Exploring Convergence of e^(-γ|x|) Integrals

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In summary, the conversation discusses a proof involving the inner product of continuous eigenstates and the use of e^{-\gamma |x|} to force the surface terms to behave at infinity. The question posed is whether \lim_{\gamma\rightarrow 0}\int_{-\infty}^{\infty}e^{-\gamma |x|}f(x)dx=\int_{-\infty}^{\infty}f(x)dx for all f(x). The speaker has shown it to be true for specific cases, but is unsure how to prove it for the general case or when the limits are from -\infty to \infty. They also question whether the integral should be evaluated before taking the limit,
  • #1
keniwas
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I am trying to understand a proof about the inner product of continuous eigenstates. Since there is no guarantee that the functions are square integrable they multiply the inner product by [tex]e^{-\gamma |x|}[/tex] then take the limit as [tex]\gamma\rightarrow 0[/tex] to force the surface terms to behave at infinity.

So my question is the following: is it true that
[tex]\lim_{\gamma\rightarrow 0}\int_{-\infty}^{\infty}e^{-\gamma |x|}f(x)dx=\int_{-\infty}^{\infty}f(x)dx[/tex]

for all f(x)?

I have shown it is true for multiple cases where f(x) = xn or f(x)=sin(x) ... etc... and the limits are finite (i.e. on some interval a<x<b) but I cannot figure out how to do it for a general case, nor for the case when the limits are from [tex]-\infty<x<\infty[/tex]

The problem I run into in the general case is I am not sure what to do with the integration by parts after 1 round of integrating since I am left with integrals.

I suppose I could also ask, do I have to evaluate the integral before taking the limit? I am not sure what the rules on this are... is
[tex]\lim_{\gamma\rightarrow 0}\int_{-\infty}^{\infty}e^{-\gamma |x|}f(x)dx=\int_{-\infty}^{\infty}\lim_{\gamma\rightarrow 0}e^{-\gamma |x|}f(x)dx[/tex]
allowed??
 
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  • #2
Perhaps look up the "Dominated Convergence Theorem"...
 

1. What is the significance of exploring the convergence of e^(-γ|x|) integrals?

The convergence of integrals is important in many areas of mathematics and physics, as it allows us to determine if a function is well-behaved and can be integrated in a certain interval. In the case of e^(-γ|x|) integrals, exploring convergence helps us understand the behavior of this specific function and its applications in various fields.

2. How do you determine the convergence of e^(-γ|x|) integrals?

To determine the convergence of e^(-γ|x|) integrals, we use mathematical techniques such as integration by parts, substitution, and comparison tests. These methods help us analyze the behavior of the integral and determine if it converges or diverges.

3. Are there any special cases where e^(-γ|x|) integrals converge or diverge?

Yes, there are special cases where e^(-γ|x|) integrals converge or diverge. For example, if the value of γ is equal to 0, the integral will always converge. On the other hand, if the value of γ is greater than 0, the integral will only converge for certain values of x.

4. What are some applications of e^(-γ|x|) integrals?

E^(-γ|x|) integrals have various applications in mathematics, physics, and engineering. They are used to model certain physical phenomena such as radioactive decay, heat transfer, and diffusion processes. They are also used in probability and statistics to calculate the probability of certain events occurring.

5. Is there a simplified form or closed-form solution for e^(-γ|x|) integrals?

No, there is no simplified form or closed-form solution for e^(-γ|x|) integrals. However, we can use numerical methods such as Monte Carlo integration to approximate the value of the integral for a given range of x and γ. In some cases, the integral can also be expressed in terms of special functions such as the error function.

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