Question Number 33345 by prof Abdo imad last updated on 14/Apr/18
![for x∈]0,+∞[ let ψ(x) = ((Γ^′ (x))/(Γ(x))) 1)prove that ψ(x) =−(1/x) −γ +x Σ_(n=1) ^∞ (1/(n(x+n))) 2)ptove that γ =−Γ^′ (1) 3) prove that ∫_0 ^∞ e^(−x) ln(x)dx =−γ .](https://www.tinkutara.com/question/Q33345.png)
$$\left.{for}\:{x}\in\right]\mathrm{0},+\infty\left[\:{let}\:\psi\left({x}\right)\:=\:\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)}\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\psi\left({x}\right)\:=−\frac{\mathrm{1}}{{x}}\:−\gamma\:+{x}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}\left({x}+{n}\right)} \\ $$$$\left.\mathrm{2}\right){ptove}\:{that}\:\gamma\:=−\Gamma^{'} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:. \\ $$