Discussing the Near Step Function for γ=0.4823241136337762

In summary: 5 to get the step function to approximate a real number, and that's only for values of x within a certain range.
  • #1
AntonVrba
92
0
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
 

Attachments

  • step.gif
    step.gif
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Last edited:
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  • #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.
 
  • #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]
 
  • #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:
 
  • #5
Data said:
Well, it's not really "simpler." You have the same number of arbitrary constants, QUOTE]


:eek: arbitrary :eek:
just play with the function and you will see that gamma or alpha has a rather very limited range for the function not to become complex for any real number:biggrin:

By proper choice of the Alpha you can make the function overshoot (become slightly larger than one) as in the example given, or the function always remaining slightly less than one i.e approach one at infinity, and that in a only very limitted range of values.
 
Last edited:
  • #6
arbitrary within a certain domain is what I should have said, of course. In this case, you need [itex]\gamma > 1/e[/itex].
 

Related to Discussing the Near Step Function for γ=0.4823241136337762

1. What is the Near Step Function for γ=0.4823241136337762?

The Near Step Function for γ=0.4823241136337762 is a mathematical function that describes a step-like behavior in a plot or graph. It is often used to represent a sudden change or transition in a system or process.

2. How is the Near Step Function calculated?

The Near Step Function is calculated by taking the limit of the Heaviside step function as γ approaches the desired value. This can be written as lim γ→0 Heaviside(x - γ).

3. What are some real-world applications of the Near Step Function?

The Near Step Function has various applications in physics, engineering, and economics. It can be used to model phase transitions, market crashes, and sudden changes in physical systems such as temperature or pressure.

4. Can the Near Step Function be generalized for different values of γ?

Yes, the Near Step Function can be generalized for any value of γ. The value of γ determines the location and sharpness of the step in the function. As γ approaches 0, the step becomes sharper and more defined.

5. How does the Near Step Function differ from the Heaviside step function?

The Near Step Function is a modified version of the Heaviside step function, where the step occurs at a specific value of γ instead of at x=0. This allows for more control and flexibility in modeling sudden changes in a system. Additionally, the Near Step Function approaches 0 at γ=0, while the Heaviside step function jumps from 0 to 1 at x=0.

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