Why is Pi in the normal distribution, asks Dave Giles. He shows us when we consider the formula.
The formula has Mr. Euler's number, the natural log thing. When that was specified, then the implicit condition is no saddle back in the ensembles, they always monotonically converge to their global statistics.. The assumption specifies Euler's number. But Euler's number is an algorithm, the numerical method to compute the next most accurate natural log from the previous. Pi appears because the maximum, no saddleback dispersion differential converges to Pi and the natural log converges to Euler.
In finite terms, normal means the system will not cycle when left to its own devices.
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