unit circle

[AntonioLao]

The important of the unit circle in mathematics could never be over emphasized as shown in the link http://en.wikipedia.org/wiki/Unit_circle

[AntonioLao]

In the first half of the 19th century, Cauchy presented an important fundamental theorem of complex analysis. This theorem states that the line integral of an analytic function around a simple closed curve is zero. His contemporary, Liouville, also presented a theorem stating that a bounded and entirely analytic complex function must be constant. Two questions could be asked. First, is zero a complex number or a real number? Second, is Liouville’s constant complex or real?

Before answering and before asking a 3rd question, it is worthwhile to note some of the important applications of these theorems for formulating physical theories. In vector analysis (as extension of complex analysis), their appearances can be found within Stoke’s theorem and Gauss’s theorem (divergence theorem). The former relates line integral to surface integral. The latter relates surface integral to volume integral. Both were crucial for the success of Maxwell’s electromagnetism.

The 3rd question is whether closed curves and closed surfaces have meaningful representations in the real number domain?

Note that closed curves in general relativity are the same as closed worldlines. Closed volumes in quantum mechanics are the same as zero-point energies. In electromagnetism, closed magnetic lines do not exist. Constant volumes only imply constant internal electric charge densities and invariant magnitudes of magnetic moments.