Solving Indefinite Integral of Normal Equation

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SUMMARY

The discussion focuses on solving the indefinite integral of the normal equation, specifically the integral \(\int e^{-\frac{(x-\mu)^2}{(2\sigma)^2}} dx\), where \(\mu\) and \(\sigma\) are constants. The integral does not have an elementary form and can be expressed in terms of the error function, erf(x). By substituting \(y = \frac{x-\mu}{2\sigma}\), the integral simplifies to \(e^{-y^2}\), which lacks a primitive. The discussion suggests using a Taylor series expansion for \(e^{-y^2}\) to integrate term by term, leading to the series with general term \(a_n = \frac{(-1)^n x^{2n+1}}{(2n + 1)n!}\).

PREREQUISITES
  • Understanding of calculus, specifically integration techniques.
  • Familiarity with the error function, erf(x).
  • Knowledge of Taylor series and their applications in integration.
  • Basic programming skills for implementing numerical integration.
NEXT STEPS
  • Study the properties and applications of the error function, erf(x).
  • Learn about Taylor series and their convergence criteria.
  • Explore numerical integration techniques for approximating integrals without elementary forms.
  • Implement a program using Python's SciPy library to compute the integral of the normal equation.
USEFUL FOR

Mathematicians, statisticians, data scientists, and anyone involved in calculus or numerical analysis, particularly those working with normal distributions and error functions.

Erienion
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I've been trying to integrate the following function but have gotten somewhat stuck doing it. The answer i managed to produce gave some bogan answers.

the integral in question is

[tex]\int e^\frac{-(x-\mu)^2}{(2\sigma)^2}[/tex]

where [tex]\mu[/tex] and [tex]\sigma[/tex] are constants.

its part of the normal equation and I've been trying to write a program to do some calculations with it.
 
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The integral can be expressed in terms of the error function, erf(x). Unfortunately, there is no elementary form.
 
As far as I can see, by setting [itex]y = x-\mu /2\sigma[/itex], we get the famous [itex]e^{-y^2}[/itex] which doesn't have a primitive. You can however develop [itex]e^{-y^2}[/itex] as a Taylor serie and integrate term by term. You get the (convergant) serie of general term

[tex]a_n=\frac{(-1)^n x^{2n+1}}{(2n + 1)n!}[/tex]
 

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