Prove that : γ=−∫10lnln(1x) dx.
where γ is Euler's constant (γ≈0.57721).
This integral was mentioned in Wikipedia as in Mathworld , but the solutions I've got uses corollaries from this theorem. Can you give me a simple solution (not using much advanced theorems) or at least some hints.
Answer
In this answer, it is shown that since Γ is log-convex, Γ′(x)Γ(x)=−γ+∞∑k=1(1k−1k+x−1) Setting x=1 yields Γ′(1)=−γ The integral definition of Γ says Γ(x)=∫∞0tx−1e−tdtΓ′(x)=∫∞0log(t)tx−1e−tdtΓ′(1)=∫∞0log(t)e−tdt Putting together (2) and (3) gives ∫∞0log(t)e−tdt=−γ Substituting t↦log(1/t) transforms (4) to ∫10log(log(1/t))dt=−γ
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