Question Number 136645 by Dwaipayan Shikari last updated on 24/Mar/21 $$\frac{\mathrm{1}}{\mathrm{1}+\frac{\eta^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{1}+..}\:\:}}}}=\frac{\mathrm{1}}{\mathrm{2}}\psi\left(\frac{\eta+\mathrm{2}}{\mathrm{2}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\psi\left(\frac{\eta+\mathrm{1}}{\mathrm{2}}\right)\:\:\left(\eta>\mathrm{0}\right) \\ $$$${Or}\:\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\boldsymbol{\mathrm{K}}}}\left(\eta+{r}\right)^{\mathrm{2}} =\frac{\mathrm{2}}{\psi\left(\frac{\eta}{\mathrm{2}}+\mathrm{1}\right)−\psi\left(\frac{\eta+\mathrm{1}}{\mathrm{2}}\right)} \\ $$ Terms of Service…
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Question Number 135647 by Dwaipayan Shikari last updated on 14/Mar/21 $$…+\frac{\mathrm{1}}{\left(\mathrm{1}−\mathrm{2}\pi^{\mathrm{2}} \right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{1}−\pi^{\mathrm{2}} \right)^{\mathrm{2}} }+\mathrm{1}+\frac{\mathrm{1}}{\left(\mathrm{1}+\pi^{\mathrm{2}} \right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{2}\pi^{\mathrm{2}} \right)^{\mathrm{2}} }+…=\frac{{csc}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\pi}\right)}{\pi^{\mathrm{2}} } \\ $$ Terms of…
Question Number 135127 by Dwaipayan Shikari last updated on 10/Mar/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} {log}^{\mathrm{2}} \left(\Gamma\left({x}\right)\right){dx} \\ $$ Answered by mathmax by abdo last updated on 11/Mar/21…
Question Number 2032 by 123456 last updated on 31/Oct/15 $${f}:\left[\mathrm{0},+\mathrm{1}\right]\rightarrow\mathbb{R} \\ $$$$\eta\left({f}\right):=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}^{\mathrm{2}} {dx} \\ $$$$\mathrm{and}\:\mathrm{if}\:\forall{x}\in\left[\mathrm{0},\mathrm{1}\right],{f}\in\left[{m},\mathrm{M}\right]\:\mathrm{for}\:\mathrm{some}\:\left({m},\mathrm{M}\right)\in\mathbb{R}^{\mathrm{2}} \\ $$$${f}\uparrow:=\mathrm{max}\left({f}\right)−{f} \\ $$$${f}\downarrow:={f}−\mathrm{min}\left({f}\right) \\ $$$$\mu\left({f}\right):=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\downarrow{f}\uparrow{dx}…
Question Number 1987 by 123456 last updated on 28/Oct/15 $$\mathrm{lets}\:{a}<{b}\:\mathrm{and}\:{f}:\left[{a},{b}\right]\rightarrow\mathbb{R}\:\mathrm{integable}\:\mathrm{into}\:\left[{a},{b}\right]\:\mathrm{and}\:\mathrm{continuous} \\ $$$$\mathrm{lets}\:\mathrm{I}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{subset}\:\mathrm{of}\:\left[{a},{b}\right] \\ $$$$\mathrm{proof}\:\mathrm{that}\:\left(\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{conter}\:\mathrm{example}\right) \\ $$$$\underset{\mathrm{I}} {\int}{fdx}=\mathrm{0}\:\forall\mathrm{I}\subset\left[{a},{b}\right]\Rightarrow{f}=\mathrm{0} \\ $$ Commented by prakash jain last updated…
Question Number 1858 by 123456 last updated on 16/Oct/15 $$\begin{bmatrix}{{x}\left(\rho,\theta,\xi\right)}\\{{y}\left(\rho,\theta,\xi\right)}\\{{z}\left(\rho,\theta,\xi\right)}\end{bmatrix}=\begin{bmatrix}{\rho{e}^{\xi} \mathrm{cos}\:\theta}\\{\rho{e}^{\xi} \mathrm{sin}\:\theta}\\{\xi}\end{bmatrix}\begin{cases}{\rho\in\left[\mathrm{0},+\infty\right)}\\{\theta\in\left[\mathrm{0},\mathrm{2}\pi\right)}\\{\xi\in\mathbb{R}}\end{cases} \\ $$$$\boldsymbol{{r}}\left(\rho,\theta,\xi\right)={x}\left(\rho,\theta,\xi\right)\boldsymbol{{i}}+{y}\left(\rho,\theta,\xi\right)\boldsymbol{{j}}+{z}\left(\rho,\theta,\xi\right)\boldsymbol{{k}} \\ $$$$\boldsymbol{{a}}_{\rho} =\frac{\partial\boldsymbol{{r}}}{\partial\rho} \\ $$$$\boldsymbol{{a}}_{\theta} =\frac{\partial\boldsymbol{{r}}}{\partial\theta} \\ $$$$\boldsymbol{{a}}_{\xi} =\frac{\partial\boldsymbol{{r}}}{\partial\xi} \\ $$$$\boldsymbol{{a}}_{\rho}…
Question Number 1448 by 123456 last updated on 05/Aug/15 $$\boldsymbol{{x}}=\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \right),\boldsymbol{{y}}=\left({y}_{\mathrm{1}} ,{y}_{\mathrm{2}} \right) \\ $$$$\eta:\left[\mathrm{0},\mathrm{1}\right)^{\mathrm{4}} \rightarrow\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\eta\left(\boldsymbol{{x}},\boldsymbol{{y}}\right):=\mathrm{med}\left[\frac{\left(\mathrm{1}−{x}_{\mathrm{1}} \right)^{{y}_{\mathrm{1}} } +\left(\mathrm{1}−{y}_{\mathrm{1}} \right)^{{x}_{\mathrm{1}} } }{\mathrm{2}},\frac{\left(\mathrm{1}−{x}_{\mathrm{2}}…
Question Number 1351 by 123456 last updated on 24/Jul/15 $$\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{f}\left({x}\right)\mathrm{ln}\left({xt}\right){dx},{t}>\mathrm{0} \\ $$$$\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)+\mathcal{W}\left\{{g}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{{cf}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}{c}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{\mathrm{1}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}^{{n}} \right\}\left({t}\right)=?,{n}\in\mathbb{N}…