Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution )
1)∫∞0e−x2lnxdx=−14(γ+2ln2)√π
2)∫∞0e−xln2xdx=γ2+π26
3) γ=∫1011+x∞∑n=1x2n−1dx
and lastly,
4) ζ(s)=e(log(2π)−1−γ/2)s2(s−1)Γ(1+s/2)∏ρ(1−sρ)es/ρ
I personally think the last is obtained from a simple use of the Weierstrass factorization theorem. I'm unsure as to what substitution is used.
γ is the Euler-Mascheroni constant and ζ(s) is the Riemannian Zeta function.
Thanks in advance.
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