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nice-calculus-0-1-0-1-1-x-1-xy-ln-xy-2019-dxdy-

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Question Number 139101 by mnjuly1970 last updated on 22/Apr/21
.......nice calculus..... 𝛗=^(???) ∫_0 ^( 1) ∫_0 ^( 1) ((1−x)/(1−xy))(−ln(xy))^(2019) dxdy .........
$$\:\:\:\:\:\:\:\:\:\:\:\:\:…….{nice}\:\:\:{calculus}….. \\ $$$$\boldsymbol{\phi}\overset{???} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{xy}}\left(−{ln}\left({xy}\right)\right)^{\mathrm{2019}} {dxdy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……… \\ $$
Answered by Dwaipayan Shikari last updated on 22/Apr/21
ϑ(α)=∫_0 ^1 ∫_0 ^1 ((1−x)/(1−xy))(xy)^α dxdy =Σ_(n=0) ^∞ ∫_0 ^1 ∫_0 ^1 (xy)^(n+α) −x^(n+α+1) y^(n+α) dxdy =Σ_(n=0) ^∞ ∫_0 ^1 (y^(n+α) /(n+α+1))−(y^(n+α) /((n+α+2)))dy =Σ_(n=0) ^∞ (1/((n+α+1)^2 ))−(1/((n+α+1)(n+α+2))) =Σ_(n=0) ^∞ (1/((n+α+1)^2 ))−ψ(α+1)+ψ(α+2)=ψ′(α+1)−ψ(α+1)+ψ(α+2) ϑ^(2019) (α)=2020!(−1)^(2019) ψ^(2021) (α+1)−2019!ψ^(2020) (α+1)+2019!ψ^(2020) (α+2) −ϑ^(2019) (0)=2020!ζ(2020)+2019!ζ(2020)−2019!(ζ(2021)−1)
$$\vartheta\left(\alpha\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{xy}}\left({xy}\right)^{\alpha} {dxdy} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \left({xy}\right)^{{n}+\alpha} −{x}^{{n}+\alpha+\mathrm{1}} {y}^{{n}+\alpha} {dxdy} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{y}^{{n}+\alpha} }{{n}+\alpha+\mathrm{1}}−\frac{{y}^{{n}+\alpha} }{\left({n}+\alpha+\mathrm{2}\right)}{dy} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}+\alpha+\mathrm{1}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\left({n}+\alpha+\mathrm{1}\right)\left({n}+\alpha+\mathrm{2}\right)} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}+\alpha+\mathrm{1}\right)^{\mathrm{2}} }−\psi\left(\alpha+\mathrm{1}\right)+\psi\left(\alpha+\mathrm{2}\right)=\psi'\left(\alpha+\mathrm{1}\right)−\psi\left(\alpha+\mathrm{1}\right)+\psi\left(\alpha+\mathrm{2}\right) \\ $$$$\vartheta^{\mathrm{2019}} \left(\alpha\right)=\mathrm{2020}!\left(−\mathrm{1}\right)^{\mathrm{2019}} \psi^{\mathrm{2021}} \left(\alpha+\mathrm{1}\right)−\mathrm{2019}!\psi^{\mathrm{2020}} \left(\alpha+\mathrm{1}\right)+\mathrm{2019}!\psi^{\mathrm{2020}} \left(\alpha+\mathrm{2}\right) \\ $$$$−\vartheta^{\mathrm{2019}} \left(\mathrm{0}\right)=\mathrm{2020}!\zeta\left(\mathrm{2020}\right)+\mathrm{2019}!\zeta\left(\mathrm{2020}\right)−\mathrm{2019}!\left(\zeta\left(\mathrm{2021}\right)−\mathrm{1}\right) \\ $$
Commented by mnjuly1970 last updated on 22/Apr/21
thanks alot
$$\:\:\:{thanks}\:{alot} \\ $$

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