Delta Function Haiku

I have proved that a Lorentzian bell curve becomes the Dirac Delta Function in the limit. Now I want to look at another representation of the Delta Function. As this is a shorter proof, a haiku will do.


Infinite numbers
of oscillations added.
Need to damp them down

attach an exponential
for each half of space.

Shut up, calculate.
Imaginaries cancel.
Again, the bell in the limit.


\displaystyle \delta(x) = \int_{-\infty}^{+\infty} \frac{dk}{2\pi} e^{ikx} = \lim_{\varepsilon \to 0} \left [ \int_{-\infty}^{0} \frac{dk}{2\pi} e^{ik(x-i\varepsilon)} + \int_{0}^{+\infty} \frac{dk}{2\pi} e^{ik(x+i\varepsilon)} \right ] =

\displaystyle  \lim_{\varepsilon \to 0} \frac{1}{2\pi} \frac{1}{i} \left [ \left. \frac{e^{ik(x-i\varepsilon)}}{x-i\varepsilon} \right |_{-\infty}^{0} + \left. \frac{e^{ik(x+i\varepsilon)}}{x+i\varepsilon} \right |_{0}^{\infty} \right ] = \lim_{\varepsilon \to 0} \frac{1}{2\pi} \frac{1}{i}  \frac{x + i\varepsilon - x + i\varepsilon}{x^2 + \varepsilon^2} = \lim_{\varepsilon \to 0} \frac{1}{\pi} \frac{\varepsilon}{x^2 + \varepsilon^2}


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