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The Integral of e^(x^2)dx

Date: 03/26/98 at 10:37:06
From: Manuel Ruiz Martinez
Subject: How should I solve the integral e^(x^2)?

First, thank you for answering my letter about Fermat's Last Theorem. 
I understood some of the concepts that you wrote about there. I am 
thinking about studying Maths or Physics in the Universidad 
Complutense in Madrid, or in Valencia.

And the question, the integral of e^(x^2). I have a program named 
Derive. With this program the solution is very strange. I don't know 
if you will be able to help me, but I think so. 


Date: 03/27/98 at 08:03:54
From: Doctor Jerry
Subject: Re: How should I solve the integral e^(x^2)?

Hi Manuel,

There are many functions - called special functions - which fail to 
have an anti-derivative expressible as a finite combination of 
elementary functions. The so-called elliptic functions, the error 
function, and the gamma function are a few examples. The error 
function, which is extremely useful in both physics and statistics, is 
defined as:

   erf(x) = (2/sqrt(pi))integral from 0 to x of e^(-t^2)dt

This is closely related to the problem you mentioned. Neither can be 
done in finite terms.

There would not exist extensive tables of the error function if the 
anti-derivative of e^(-t^2) were expressible as a finite combination 
of elementary functions.

The Annals can be found in the library of any good university.

-Doctor Jerry,  The Math Forum   
Associated Topics:
High School Calculus

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