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Is that really how the exponential function is introduced? The most common ones as far as I know are: - "The Classic": There exists a unique function equal to
by qsort 1y ago
Is that really how the exponential function is introduced? The most common ones as far as I know are:
- "The Classic": There exists a unique function equal to its own derivative up to a constant.
- "I can't bothered with this": Have a series. It's obviously absolutely convergent. kthxbye.
- "My name is Hardy, G.H. Hardy.": A unique function satisfies exp(x+y) = exp(x)exp(y).
- gthompson512 1y ago> - "My name is Hardy, G.H. Hardy.": A unique function satisfies exp(x+y) = exp(x)exp(y). This has nothing to do with e and is satified by 2^x or any a^x, so this wouldn't work for introducing e in particular. - "The Classic": There exists a unique function equal to its own derivative up to a constant. Same for this, but if you fix the constant to be 1, then e^x is the only one that works. I will give the series works too.
- qsort 1y ago> This has nothing to do with e and is satified by 2^x or any a^x, so this wouldn't work for introducing e in particular. You need to impose f'(0) = 1. (If you want to be really technical, also at least some regularity condition, I'll be honest, I don't remember what's the minimal one, let's say continuity) > Same for this I did say up to a constant.
- mthiim 1y agoI imagine G.H. Hardy might be like: e is the unique base of a logarithm function, ln(n), so so that the average distance between prime numbers less than n, converges proportionally towards ln(n) - the ratio between ln(n) and the actual average distance between primes < n converges to 1 as n goes to infinity.