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How can $$\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $$ possibly be the Heaviside Step Function?

What I'm looking for is a direct visualization or maybe an approximate numerical calculation that confirms that this weird thing is the unit step function.

To give some context:

I tried to prove $\frac{\mathrm{d} }{\mathrm{d} x}\Theta =\delta (x)$ using this representation of the delta function: $\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ This should be easy. I just need to integrate $\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ with respect to x to get $ \Theta(x) $

This was the first step:

$$\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $$

Then I gave up because this did't look like the step function to me.

I tried to do integration by parts, but I just have a more complicated mess. A few people showed me an alternative method of proof. Today, I was reading The Six Core Theories of Modern Physics by Charles Stevens and I saw that, according to Stevens, what I originally wrote is the step function. How could this be? All he says is "This must be the unit step function because its derivative is the dirac delta.

The Step Function:

$\theta(x) = 1$ if $x>0$

$\theta(x) = 0$ if $x \leq 0$

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Hint: Split it into real and imaginary parts. You'll need to evaluate $\int_{-\infty}^\infty \frac{\cos(u)}{u}du$ which is 0 because you're integrating an odd function, though the integral is divergent. You'll also need to evaluate $\int_{-\infty}^\infty \frac{\sin u}{u}du$ which is twice the Dirichlet integral, and is therefore equal to $\pi$.

It won't be exactly those, but armed with that you should be able to prove it.

Correction: $\int_{-\infty}^\infty \frac{\cos(u)}{u}du$ is actually divergent, so you'll need to take the principle value.

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  • $\begingroup$ alright, I will try is. I started to do it, but stopped because I felt I was possibly wasting time trying to do something that cant be done. $\endgroup$ – mathmath12 Apr 28 '15 at 19:11
  • $\begingroup$ @mathmath12 wolframalpha.com/input/… $\endgroup$ – man and laptop Apr 28 '15 at 19:36
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Hint: you have to consider test functions.

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  • $\begingroup$ What I want to do is actually evaluate $\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $ I don't see why it should be impossible to integrate this. $\endgroup$ – mathmath12 Apr 28 '15 at 18:56
  • $\begingroup$ Can't I at least do this numerically? $\endgroup$ – mathmath12 Apr 28 '15 at 18:58
  • $\begingroup$ this is impossible in the sense that the integral is not convergent in the classic sense. $\endgroup$ – mookid Apr 28 '15 at 18:59

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