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Topic: The Many Roles of e
Replies: 8   Last Post: Apr 23, 2013 8:39 AM

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Posts: 234
Registered: 12/4/04
The Many Roles of e
Posted: Apr 20, 2013 11:06 AM
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I am collecting results for a paper or monograph on the appearance of the constant e in diverse aeas of mathematics, and am asking anyone with examples to let me know about them so they may be included. Here are a few, from well-known to rather exotic:

1: The classical definition found in attempts to differentiate the logarithm function can also be computed and shown to exist using the "area" definition.
2: Probablility (e.g., "If a diner at random picks a hat at random, ...") and statistics (e.g., limits of binomial distributions are Poisson distributions).
3: It is a "factorially stable" irrational number, meaning that the sequence {n!e-[n!e]} has a limit (it's 0).
4: Here's my favorite: The "exponential tower" x^(x^(x^(...))) converges on the interval [e^(-e), e^(1/e)] and has range [1/e, e].

Of course, there are many other expressions (limits, definite integrals, ...) that perhaps surprisingly be expressed in terms of e, and I would like to "collect them all". At present, the interest is only in real analysis, but any complex results are also solicited. Wherever results are used, credit will be given to the the contributor. Thanks for your attention - I look forward to a lively dialogue and more surprises about this remarkable number.

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