Here are some integration problems submitted by "Anonymous". In at least one example, Axiom generated a segmentation fault. We will see if we can reproduce the problem here in a more controlled environment.

Test1

Start here:

integrate(x/sqrt(2*%pi)*exp(-1/2*log(x)**2),x=0..%plusInfinity)

(1)

Test2

Might not work:

integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)**2),x=0..%plusInfinity, noPole)

   There are 4 exposed and 1 unexposed library operations named
      integrate having 3 argument(s) but none was determined to be
      applicable. Use HyperDoc Browse, or issue
                            )display op integrate
      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
      integrate with argument type(s)
                             Expression Integer
                  SegmentBinding OrderedCompletion Integer
                               Variable noPole

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

Test3

A simpler integration:

integrate(1/sqrt(2*%pi)*exp(-1/2*x**2),x=%minusInfinity..%plusInfinity)

(2)

We are using the following version of Axiom:

)version

Value = "Friday November 9, 2007 at 19:35:06 "

Another definite integral

integrate(x^n*exp(-x^2/2), x=0..%plusInfinity,"noPole")

(3)

What if the positive Integer is declare as a positiveInteger or even chosen, say, as 2:

n:PositiveInteger

integrate(x^n*exp(-x^2/2), x=0..%plusInfinity)

   n is declared as being in PositiveInteger but has not been given a
      value.
integrate(x^2*exp(-x^2/2), x=0..%plusInfinity)

(4)

I think the idea that one should be able to "declare the type" of a variable in Axiom by the command

n:PositiveInteger

is a frequent expectation of new users of Axiom - especially if one have used other computer algebra systems, after all Axiom is supposed to be a "strongly typed" system, right? Certainly I was surprized (and disappointed) by Axiom's limitations in this reguard.

Unfortunately Axiom does not attempt to use this type information when forming expressions - but worse - declaring the type actually interferes with the use of the variable to form expressions!

When you write

n:PositiveInteger

what this tells Axiom is that n will be assigned an integer value greater than 0 - only that. After it is actually assigned some value, then it can be used exactly like that value, but not before.

To me, this is a tremedous waste of an opportunity in Axiom to to deal with "domain of computation" issues such as are addressed in other untyped computer algebra systems by the use of "assumptions" such as:

  assume(x,PositiveInteger);

Such knowledge can be used to considerably improve the quality and generality of the computations.

Another problem with integrate

Sat, 12 Mar 2005 08:25:13 -0600 reply

f(x | (x >=0) and (x <=1) ) == 1

f(x | (x<0) or (x > 1)) == 0

integrate(f(t), t=-1..2)

Compiling function f with type Variable t -> NonNegativeInteger