Dichotomy
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| dichotomy summary |
The normal world, according to the way we learn maths at school, exists on a continuum of infinitesimally small to infinitely large. There are instantaneous and eternal concepts, but one doesn't hear much of them: this is because time inheres in the structure of what we all perceive to be the world of objects moving through time as a one-way arrow into the future.
In the Fourier domain that axis is radically shifted away from space to become one of time, where infinintesimally small times become the ultimate of fine-grainedness in perceiving the frequency spectrum, and the infinitely long period being the biggest window, or, if you like, the standing wave of the universe (see Hubble constant).
Given that the Heisenberg Uncertainty Principle, or at the very least observer error, seems to defy all attempts to inquire into the nature of things, it is instructive to know where our observations must be incomplete. In the Fourier domain, our information is going to be limited both by an inability to fine-grain the frequencies enough (can't plot infinitesimals) or to observe over a sufficiently long wavelength (window problems). In the space-time world, these roughly equate to the problems respectively of quantum mechanics and cosmology.
Dealing with the continuous Fourier transform, though alien at first, does not pose any special problems that are not normally present. Continuous observation in spacetime requires a perception that does not reduce the content into the spacetime artifacts of things in time - continuous observation of the Fourier domain requires that we do not grade frequencies.
While it is customary to note that the trick of continuous perception is easier said than done, in either case it is not so - it absolutely cannot be said (by its very nature) and hence can only be done! It is instructive to note that each visual perception actually constitutes a reverse Fourier transform,
For a person, however we look at the world, we are constrained perception-wise by the sum total of knowable local information. The beauty is, of course, that our species has developed the capacity to generate information by description (incomplete though that must necessarily be) as well as the ability to theorise about such things.
Heaven & Earth
Lets have a closer look at the eix formular to see how it splits into by odd and even into the two domains. First here's its taylor-series expansion (I've separated the i and x powers out for clarity)...
| [math] e^{ix} = \sum_{n = 0}^{\infty} {ix^n \over n!} = {i^0 \cdot x^0 \over 0!} + {i^1 \cdot x^1 \over 1!} + {i^2 \cdot x^2 \over 2!} + {i^3 \cdot x^3 \over 3!} + {i^4 \cdot x^4 \over 4!} + \cdots [/math] |
Now because i2 is -1, successive powers of i lead to the terms alternating between real and imaginary as follows...
| [math] = {1 \cdot x^0 \over 0!} + {i \cdot x^1 \over 1!} + {-1 \cdot x^2 \over 2!} + {-i \cdot x^3 \over 3!} + {1 \cdot x^4 \over 4!} + {i \cdot x^5 \over 5!} + {-1 \cdot x^6 \over 6!} + \cdots [/math] |
As you can see, all the terms have become a sequence of alternating positive and negative i or 1, so this means we can separate all the terms containing i and factorise them out,
| [math] = \left( {x^0 \over 0!} - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} \cdots \right) + i \left( {x^1 \over 1!} - {x^3 \over 3!} + {x^5 \over 5!} - {x^7 \over 7!} \cdots \right) [/math] |
And so you can see that all the even terms have grouped together to make the real component, and all the odds make the imaginary, so where x is angular, eix yields the rectangular components, hence:
| [math] \!\, e^{ix} = \cos x + i \sin x [/math] |
or more generally, when e is raised to the power of any arbitrary complex number,
| [math] \!\, e^{a + bi} = e^a (\cos b + i \sin b) [/math] |
