Γ(1/n)
by John from John D. Cook on (#73BER)
If n is a positive integer, then rounding (1/n) up to the nearest integer givesn. In symbols,
We an illustrate this with the following Python code.
>>> from scipy.special import gamma>>> from math import ceil>>> for n in range(1, 101): ... assert(ceil(gamma(1/n)) == n)
You can find a full proof in [1]. I'll give a partial proof that may be more informative than the full proof.
The asymptotic expansion of the gamma function near zero is
where is the Euler-Mascheroni constant.
So when we setz = 1/n we find (1/n) n - +O(1/n^2). Since 0 < < 1, the theorem above is true for sufficiently largen. And it turns out sufficiently large" can be replaced withn >= 1.
[1] Gamma at reciprocals of integers: 12225. American Mathematical Monthly. October 2022. pp 789-790.
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