MathAction AxiomTutorial

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- AxiomDocumentation
  - AxiomTutorial <-- You are here.

The Axiom Volume 1: Tutorial is available in printed format at http://lulu.com

There is a good introduction to Axiom by Martin N. Dunstan, in English, at http://www.dcs.st-andrews.ac.uk/~mnd/documentation/axiom_tutorial
a complete introduction to Axiom by Daniel Augot, en FranÃ§ais, is available at http://www-rocq.inria.fr/codes/Daniel.Augot/axiom_intro.pdf
an Axiom tutorial and a collection of exercises by Christophe Conil and Quentin Carpent at http://la.riverotte.free.fr/axiom
and a blog by Alasdair McAndrew containing an excellent introduction to Axiom.

A Quick Tutorial

On this web site you can enter Axiom commands and see the output in proper mathematical form. These commands can also be typed directly into Axiom installed on your own computer.

You must enclose each set of Axiom commands in a \begin{axiom} \end{axiom} section. For example:

    \begin{axiom}
    [1/2, 3/4, 2/3]
    \end{axiom}

When you save the changes you make to a page, Axiom processes the contents of each section you have marked and the Axiom output is displayed.

Try this command:

axiom

[1/7, 3/4, 5/6]

$\label{eq1}\left[{1 \over 7}, \:{3 \over 4}, \:{5 \over 6}\right]$

(1)

Type: List(Fraction(Integer))

Please feel free to play with this in the SandBox!

Advanced Examples

Let's see if we can perform a finite window Fourier inversion.

axiom

simplify(integrate(exp(%i*_\omega*t) * exp(-%i*_\omega *
_\tau), _\omega=-_\Omega .. _\Omega))

$\label{eq2}{\left( \begin{array}{@{}l} \displaystyle -{i \ {e^{\left({i \ \Omega \ t}-{i \ \Omega \ \tau}\right)}}}+ \ \ \displaystyle {i \ {e^{\left(-{i \ \Omega \ t}+{i \ \Omega \ \tau}\right)}}}$

(2)

Type: Expression(Complex(Integer))

Try this integrating this

axiom

2*x/sin(x)^2

$\label{eq3}{2 \ x}\over{{\sin \left({x}\right)}^2}$

(3)

Type: Expression(Integer)

axiom

integrate(%,x)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle {2 \ {\sin \left({x}\right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{2 \ {\sin \left({x}\right)}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {2 \ x \ {\cos \left({x}\right)}}$

(4)

Type: Union(Expression(Integer),...)

Now differentiate it

axiom

D(%,x)

$\label{eq5}{{2 \ x \ {{\sin \left({x}\right)}^2}}+{2 \ x \ {{\cos \left({x}\right)}^2}}}\over{{\sin \left({x}\right)}^2}$

(5)

Type: Expression(Integer)

axiom

simplify(%)

$\label{eq6}-{{2 \ x}\over{{{\cos \left({x}\right)}^2}- 1}}$

(6)

Type: Expression(Integer)

Expressions (3) and (6) are really that same.

axiom

simplify(%%(3) - %%(6))

$\label{eq7}0$

(7)

Type: Expression(Integer)

axiom

D(x^x,x)

$\label{eq8}{{\log \left({x}\right)}\ {x^x}}+{x \ {x^{\left(x - 1 \right)}}}$

(8)

Type: Expression(Integer)

See, that was easy!

Here are some more things to try.

axiom

integrate(log(sin(x)),x=0..%pi)

$\label{eq9}potentialPole$

(9)

Type: Union(pole: potentialPole,...)

axiom

limit(sum(1/i^2,i=1..n),n=%plusInfinity)

$\label{eq10}\mbox{\tt "failed"}$

(10)

Type: Union("failed",...)

Can you explain these results?

Advanced Programming

For more details see: AxiomProgramming .

Spad

To include Spad programs on web pages in this website click edit (or when adding comments or in bug reports) use:

       \begin{spad}
       )abbrev ...
       ...
       \end{spad}

Aldor

To include Aldor programs on pages in this website use:

       \begin{aldor}
       ...
       \end{aldor}

or:

       \begin{aldor}[name]
       ...
       \end{aldor}

LaTeX

You can add LaTeX formatted equations to pages on this web site by:

       $$
       ...
       $$

and inline symbols are written like this: $\alpha +\beta$

To include arbitrary LaTeX on pages in this website use:

       \begin{latex}
       ...
       \end{latex}

Lisp

To include Lisp programs on pages in this website use:

       \begin{lisp}
       ...
       \end{lisp}

Boot

To include Boot programs on pages in this website use:

       \begin{boot}
       ...
       \end{boot}

Subject: Be Bold !!

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