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Editor: page
Time: 2007/09/11 18:48:06 GMT-7
Note:

changed:
-\begin{reduce}
-load_package sfgamma;
-load_package defint;
-int(1/(a+z^3), z,0,1);
-\end{reduce}
  !\begin{reduce}
  load_package sfgamma;
  load_package defint;
  int(1/(a+z^3), z,0,1);
  \end{reduce}

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[SandBox Aldor Foreign]?
Using Aldor to call external C routines
[SandBox Aldor Generator]?
Aldor defines a generator for type Vector
[SandBox Aldor Sieve]?
A prime number sieve in Aldor to count primes <= n.
[SandBox Aldor Testing]?
Using Aldor to write Axiom library routines
[SandBox Arrays]?
How fast is array access in Axiom?
[SandBox Axiom Syntax]?
Syntax of if then else

[SandboxBiblography]?

[SandBox Boolean]?
evaluating Boolean expressions and conditions
[SandBox Cast]?
Meaning and use of pretend vs. strong typing
[SandBox Categorical Relativity]?
Special relativity without the Lorentz group
[SandBox Category of Graphs]?
Graph theory in Axiom
[SandBox CL-WEB]?
Tangle operation for literate programming implemented in Common Lisp
[SandBox Combinat]?
A{ld,xi}o{r,m}Combinat
[SandBox Content MathML]?
Content vs. presentation MathML?
[SandBox Direct Product]?
A x B
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expression in sum-of-products form
[SandBox Domains and Types]?
What is the difference?
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Beginnings of an Emacs mode for Axiom based off of Jay's work and others
[SandBox Embeded PDF]?
pdf format documents can be displayed inline
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Algebra and Data Structure Hierarchy (lattice) diagrams
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experiments with DHTML, javascript, etc.
[SandBox Functional Addition]?
"adding" two functions
[SandBox Functions]?
How do they work?
[SandBox Functors]?
What are they? In Axiom functors are also called domain constructors.
[SandBox Gamma]?
Numerical evaluation of the incomplete Gamma function
[SandBox GuessingSequence]?
Guessing integer sequences
[SandBox Integration]?
Examples of integration in Axiom and Reduce
[SandBox Kernel]?
What is a "kernel"?

[SandBox kaveh]?

[SandBox LaTeX]?
LaTeX? commands allowed in MathAction?
[SandBox Lisp]?
Using Lisp in Axiom
[SandBox Manip]?
expression manipulations
[SandBox Manipulating Domains]?
testing the domain of an expression
[SandBox Mapping]?
A->B is a type in Axiom

[MathMLFormat]?

[SandBox Matrix]?
Examples of working with matrices in Axiom
[SandBox Maxima]?
Testing the Maxima interface
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Rings and things
[SandBox Monoid Extend]?
Martin Rubey's beautiful idea about using extend to add a category to a previously defined domain.
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XPOLY and friends
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Simpson method
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NonNegative? Integer without using SubDomain?
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Trigonometric expansion example
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an interface between Axiom and PolyMake?
[SandBox Polynomials]?
Axiom's polynomial domains are certainly rich and complex!
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Test page for educational purposes
[SandBox Qubic]?
Solving cubic polynomials
[SandBox Reduce And MathML]?
Reduce can use MathML? for both input and output
[SandBoxRelativeVelocity]?
Slides for IARD 2006: Addition of Relative Velocites is Associative
[SandBox RenameTitle]?
trying to re-create a crash due to renaming pages
[SandBox Sage]?
This is a test of Sage in MathAction?
[SandBox Shortcoming]?
Implementation of solve
[SandBox Solve]?
Solving equations
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calculating statistics in Axiom
[SandBox SubDomain]?
What is a SubDomain??
[SandBox Tail Recursion]?
When does Axiom replace recursion with iteration?
[SandBox Text Files]?
How to access text files in Axiom
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Tracing can affect output of 1::EXPR INT or 1::FRAC INT
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Basic structured data types in Axiom
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Scientific units and dimensions
[SandBox Spad]?
Domain construction
[SandBox Speed]?
Compilation speed

[SandBox Zero]?

