Could this function be approximated by Dirac delta function?

[Haorong Wu]

TL;DR Summary

Under what conditions, could a function be replaced by a delta function?

hi, there. I am doing some frequency analysis. Suppose I have a function defined in frequency space $$N(k)=\frac {-1} {|k|} e^{-c|k|}$$ where $c$ is some very large positive number, and another function in frequency space $P(k)$. Now I need integrate them as $$ \int \frac {dk}{2 \pi} N(k) P(k).$$However, the integration wil be too complicated to be solved.

Meanwhile, I notice that, $N(k)$ is infinite when $k\rightarrow 0$, and decreases to zero rapidly when $k \ne 0$. However, the integral of $$ \int dk N(k)$$ is also infinite. I am not sure whether I could approximate $N(k)$ by $\delta(k)$ or not. Would it yield problems in physics?

Thanks!

 

[stevendaryl]

No. There are three facts that you want for a function that acts like a delta function are:

1. $f(k) \rightarrow \text{very large}$ when $k \rightarrow 0$
2. $f(k) \rightarrow 0$ when $k \rightarrow \infty$
3. $\int_{-\infty}^{+\infty} f(k) dk = 1$

Your function has properties 1 and 2, but not 3.

 

[Office_Shredder]

You could shuffle some of N onto P. Like if you said $\tilde{N}(k)= e^{-c|k|}/\sqrt{|k|}$ and $\tilde{P}(k)=P(k)/\sqrt{|k|}$ you might be able to do something.

As c goes to infinity the integral of $\tilde{N}$ is not constant, so you would also need to shuffle around some $c$ stuff in order to get this to totally work.