The integration of e^(x^2)

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Discussion Overview

The discussion revolves around the integral of the function e^(x^2), exploring various methods of integration, the nature of its anti-derivative, and the implications of existing mathematical theories. Participants engage in technical reasoning, propose different approaches, and clarify misconceptions regarding the integral's properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about their method of integrating e^(x^2) using the derivative of the exponent.
  • Another participant asserts that e^(x^2) does not have a nice anti-derivative that can be expressed in elementary functions.
  • A participant mentions that the anti-derivative of e^(-x^2) is known as the error function (erf), while noting that no similar name exists for e^(x^2).
  • Some participants suggest using polar coordinates or series expansions to approach the integral.
  • There is a proposal to use bilateral Laplace transforms to potentially express the integral in a different domain, though doubts are raised about the feasibility of this method.
  • Concerns are raised about the applicability of series expansions for large values of x and the limitations of numerical methods in approximating the integral.
  • One participant discusses the implications of Differential Galois Theory on the inexpressibility of the integral in terms of elementary functions.
  • Multiple participants challenge each other's reasoning regarding the integration techniques and the nature of the functions involved.

Areas of Agreement / Disagreement

Participants generally agree that the integral of e^(x^2) cannot be expressed in terms of elementary functions. However, there are competing views on the methods that could be used to approximate or analyze the integral, and the discussion remains unresolved regarding the best approach.

Contextual Notes

Some methods proposed, such as series expansions and Laplace transforms, depend on specific conditions and may not yield satisfactory results for all values of x. The discussion also highlights the limitations of numerical methods in providing accurate approximations for the integral.

  • #31
HallsofIvy said:
I don't know how many times we have to say this- it's given in any good calculus book. The anti-derivative of e^{x^2} is not an "elementary" function.
My best guess, this number is aleph-naught: infinite, but countable.

I have assumed an infinite lifetime for the forum. :wink:
 
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  • #32
Redbelly98 said:
I have assumed an infinite lifetime for the forum. :wink:

lol :biggrin:

a nice point of view :wink:
 
  • #33
thepatient said:
No.


If u = x^2. du = 2xdx, not 2dx.

You're right. What a stupid error I made.

u = x^2, du = 2x
multiply du * e^u = 2x e^x^2.
 
  • #34
LTA85 said:
You're right. What a stupid error I made.

u = x^2, du = 2x
multiply du * e^u = 2x e^x^2.
\intex2dx If:

u = x2
du = 2xdx

You can't substitute dx by du, since dx = du/(2x), not just du. You would have dx in terms of two variables, you can't integrate x in terms of u. The anti-derivative of apples can't be oranges.In no nice way is this function integrable.
 

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