[SandBox Axiom Strengths]?

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Experiments with matrices and various other stuff
SandBox2?
Experiments
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Experiments
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Experiments
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[SandBox7]?
[SandBox8]?
Here you can create your own SandBox?.
[SandBox9]?
Experiments with JET Bundles

[SandBox10]?

[SandBox DoOps]?
used to run Axiom without actually have to have it installed!

[SandBoxKMG]?

Click on the ? to create a new page. You should also edit this page to include a description and a new empty link for the next person.


Examples

Here is a simple Axiom command:

    \begin{axiom}
    integrate(1/(a+z^3), z=0..1,"noPole")
    \end{axiom}

axiom
integrate(1/(a+z^3), z=0..1,"noPole")
LatexWiki Image(1)
Type: Union(f1: OrderedCompletion? Expression Integer,...)

And here is a REDUCE command:

  \begin{reduce}
  load_package sfgamma;
  load_package defint;
  int(1/(a+z^3), z,0,1);
  \end{reduce}

  \begin{reduce}
  load_package sfgamma;
  load_package defint;
  int(1/(a+z^3), z,0,1);
  \end{reduce}
<hr />

Common Mistakes

Please review the list of [Common Mistakes]? and the list of [MathAction Problems]? if you are have never used MathAction? before. If you are learning to use Axiom and think that someone must have solved some particular problem before you, check this list of Common [Axiom Problems]?.

Works with ASCII text output formatting.
axiom
)set output tex off )set output algebra on

axiom
solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y]) (2) [ (- 2cy + 2ay)y - l2 + l1 [x= ------------------------, 2cx - 2ax 2 2 2 2 2 (4cy - 8ay cy + 4cx - 8ax cx + 4ay + 4ax )y + 2 2 (4cy - 4ay)l2 + (- 4cy + 4ay)l1 + (8ax cx - 8ax )cy - 8ay cx + 8ax ay cx * y + 2 2 2 2 l2 + (- 2l1 + 4ax cx - 4ax )l2 + l1 + (- 4cx + 4ax cx)l1 = 0 ] ]
Type: List List Equation Fraction Polynomial Integer

But fails with LaTeX?.

axiom
)set output tex on )set output algebra off

The result of 0**0 depends on the type of '0':

axiom
(0::Float)**(0::Float) >> Error detected within library code: 0**0 is undefined

The idea was, that defining LatexWiki Image as 1 is ok whenever there is no notion of limit. However,

axiom
(0::EXPR INT)**(0::EXPR INT)
LatexWiki Image(2)
Type: Expression Integer

is not quite in line with this, I think. There has been some discussion on this subject on axiom-developer.

It is easy to change this behaviour, if we know better...

Let's see if the same happens here:
axiom
sinCosProducts := rule sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2 cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2 sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2
LatexWiki Image(3)
Type: Ruleset(Integer,Integer,Expression Integer)

when typing --Bill Page, Mon, 30 Jan 2006 09:00:02 -0600 reply
When you are typing or when you cut-and-paste commands directly into the Axiom interpreter you must use an underscore character at the end of each incomplete line, and you must use the ( ) syntax instead of identation, like this:
  sinCosProducts := rule (_
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)

Alternatively, using a text editor you can enter the commands into a file called, for example sincos.input exactly as in MathActon? above and the use the command:

  )read sincos.input

axiom
)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP )library cannot find the file RINTERPA. )library cannot find the file RINTERP. )library cannot find the file PCDEN. )library cannot find the file GUESS. )library cannot find the file GUESSINT. )library cannot find the file GUESSP. guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)$GuessInteger GuessInteger is not a valid type.

conversion failed --Bill Page, Thu, 23 Mar 2006 22:21:41 -0600 reply
Unknown wrote:
z:=sum(myfn(x),x=1..10) -- This fails, why?

