find-ln-x-

Question Number 153105 by alisiao last updated on 04/Sep/21

f i n d l n Γ (x) ?

Answered by puissant last updated on 04/Sep/21

l n (Γ (x)) = - γ x - l n x + \underset{k = 1}{\sum^{\infty}} {\frac{x}{k} - l n (1 + \frac{x}{k})}

= (x - \frac{1}{2}) l n x - x + \frac{1}{2} l n (2 π) + \frac{1}{2} \underset{n = 2}{\sum^{\infty}} \frac{(n - 1)}{n (n + 1)} ζ (n, x + 1)

= (x - \frac{1}{2}) l n x - x + \frac{1}{2} l n (2 π) + 2 \int_{0}^{\infty} \frac{a r c t a n (\frac{x}{a})}{e^{2 π x} - 1} d x .

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