it should be possible to automate conversion between these forms in Computer Algebra Systems (CASs). However, only Macsyma provides a rather limited procedure powerseries to calculate FPS from analytic expressions in certain special cases.
We present an algorithmic approach to compute an FPS, which has been implemented by the author and A. Rennoch in Mathematica, and by D. Gruntz in Maple. Moreover, the same algorithm can be reversed to calculate a function that corresponds to a given FPS, in those cases when an initial value problem for a certain ordinary differential equation can be solved.
Further topics of application like infinite summation, and asymptotic expansion are presented.
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References
Axiom: User Guide. The Numerical Algorithms Group Ltd., 1991.
Davenport, J.H., Siret, Y., Tournier, E.: Computer-Algebra: Systems and algorithms for algebraic computation. Academic Press, 1988.
Gosper Jr., R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75 (1978) 40–42.
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products — corrected and enlarged version. Academic Press, New York-London, 1980.
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Publ. Co., Reading, Massachusetts, 1988.
Hansen, E.R.: A table of series and products. Prentice-Hall, Englewood Cliffs, NJ, 1975.
Hearn, A.: Reduce User's manual, Version 3.4. The RAND Corp., Santa Monica, CA, 1987.
Koepf, W.: Power series in Computer Algebra. J. Symb. Comp. 13 (1992) 581–603.
Koepf, W.: Examples for the algorithmic calculation of formal Puiseux, Laurent and power series (to appear).
Macsyma: Reference Manual, Version 13. Symbolics, USA, 1988.
Maple: Reference Manual, fifth edition. Watcom publications, Waterloo, 1988.
Norman, A. C.: Computing with formal power series. Transactions on mathematical software 1, ACM Press, New York (1975) 346–356.
Singer, M. F.: Liouvillian solutions of n-th order homogeneous linear differential equations. Amer. J. Math. 103 (1981) 661–682.
Walter, W.: Analysis I. Springer Grundwissen Mathematik 3, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.
Wilf, H.S.: Generatingfunctionology. Academic Press, Boston, 1990.
Wolfram, St.: Mathematica. A system for doing mathematics by Computer. Addison-Wesley Publ. Comp., Redwood City, CA, 1991.
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© 1993 Springer-Verlag Berlin Heidelberg
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Koepf, W. (1993). Algorithmic development of power series. In: Calmet, J., Campbell, J.A. (eds) Artificial Intelligence and Symbolic Mathematical Computing. AISMC 1992. Lecture Notes in Computer Science, vol 737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57322-4_14
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DOI: https://doi.org/10.1007/3-540-57322-4_14
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