The reason this fails is because Axiom tries to evaluate myfn(x) first. But x is not yet an Integer so Axiom cannot compute myfn(x). I guess you were expecting Axiom to "wait" and not evaluate myfn(x) until after x has been assigned the value 1, right? But Axiom does not work this way.

The solution is to write myfn(x) so that is can be applied to something symbolic like x. For example something this:

axiom
myfn(i : Expression Integer) : Expression Integer == i Function declaration myfn : Expression Integer -> Expression Integer has been added to workspace.
Type: Void
axiom
myfn(x)
axiom
Compiling function myfn with type Expression Integer -> Expression 
      Integer
LatexWiki Image(4)
Type: Expression Integer
axiom
z:=sum(myfn(x),x=1..10)
LatexWiki Image(5)
Type: Expression Integer

Any hints for multivariate functions? --Bill (Name omitted), Fri, 24 Mar 2006 21:45:25 -0600 reply
Hi Bill:

Thanks for your quick response. I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.

I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function, and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions. Please consider the following example.

axiom
a(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p**(k) * (1.0-p)**(n-k) Function declaration a : (Expression Integer,Expression Integer, Expression Float) -> Expression Float has been added to workspace.
Type: Void
axiom
output(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
axiom
Compiling function a with type (Expression Integer,Expression 
      Integer,Expression Float) -> Expression Float 
   0.046875
Type: Void
axiom
z := sum(a(4,i,0.25), i=1..3) --- this fails There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sum with argument type(s) Expression Float SegmentBinding PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z) 55
Type: Void

I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do inference on parameter and result type.

axiom
b(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
Type: Void
axiom
output(b(4,3,0.25)) -- see that the function actually evaluates for sensible values
axiom
Compiling function b with type (PositiveInteger,PositiveInteger,
      Float) -> Float 
   0.046875
Type: Void
axiom
z := sum(b(4,i,0.25), i=1..3) --- this fails
axiom
Compiling function b with type (PositiveInteger,Variable i,Float)
       -> Expression Float 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named sum 
      with argument type(s) 
                              Expression Float
                       SegmentBinding PositiveInteger
      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
output(z)
   55
Type: Void

For univariate functions the approach

axiom
c(k) == binomial(4,k) * 0.25**k * (1.0 - 0.25)**(4-k) -- This approach is only a test, but is not suitable for my program
Type: Void
axiom
output(c(3)) -- test to see if function can be evaluated for sensible arguments
axiom
Compiling function c with type PositiveInteger -> Float 
   0.046875
Type: Void
axiom
z := sum(c(i), i=1..3) -- still doesn't work
axiom
Compiling function c with type Variable i -> Expression Float 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named sum 
      with argument type(s) 
                              Expression Float
                       SegmentBinding PositiveInteger
      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
output(z)
   55
Type: Void

But interestingly something like

axiom
d(k) == 1.5 * k -- coerce uotput to be a Float
Type: Void
axiom
z := sum(d(i), i=1..3) -- This works!
axiom
Compiling function d with type Variable i -> Polynomial Float
LatexWiki Image(6)
Type: Fraction Polynomial Float
axiom
output(z) 9.0
Type: Void

Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.

reduce(+,[...]?) = sum(...) --billpage, Sat, 25 Mar 2006 16:01:07 -0600 reply
Try this
axiom
z := reduce(+,[b(4,i,0.25) for i in 1..3])
LatexWiki Image(7)
Type: Float

Handling the result from functions returning a matrix --Bill (Name omitted), Mon, 27 Mar 2006 08:10:26 -0600 reply
Hi all:

Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).

