Section15.5

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## 15.5 Library

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The FullPartialFractionExpansion domain has been added. This domain computes factor-free full partial fraction expansions. See section FullPartialFractionExpansion for examples.

We have implemented the Bertrand/Cantor algorithm for integrals of hyperelliptic functions. This brings a major speedup for some classes of algebraic integrals.

We have implemented a new (direct) algorithm for integrating trigonometric functions. This brings a speedup and an improvement in the answer quality.

The SmallFloat domain has been renamed DoubleFloat and SmallInteger has been renamed SingleInteger. The new abbreviations as DFLOAT and SINT, respectively. We have defined the macro SF, the old abbreviation for {\sf SmallFloat}, to expand to DoubleFloat and modified the documentation and input file examples to use the new names and abbreviations. You should do the same in any private Axiom files you have.

There are many new categories, domains and packages related to the NAG Library Link facility. See the file

src/algebra/exposed.lsp

for a list of constructors in the naglink Axiom exposure group.

We have made improvements to the differential equation solvers and there is a new facility for solving systems of first-order linear differential equations. In particular, an important fix was made to the solver for inhomogeneous linear ordinary differential equations that corrected the calculation of particular solutions. We also made improvements to the polynomial and transcendental equation solvers including the ability to solve some classes of systems of transcendental equations.

The efficiency of power series have been improved and left and right expansions of tan(f(x)) at x= a pole of f(x) can now be computed. A number of power series bugs were fixed and the GeneralUnivariatePowerSeries domain was added. The power series variable can appear in the coefficients and when this happens, you cannot differentiate or integrate the series. Differentiation and integration with respect to other variables is supported.

A domain was added for representing asymptotic expansions of a function at an exponential singularity.

For limits, the main new feature is the exponential expansion domain used to treat certain exponential singularities. Previously, such singularities were treated in an ad hoc way and only a few cases were covered. Now Axiom can do things like

limit( (x+1)**(x+1)/x**x - x**x/(x-1)**(x-1), x = %plusInfinity)

in a systematic way. It only does one level of nesting, though. In other words, we can handle exp(some function with a pole), but not exp(exp(some function with a pole)).

The computation of integral bases has been improved through careful use of Hermite row reduction. A P-adic algorithm for function fields of algebraic curves in finite characteristic has also been developed.

Miscellaneous: There is improved conversion of definite and indefinite integrals to InputForm; binomial coefficients are displayed in a new way; some new simplifications of radicals have been implemented; the operation complexForm for converting to rectangular coordinates has been added; symmetric product operations have been added to LinearOrdinaryDifferentialOperator.

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