Can erf(x) be used to solve e^(x^2)?

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Discussion Overview

The discussion revolves around the integration of the function e^(x^2) and whether it can be expressed in terms of elementary functions or if it relates to the error function, erf(x). The scope includes theoretical considerations about anti-derivatives and the nature of continuous functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the integral of e^(x^2) can be solved, expressing doubt about its solvability.
  • Another participant clarifies that while e^(x^2) has an integral as a continuous function, it is not necessarily an elementary function.
  • This participant notes that many functions do not have elementary anti-derivatives and introduces the concept of defining new functions for integration purposes.
  • The error function, erf(x), is mentioned as an anti-derivative of e^(-x^2), suggesting a potential connection to the discussion.
  • A participant requests clarification on what the error function is, indicating a need for further explanation.
  • A suggestion to look up the error function on Google is provided, directing to an external resource for more information.

Areas of Agreement / Disagreement

Participants express differing views on the solvability of the integral of e^(x^2). There is no consensus on whether it can be expressed in terms of elementary functions, and the discussion remains unresolved regarding the relationship to the error function.

Contextual Notes

The discussion does not resolve the assumptions about the definitions of elementary functions or the specific properties of the error function in relation to e^(x^2).

newton1
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does the integrate e^(x^2) can solve??
i think is no...
but why??
 
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That depends upon exactly what you mean.

Since e^(x^2) is a continuous function, yes, it HAS an integral (anti-derivative). Every continuous function (and many non-continuous functions) is the derivative of some function and therefore has an anti-derivative.

Is that anti-derivative any "elementary function" (defined as polynomials, rational functions, exponentials, logarithms, trig functions and combinations of them)? No, if fact for most functions the anti-derivative is not an elementary function. (There are more functions in heaven and Earth than are dreamed of in your philosophy, Horatio!)

Of course one can always DEFINE a new function to do the job. I don't know specifically about e^(x^2) but the ERROR FUNCTION, Erf(x) is defined as an anti-derivative of e^(-x^2).
 
eh...

may i ask what is Error Function??
 

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