An introduction to Axiom (3): Calculus

This is the third in a sequence of posts about the open source CAS Axiom. The previous post can be found at http://amca01.wordpress.com/2008/05/25/an-introduction-to-axiom-2/.

In this post I shall be looking at Axiom’s calculus abilities, starting with every beginning student’s love, limits:

Note here the use of %plusInfinity which is positive infinity. Axiom also has %minusInfinity. Limits are also possible for complex variables, using complexLimit, and %infinity for complex infinity.

After limits there are derivatives, implemented in Axiom with D. A third optional parameter gives the order of the derivative:

Integration is performed with the integrate command. Sometimes Axiom gets worried that an integrand may have a pole in the region of integration, in which case the addition of the string "noPole" will alleviate its fears:

Now for some series. There are several different commands for producing a power series, of which series is the most general. The following three commands show some of the power of Axiom’s type system. Since the first two outputs are of type UnivariatePuiseuxSeries, arithmetic on them will produce an output also of that type. Thus the division command produces the series for :

As an example of Axiom’s calculus power, let’s use the series command as a generating function. In particular we shall construct the Euler polynomials
defined by

So first create the series, and then extract a particular polynomial from it:

Finally, some differential equations. Note the use of the solve command – since the equation to be solved is a differential equation, solveproduces an appropriate solution. The main thing to note here is the prior definition of the unknown function as an operator: