Integrating The Delta Function (Again and Again) – Penrose Version

I quoted Nobel prize winner Paul Dirac’s book, now I will quote this year’s physics Nobel prize winner Roger Penrose.

In his book The Road to Reality Penrose discusses not-so-well-behaved functions like the Delta Function: They belong in the category of  Hyperfunctions. A Hyperfunction is the difference of two complex functions: Each of the complex functions is defined on either complex half-plane, and the difference is interesting in the place where the two functions ‘meet’ in the limit: on the real axis. The difference of these functions is finally a real function of real arguments.

The Delta Function seems to be a canonical example for this concept: Its Hyperfunction representation is the pair of  functions \left(-\frac{1}{2\pi iz},-\frac {1}{2\pi iz}\right)

I re-use my previous calculation to show why the Delta function is the difference of two reciprocal functions, starting again from its spectral representation. In the Haiku post, my intent was to show that the sum of this integrals is equal to a Lorentzian bell curve – of which I have shown before that is has the required properties of the Delta Function (See formulas below the poetry :-)). This time I try to make this difference of functions obvious.

The integral over the real axis is split into a negative and positive part, and a small complex number is added to make sure the integral converges. In the limit this number tends to zero:

\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 ]

I am now adding one more step. Having integrated the exponential function and inserted the values at the boundaries we see indeed the difference of two reciprocal functions. These are actually the same function, just shifted ‘up’ and ‘down’, away from the real axis:

\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}  \left [ \frac{1}{x-i\varepsilon } - \frac{1}{ x+i\varepsilon} \right ] = \lim_{\varepsilon \to 0} \frac{1}{\pi} \frac{\varepsilon}{x^2 + \varepsilon^2}

Penrose discussed yet another way of looking at these two ‘parts’ of a Hyperfunctions – by considering their Fourier analysis, their representations in terms of harmonic functions: The two ‘parts’ can be distinguished by their negative versus positive frequencies.

In the calculation above, I started from the Delta Function’s spectral representation and split the integral in a part from minus infinity to zero, and another from zero to infinity, each using the same sign of ‘frequencies’ I called k. But you could combine these integrals again (and use limits zero to infinity for each of them), by substituting k by -k. Then the integrals could be extended to minus infinity again by adding a factor 1/2 – for the same symmetry reasons Dirac did the reverse in his way of integrating the Delta Function. Or you could just look at the spectral representation of the Delta Function at the beginning, substitute k by its negative. So you have two integrals with the same value, and the Delta Function is equal to half of the sum of those. In order to damp down the exponential function, you add the small imaginary numbers. It’s always the same calculation, but this version makes more obvious that we separate positive and negative frequencies.

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For completeness: All that stuff only works with holomorphic functions – complex functions of complex arguments that can differentiated by complex values in a meaningful way … which is a strong constraint.

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For overarching context: COVID cases surging here, alarmingly fast, ICU occupancy +78% in the last week. Many data nerds and science commentators play with latest published data. I resort to well-known mathematical physics.

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