Tuesday, October 31, 2017

real analysis - Integral representation of Euler's constant


Prove that : γ=10lnln(1x) dx.


where γ is Euler's constant (γ0.57721).




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(1k1k+x1) Setting x=1 yields Γ(1)=γ The integral definition of Γ says Γ(x)=0tx1etdtΓ(x)=0log(t)tx1etdtΓ(1)=0log(t)etdt Putting together (2) and (3) gives 0log(t)etdt=γ Substituting tlog(1/t) transforms (4) to 10log(log(1/t))dt=γ


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