Finding the inverse of a nasty 1-to-1 function.

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Discussion Overview

The discussion revolves around finding the inverse of a specific integral function involving the exponential function, particularly focusing on numerical methods for evaluation. The context includes both theoretical and practical approaches to solving the problem, with participants exploring various mathematical techniques.

Discussion Character

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

Main Points Raised

  • One participant presents the integral equation and seeks a numerical method to find x given y and c, noting the complexity due to the limits of integration.
  • Another participant suggests using a Taylor expansion of exp(-t²) for values of x smaller than 1, leading to a specific approximation.
  • A different participant proposes a Fourier inversion expansion as a rigorous method, although they express uncertainty about the details.
  • One participant points out that the function does not have a proper inverse unless x is restricted to greater than zero, highlighting a potential limitation.
  • Another participant suggests viewing the problem as a differential equation and mentions numerical methods like the midpoint method or Runge-Kutta, but expresses uncertainty about the outcome.
  • One participant recommends using the Newton-Raphson method for numerical evaluation, which is acknowledged positively by another participant.
  • There is a discussion about the necessity of evaluating the integral, which cannot be done analytically, but tables of data for the error function are mentioned as a resource.

Areas of Agreement / Disagreement

Participants express a variety of methods and approaches, with no consensus on a single solution or method. Some participants agree on the utility of numerical methods, while others raise concerns about the function's properties and limitations.

Contextual Notes

There are limitations regarding the assumptions made about the values of x, particularly the restriction to values less than 1 and greater than zero for the existence of an inverse. The discussion also reflects uncertainty about the effectiveness of proposed methods.

Who May Find This Useful

Individuals interested in numerical methods for solving integrals, particularly in the context of inverse functions, as well as those exploring advanced mathematical techniques such as Taylor expansions and Fourier analysis.

jforeman83
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Hey, everybody.

I have a function:
[tex] \int\limits_{x}^{x+c} exp(-t^2) dt = y[/tex]

c is a known constant here.

I am beating my head against the wall trying to find a good way to numerically evaluate the inverse here, i.e. I have y and c and I want to know x. I know that erf^-1 is readily available in mathematica and maple and the like but the limits of integration here make this a bit nastier. Any ideas? I don't need a perfect evaluation, just a moderately good approximation will work.
 
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If you're interested only in values of x smaller than 1, then you can always Taylor expand exp(-t²) and integrate term by term and keep only the first 2 terms. I get y=c-xc+c²/2.
 
i think the rigorous method would be to use a Fourier inversion expansion; can't remember the details of how to do that - but with a smooth gaussian function i think its not that bad.
 
quasar987 said:
If you're interested only in values of x smaller than 1, then you can always Taylor expand exp(-t²) and integrate term by term and keep only the first 2 terms. I get y=c-xc+c²/2.

I wish I could say with certainty that this was the case, because you're right, that would be a good idea. Unfortunetly, I think I'll need something that covers a bit more values.
 
lzkelley said:
i think the rigorous method would be to use a Fourier inversion expansion; can't remember the details of how to do that - but with a smooth gaussian function i think its not that bad.

Do you have any recommendations on sources where I might read up on this?
 
This function does not have a proper inverse as y(x)=y(-x-c)

I believe that the inverse may exist if you restrict x to be greater than zero.
 
I think we can view this problem as a differential equation, and use some method of numerically solving ODE's such as the midpoint method or Runge-Kutta? Not 100% sure how it'll work out though.
 
If you only want to evaluate it numerically, then why don't you just use Newton-Rhapson to determine when y(x) = K ?
 
That involves evaluating the integral, which can't be done analytically though I do know there are many tables of data for that particular function (The error function). So Yes, I guess that method can do it numerically, good idea =]
 
  • #10
daudaudaudau said:
If you only want to evaluate it numerically, then why don't you just use Newton-Rhapson to determine when y(x) = K ?

This is a fantastic idea. Thanks!
 

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