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Re: Integration with non-numeric parameters

  • To: mathgroup at
  • Subject: [mg79767] Re: Integration with non-numeric parameters
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at>
  • Date: Sat, 4 Aug 2007 05:47:33 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f8v1cb$ded$>

ingramfinance at wrote:


> But when I try
>  q=Exp[-(x1-t)^2/2*sigma^2*t]
> Integrate[q, {t, .5,1}]
> Now Mathematica does not solve this integral, it just repeats the
> command
> I am trying to get an expression in terms of x1. Why do I get a
> statement like this instead of an answer? 


It is conventional (by design) that Mathematica returns an expression 
unevaluated when Mathematica does not know how to evaluate this 
expression. This can happen for user-defined functions as well as 
built-in functions (though in special circumstances).

For instance, having started a new Mathematica session, if we try to 
evaluate f[2], Mathematica just returns f[2] since it has not  the 
slightest idea of what the function f can possibly do.

In[1]:= f[2]

Out[1]= f[2]

Now, we give a definition (a meaning) to the symbol f.

In[2]:= f[x_] = 2 x

Out[2]= 2 x

 From now on, evaluating f will return a value.

In[3]:= f[2]

Out[3]= 4

Of course, *Integrate* is a built-in function that has already a 
meaning. Still, if Mathematica does not know how to find a definite or 
indefinite integral, it returns the original expression as answer.

For instance, Mathematica knows how to integrate E^(-x^2) (in terms of 
error function) and E^(-x^3) (in terms of gamma function) but not 
E^(-x^3 - x^2) (the expression is returned unevaluated).

In[1]:= Integrate[Exp[-x^2], x]

Out[1]= 1/2 Sqrt[\[Pi]] Erf[x]

In[2]:= Integrate[Exp[-x^3], x]

Out[2]= -((x Gamma[1/3, x^3])/(3 (x^3)^(1/3)))

In[3]:= Integrate[Exp[-x^3 - x^2], x]

Out[3]= Integrate[E^(-x^2 - x^3), x]


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