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

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The conversation discusses the possibility of using the integrate function to solve e^(x^2). The responder explains that while e^(x^2) does have an anti-derivative, it is not an elementary function. They also mention the option of defining a new function to solve it, specifically the error function, and provide a link for more information on it. In summary, the conversation concludes that while e^(x^2) does have an anti-derivative, it is not an elementary function and the error function can potentially be used to solve it.
  • #1
newton1
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does the integrate e^(x^2) can solve??
i think is no...
but why??
 
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  • #2
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).
 
  • #3
eh...

may i ask what is Error Function??
 

What is e^(x^2)?

e^(x^2) is a mathematical expression that represents the exponential function where the base is the mathematical constant e (approximately equal to 2.71828) and the exponent is x^2, or x multiplied by itself.

Can e^(x^2) be integrated?

Yes, e^(x^2) can be integrated. It is a continuous function and can be integrated using various methods such as substitution, integration by parts, or using special functions like the error function.

What is the general form of the integral of e^(x^2)?

The integral of e^(x^2) has no closed-form solution and cannot be expressed in terms of elementary functions. However, it can be written in terms of the error function as ∫e^(x^2)dx = (√π/2)erfi(x) + C, where erfi(x) is the imaginary error function.

Why is the integration of e^(x^2) important?

The integration of e^(x^2) is important in both mathematics and science. It is used in various fields such as statistics, physics, and engineering to solve problems involving normal distributions, heat transfer, and wave propagation, among others.

Are there any special techniques for integrating e^(x^2)?

Yes, there are special techniques like the Gaussian quadrature method and the Laplace method that can be used to approximate the integral of e^(x^2) with high accuracy. These techniques are particularly useful when the limits of integration are infinite or when the function is highly oscillatory.

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