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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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