Edit detail for Definite Integration revision 4 of 8

Here are some integration problems.

Test1

Start here:

fricas

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

(1)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Test2

A simpler integration:

fricas

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

(2)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

We are using the following version of FriCAS?:

fricas

)version

Value = "FriCAS 1.3.0 compiled at Thu Sep  1 17:01:37 UTC 2016"

... --HPW, Sat, 22 Jan 2005 06:14:31 -0600 reply

Another definite integral

fricas

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

(3)

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

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

fricas

n:PositiveInteger

Type: Void

fricas

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

   n is declared as being in PositiveInteger but has not been given a 
      value.

declaring n as PositiveInteger? --Bill Page, Mon, 24 Jan 2005 14:11:55 -0600 reply

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

fricas

n:PositiveInteger

Type: Void

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

fricas

n:PositiveInteger

Type: Void

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

It is obvious that , while FriCAS? signals error.

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

   There are 1 exposed and 1 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) 
                                 Variable(t)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 1 exposed and 1 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) 
                                 Variable(t)
                             NonNegativeInteger

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

Currently conditional expressions are not supported. Axiom used to evaluate such examples using its internal order on expressions, which gave unexpected results. From users point of view the results were wrong.