Construction of Rational and Negative Powers of a Formal Series

Some methods are described for the generation
of fractional and negative powers of any formal 
series, such as Poisson series or Chebyshev series.  It
is shown that, with the use of the three elementary 
operations of addition, subtraction, and multiplication,
all rational (positive and negative) powers 
of a series can be constructed.  There are basically two
approaches: the binomial theorem and the iteration 
methods.  Both methods are described here, and the relationship
between them is pointed out.  Some well-known 
classical formulas are obtained as particular cases,
and it is shown how the convergence properties of 
these formulas can be improved with very little additional
computations.  Finally, at the end of the 
article, some numerical experiments are described
with Chebyshev series and with Fourier series.

CACM January, 1971

Brucke, R. A.

series expansion, series inversion, root extraction,
binomial theorem, Newton iterations, Chebyshev 
series, Poisson series, Fourier series

3.11 3.15 3.21 5.0

CA710105 JB February 8, 1978  10:57 AM

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