- #1
rajetk
- 1
- 0
I am looking at a problem, part of which deals with expressing delta dirac as a limiting case of gaussian function. I am aware of the standard ways of doing it. In addition, I would also like to know if the following are correct -
[tex]
\delta(x-a) = \lim_{\sigma \rightarrow{0}} \int_{a - \sigma}^{{a + \sigma}} \sqrt{\frac{1}{2\pi \sigma^{2}}}e^{-((x-a)^{2})/(2\sigma^{2})} dx
[/tex]
Or can I say something like the following -
[tex]
\lim_{\sigma \rightarrow{0+}} \int_{a - \sigma}^{{a + \sigma}} \sqrt{\frac{1}{2\pi \sigma^{2}}}e^{-((x-a)^{2})/(2\sigma^{2})} dx
[/tex]
where the second expression is not a delta function but its approximation (since both the a+ and a- regions are considered in the integral)?
Please do excuse me if I am seriously wrong :(. Thanks in advance.
[tex]
\delta(x-a) = \lim_{\sigma \rightarrow{0}} \int_{a - \sigma}^{{a + \sigma}} \sqrt{\frac{1}{2\pi \sigma^{2}}}e^{-((x-a)^{2})/(2\sigma^{2})} dx
[/tex]
Or can I say something like the following -
[tex]
\lim_{\sigma \rightarrow{0+}} \int_{a - \sigma}^{{a + \sigma}} \sqrt{\frac{1}{2\pi \sigma^{2}}}e^{-((x-a)^{2})/(2\sigma^{2})} dx
[/tex]
where the second expression is not a delta function but its approximation (since both the a+ and a- regions are considered in the integral)?
Please do excuse me if I am seriously wrong :(. Thanks in advance.