Limiting Case of Dirac Function: delta(x) = lim_{a->0} (1/sqrt[pi*a])*e^(-x^2/a), |x|<a

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mario99 said:
It is defined like that in the book.

What are you trying to say?

[math]\displaystyle \int _{-\infty}^{\infty} f(x) \delta(x) \ dx = \displaystyle \int _{-\infty}^{\infty} f(x) \left(\lim_{a\rightarrow 0} \frac{1}{\sqrt{\pi a}}e^{-x^2/a}\right) \ dx[/math]
But

[math]\displaystyle \int _{-a}^{a} f(x) \delta(x) \ dx \neq \displaystyle \int _{-a}^{a} f(x) \left(\lim_{a\rightarrow 0} \frac{1}{\sqrt{\pi a}}e^{-x^2/a}\right) \ dx[/math]
Or

[math]\displaystyle \int _{-\infty}^{\infty} f(x) \delta(x) \ dx \neq \displaystyle \int _{-\infty}^{\infty} f(x) \left(\lim_{a\rightarrow 0} \frac{1}{\sqrt{\pi a}}e^{-x^2/a}\right) \ dx[/math]
[math]\displaystyle \int _{-a}^{a} f(x) \delta(x) \ dx \neq \displaystyle \int _{-a}^{a} f(x) \left(\lim_{a\rightarrow 0} \frac{1}{\sqrt{\pi a}}e^{-x^2/a}\right) \ dx[/math]
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I find the whole expression [imath]\lim_{a\rightarrow 0} \frac{1}{\sqrt{\pi a}}e^{-x^2/a}[/imath] misleading, at least in the point-wise sense of the limits. But I just saw that its equivalent is mentioned in the Wikipedia page on Dirac function, where they define it as a weak limit -- is this how it is described in your course?

On a more general note: It would make this discussion more productive if you provided more context, i.e., full, verbatim statement of the problem as it is given to you, plus some info about the subject you are studying.
 
topsquark said:
It is defined like which definition in your book? I gave you two.

Since you seemed to be confused about the definition, I gave you the definition. Your post here is talking about applying a limiting form. The first line is correct, so long as the integral exists.

-Dan
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What you have written in post #18 is exactly the definition and properties of Dirac delta function in the book. To see a picture from the book is better right? I might be lying.


blamocur said:
Try answering the question about [imath]\delta(x)[/imath] vs. [imath]2\delta(x)[/imath] -- it might make you think likewise.
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If I answered your question here, would you answer my question in post #1 fully as you are not convinced by my answer?


blamocur said:
On a more general note: It would make this discussion more productive if you provided more context, i.e., full, verbatim statement of the problem as it is given to you, plus some info about the subject you are studying.
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What makes you think that I didn't post the exact statement of the question? You want to see a picture to be convinced? May be I am lying and wants to confuse you Seniors! Is that what you think?

blamocur said:
as it is given to you, plus some info about the subject you are studying.
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How to solve differential equations by Green Function.
 
mario99 said:
If I answered your question here, would you answer my question in post #1 fully as you are not convinced by my answer?
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Does not sound like a fair deal to me since A) I wanted you to answer my question for your own benefit, hoping that trying to answer it you might see the issues with your OP; and B) I've already done my best to help you with the question, and don't see what else I can add.
mario99 said:
What makes you think that I didn't post the exact statement of the question? You want to see a picture to be convinced? May be I am lying and wants to confuse you Seniors! Is that what you think?
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I meant no offense there. What I meant is that I didn't see enough context for me being able to help you better.
 
blamocur said:
Does not sound like a fair deal to me since A) I wanted you to answer my question for your own benefit, hoping that trying to answer it you might see the issues with your OP; and B) I've already done my best to help you with the question, and don't see what else I can add.
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I know the answer, but I don't want to share it for no reason.


blamocur said:
I meant no offense there. What I meant is that I didn't see enough context for me being able to help you better.
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Now you know that the statement is complete and you know what I am studying.


Would you correct my answer as you said:
blamocur said:
No, I don't see this as a valid proof.
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mario99 said:
I did.



I didn't say I don't need help. I said I know the answer for:

[math]\delta(x) [/math] vs [math]2\delta(x)[/math]
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In post #25 you said you know the answer to blamocur's question. If you don't want to share it, then this thread no longer serves a useful purpose.

-Dan
 
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