I'm having a bit of difficulty getting a Function returning a matrix to work as expected, perhaps it is just cockpit error, but I don't see the error of my ways.

axiom
CFM(Q : Matrix(Float)): Matrix(Float) == x := nrows(Q) MyIdentityMatrix : Matrix(Float) := new(x, x, 0) for i in 1..nrows(MyIdentityMatrix) repeat MyIdnetityMatrix(i,i) := 1.0 Ninv := MyIdnetityMatrix - Q N := inverse(Ninv) N Function declaration CFM : Matrix Float -> Matrix Float has been added to workspace.
Type: Void
axiom
--test ComputeFundamentalMatrix X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
LatexWiki Image(8)
Type: Matrix Float
axiom
output(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
Type: Void
axiom
N := CFM(X) The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N) N
Type: Void

Any ideas where I'm blowing it here? I tried explicitly setting N to be a Matrix type but that failed too.

axiom
CFM(Q : Matrix(Float)): Matrix(Float) == x := nrows(Q) MyIdentityMatrix : Matrix(Float) := new(x, x, 0) for i in 1..nrows(MyIdentityMatrix) repeat MyIdnetityMatrix(i,i) := 1.0 Ninv := MyIdnetityMatrix - Q N := inverse(Ninv) N Function declaration CFM : Matrix Float -> Matrix Float has been added to workspace. Compiled code for CFM has been cleared. 1 old definition(s) deleted for function or rule CFM
Type: Void
axiom
--test ComputeFundamentalMatrix X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
LatexWiki Image(9)
Type: Matrix Float
axiom
output(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
Type: Void
axiom
N : Matrix(Float) := CFM(X) The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N) N is declared as being in Matrix Float but has not been given a value.

Thanks again for all your help.

Regards:

Bill M. (Sorry, my unique last name attracts too much spam).

typo and identity --billpage, Mon, 27 Mar 2006 09:34:35 -0600 reply
although I used a for loop and not reduce :-)

Good thinking. ;)

You have a simple typographical error. You have written both:

  MyIdentityMatrix

and :

  MyIdnetityMatrix

BTW, instead of the complicated construction of the identify matrix you should just write:

  Ninv := 1 - Q

For matrices 1 denotes the identity.

axiom
)set output tex off )set output algebra on FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1 4 3 2 (28) x - 6x + 11x + 2x + 1
Type: Polynomial Integer
axiom
radicalSolve(FunFun) (29) [ x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ]
Type: List Equation Expression Integer
axiom
)set output tex on )set output algebra off

Matthias

axiom
t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])
LatexWiki Image(10)
Type: Matrix Integer

We cat diagonalise t by finding it's eigenvalues.

axiom
)set output tex off )set output algebra on e:=radicalEigenvectors(t) (31) [ +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [radval= --------------------------------------------------, radmult= 1, +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3 radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] , [ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1, radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] , [ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1, radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] ]
Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer)
axiom
d:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval]) Function definition for d is being overwritten. Compiled code for d has been cleared. (32) +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [[--------------------------------------------------,0,0], +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3 [0, +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , 0] , [0, 0, +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 ] ]
Type: Matrix Expression Integer

Now prove it by constructing the simularity transformation from the eigenvectors:

axiom
p:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1) (33) [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1,1,1]]
Type: Matrix Expression Integer
axiom
p*d*inverse(p) +0 1 1 + | | (34) |1 - 2 2 | | | +1 2 - 1+
Type: Matrix Expression Integer
axiom
)set output tex on )set output algebra off

\end{axiom}

Axiom can't integrame exp(x^4) ;( --unknown, Fri, 28 Apr 2006 14:03:28 -0500 reply
Axiom can't integrame exp(x^4) ;(

axiom
integrate(exp(x**4),x)
LatexWiki Image(11)
Type: Union(Expression Integer,...)

But Maple can...

axiom
f(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4))
Type: Void
axiom
D(f(x),x)
axiom
Compiling function f with type Variable x -> Expression DoubleFloat
LatexWiki Image(12)
Type: Expression DoubleFloat?

Axiom cannot integrate e^(4*x) --kratt6, Fri, 28 Apr 2006 16:34:16 -0500 reply
This is not a big surprise: note that Gamma(x,y) is not an elementary function.

