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Perturbation theory and asymptotics

  1. Oct 28, 2010 #1
    I need to find the roots of the transcendental function,

    f(x;a)=x^2-3ax-1-a+exp(-x/a)=0;

    I've done many problems like this before and am fairly sure this is just a regular perturbation problem. The difficulty I'm having is with the exponential term.

    Could anyone give me an idea of how to tackle this problem?

    Thanks.
     
  2. jcsd
  3. Oct 29, 2010 #2

    HallsofIvy

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    Approximate the exponential with a Taylor's polynomial:
    [tex]exp(-x/a)= 1- x/a+ \frac{x^2}{2a^2}+ \cdot\cdot\cdot+ (-1)^n\frac{x^n}{n!a^n}[/tex]
     
  4. Oct 29, 2010 #3
    Sorry I didn't make it clear in my first post, I'm finding the roots of the equation as a tends to 0. I thought about the Taylor expansion but then the terms are being divided by 0 (as a tends to zero). Is there anyway around this?

    Thanks.
     
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