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MathGroup Archive 2007

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Re: Integration with non-numeric parameters

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79767] Re: Integration with non-numeric parameters
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sat, 4 Aug 2007 05:47:33 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f8v1cb$ded$1@smc.vnet.net>

ingramfinance at gmail.com wrote:

(*snip*)

> 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? 

(*snip*)

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]

Regards,
Jean-Marc


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