Martin

This is both obviously wrong since the integrand is a positive function:
axiom
integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)
LatexWiki Image(13)
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiom
numeric(integrate(1/(1+x^4),x=0..1))
LatexWiki Image(14)
Type: Float

axiom
)clear co All user variables and function definitions have been cleared. All )browse facility databases have been cleared. Internally cached functions and constructors have been cleared. )clear completely is finished. n := 32
LatexWiki Image(15)
Type: PositiveInteger?
axiom
y : FARRAY INT := new(n,1)
LatexWiki Image(16)
Type: FlexibleArray? Integer
axiom
n0 := n
LatexWiki Image(17)
Type: PositiveInteger?
axiom
n1 := sum(x^1, x=0..n-1)
LatexWiki Image(18)
Type: Fraction Polynomial Integer
axiom
n2 := sum(x^2, x=0..n-1)
LatexWiki Image(19)
Type: Fraction Polynomial Integer
axiom
n3 := sum(x^3, x=0..n-1)
LatexWiki Image(20)
Type: Fraction Polynomial Integer
axiom
n4 := sum(x^4, x=0..n-1)
LatexWiki Image(21)
Type: Fraction Polynomial Integer
axiom
A := matrix([[n4, n3, n2],_ [n3, n2, n1],_ [n2, n1, n0]])
LatexWiki Image(22)
Type: Matrix Fraction Polynomial Integer
axiom
X := vector([x1, x2, x3])
LatexWiki Image(23)
Type: Vector OrderedVariableList? [x1,x2,x3]?
axiom
B := vector([sum(x^2* u, x=0..n-1),_ sum(x* v, x=0..n-1),_ sum( w, x=0..n-1)])
LatexWiki Image(24)
Type: Vector Fraction Polynomial Integer
axiom
solve([A * X = B], [x1, x2, x3]) There are 20 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) List Equation Vector Fraction Polynomial Integer List OrderedVariableList [x1,x2,x3] Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

can this be correct? --unknown, Tue, 30 May 2006 23:51:26 -0500 reply
axiom
integrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")
LatexWiki Image(25)
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiom
subst(%,x=1)
LatexWiki Image(26)
Type: Expression Integer
axiom
integrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")
LatexWiki Image(27)
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiom
simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))
LatexWiki Image(28)
Type: Expression Integer
axiom
%::Expression Float
LatexWiki Image(29)
Type: Expression Float

axiom
a := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]])
LatexWiki Image(30)
Type: Matrix Integer
axiom
determinant(a)
LatexWiki Image(31)
Type: NonNegativeInteger?
axiom
inverse(a)
LatexWiki Image(32)
Type: Union("failed",...)

a := matrix([ [-3,1,1,1]?, [1,1,1,1]?, [1,1,1,1]?, [1,1,1,1]]?)

