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Step function

  1. Apr 29, 2005 #1
    I would like to invite comment to the near step function below.

    [tex]
    \Mvariable{step(x)}=\frac{{e^{-\frac{\gamma }{{x^2}}}}}{{\sqrt{1-\frac{{e^{-\frac{\gamma }{{x^2}}}}}{{x^2}}}}}
    [/tex]


    the above function evaluates to nearly 1 for |x|>1 and nearly zero for |x|<0.3

    for gamma = 0.4823241136337762 I have attached the plots of step and 1-step

    here are some spot values for
    x and step(x)
    0.05 1.15E-83
    0.1 1.84E-21
    0.2 6.55E-6
    0.3 5.11E-3
    0.4 6.11E-2
    0.5 2.32E-1
    0.6 0.518
    0.8 0.932
    1.0 1.0068
    2.0 1.006
    5.0 1.00092
    10 1.00022
    50 1.0000090
     

    Attached Files:

    Last edited: Apr 29, 2005
  2. jcsd
  3. Apr 29, 2005 #2
    Indeed. If you change all the [itex]x^2[/itex]s to [itex]x^4[/itex]s (or [itex]x^{1000}[/itex]s), then it'll get even closer to a step function.
     
  4. Apr 30, 2005 #3
    I just noticed that the step function can be simplified even more
    [tex]
    \Mvariable{step}(x)=
    \frac{{e^{-\frac{\gamma }{{x^2}}}}}{{\sqrt{1-\frac{{e^{-\frac{\gamma }{{x^2}}}}}{{x^2}}}}}

    =\frac{{{\alpha}^{\frac{1}{{x^2}}}}}{{\sqrt{1-\frac{{{\alpha}^{\frac{1}{{x^2}}}}}{{x^2}}}}}\\
    [/tex]

    for above gamma= 0.4823241136337762 , alpha =0.61734693877551, hence

    [tex]
    \Mvariable{step}(x)=
    \frac{{{0.61734693877551}^{\frac{1}{{x^2}}}}}{{\sqrt{1-\frac{{{0.61734693877551}^{\frac{1}{{x^2}}}}}{{x^2}}}}}\\
    [/tex]
     
  5. Apr 30, 2005 #4
    Well, it's not really "simpler." You have the same number of arbitrary constants, though I guess you have a couple fewer symbols in general (you got rid of two minus signs - but you could do that just by specifying that [itex]\gamma[/itex] must be negative). I'd usually just leave it in the exponential form, but that's a subjective choice based on the fact that I like the letter e :wink:
     
  6. Apr 30, 2005 #5
     
    Last edited: Apr 30, 2005
  7. Apr 30, 2005 #6
    arbitrary within a certain domain is what I should have said, of course. In this case, you need [itex]\gamma > 1/e[/itex].
     
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