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

Symmetrically
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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