axiom
As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
LatexWiki Image(33)
Type: Matrix Integer
axiom
A := subMatrix(As, 2,4,2,4)
LatexWiki Image(34)
Type: Matrix Integer
axiom
ob := orthonormalBasis(A)
LatexWiki Image(35)
Type: List Matrix Expression Integer
axiom
P : Matrix(Expression Integer) := new(3,3,0)
LatexWiki Image(36)
Type: Matrix Expression Integer
axiom
setsubMatrix!(P,1,1,ob.3)
LatexWiki Image(37)
Type: Matrix Expression Integer
axiom
setsubMatrix!(P,1,2,ob.1)
LatexWiki Image(38)
Type: Matrix Expression Integer
axiom
setsubMatrix!(P,1,3,ob.2)
LatexWiki Image(39)
Type: Matrix Expression Integer
axiom
Pt := transpose(P)
LatexWiki Image(40)
Type: Matrix Expression Integer
axiom
Ps : Matrix(Expression Integer) := new(4,4,0)
LatexWiki Image(41)
Type: Matrix Expression Integer
axiom
Ps(1,1) := 1
LatexWiki Image(42)
Type: Expression Integer
axiom
setsubMatrix!(Ps,2,2,P)
LatexWiki Image(43)
Type: Matrix Expression Integer
axiom
PsT := transpose(Ps)
LatexWiki Image(44)
Type: Matrix Expression Integer
axiom
PsTAsPs := PsT * As * Ps
LatexWiki Image(45)
Type: Matrix Expression Integer
axiom
b1 := PsTAsPs(2,1)
LatexWiki Image(46)
Type: Expression Integer
axiom
l1 := PsTAsPs(2,2)
LatexWiki Image(47)
Type: Expression Integer
axiom
Us : Matrix(Expression Integer) := new(4,4,0)
LatexWiki Image(48)
Type: Matrix Expression Integer
axiom
Us(1,1) := 1
LatexWiki Image(49)
Type: Expression Integer
axiom
Us(2,2) := 1
LatexWiki Image(50)
Type: Expression Integer
axiom
Us(3,3) := 1
LatexWiki Image(51)
Type: Expression Integer
axiom
Us(4,4) := 1
LatexWiki Image(52)
Type: Expression Integer
axiom
Us(2,1) := -b1 / l1
LatexWiki Image(53)
Type: Expression Integer
axiom
PsUs := Ps * Us
LatexWiki Image(54)
Type: Matrix Expression Integer
axiom
PsUsT := transpose(PsUs)
LatexWiki Image(55)
Type: Matrix Expression Integer
axiom
PsUsTAsPsUs := PsUsT * As * PsUs
LatexWiki Image(56)
Type: Matrix Expression Integer
axiom
C := inverse(PsUs)
LatexWiki Image(57)
Type: Union(Matrix Expression Integer,...)
axiom
c := PsUsTAsPsUs(1,1)
LatexWiki Image(58)
Type: Expression Integer
axiom
gQ := PsUsTAsPsUs / c
LatexWiki Image(59)
Type: Matrix Expression Integer
axiom
x1 := transpose(matrix([[1,2,3,4]]))
LatexWiki Image(60)
Type: Matrix Integer
axiom
v1 := transpose(x1) * As * x1
LatexWiki Image(61)
Type: Matrix Integer
axiom
x2 := C * x1
LatexWiki Image(62)
Type: Matrix Expression Integer
axiom
v2 := transpose(x2) * PsUsTAsPsUs * x2
LatexWiki Image(63)
Type: Matrix Expression Integer

axiom
draw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1]) There are 20 exposed and 18 unexposed library operations named ** having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ** to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named ** with argument type(s) FlexibleArray Integer PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiom
f1 := taylor(1 - x**2,x = 0)
LatexWiki Image(64)
Type: UnivariateTaylorSeries?(Expression Integer,x,0)
axiom
asin f1
LatexWiki Image(65)
Type: UnivariateTaylorSeries?(Expression Integer,x,0)
axiom
sin %
LatexWiki Image(66)
Type: UnivariateTaylorSeries?(Expression Integer,x,0)

SandboxMSkuce?

axiom
1+1
LatexWiki Image(67)
Type: PositiveInteger?

SandBoxCS224?

axiom
integrate((x-1)/log(x), x)
LatexWiki Image(68)
Type: Union(Expression Integer,...)
axiom
integrate(x*exp(x)*sin(x),x)
LatexWiki Image(69)
Type: Union(Expression Integer,...)

Working With Lists --daneshpajouh, Sat, 16 Jun 2007 07:00:00 -0500 reply
axiom
[p for p in primes(2,1000)|(p rem 16)=1]
LatexWiki Image(70)
Type: List Integer
axiom
[p**2+1 for p in primes(2,100)]
LatexWiki Image(71)
Type: List Integer

axiom
integrate (2x^2 + 2x, x) Cannot find a definition or applicable library operation named 2 with argument type(s) Variable x Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

\end {axiom}

axiom
radix(36,37) >> Error detected within library code: index out of range

Is it error?

example from my daughter's college calc --pbwagner, Mon, 10 Sep 2007 13:00:06 -0500 reply
integrate(log(log(x)),x)

(better) example (with axiom markers this time) ;-) --pbwagner, Mon, 10 Sep 2007 13:01:48 -0500 reply
axiom
integrate(log(log(x)),x)
LatexWiki Image(72)
Type: Union(Expression Integer,...)