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Question Number 134102 by mnjuly1970 last updated on 27/Feb/21

prove that : 𝛗=∫_0 ^( ∞) ((cos((√x) ))/(e^(2Ο€(√x) ) βˆ’1))dx=1βˆ’(e/((eβˆ’1)_ ^2 ))

$$\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left(\sqrt{{x}}\:\right)}{{e}^{\mathrm{2}\pi\sqrt{{x}}\:} βˆ’\mathrm{1}}{dx}=\mathrm{1}βˆ’\frac{{e}}{\left({e}βˆ’\mathrm{1}\right)_{} ^{\mathrm{2}} } \\ $$

Answered by Dwaipayan Shikari last updated on 27/Feb/21

x=t^2 2∫_0 ^∞ t((cos(t))/(e^(2Ο€t) βˆ’1))dt=Ξ£_(n=0) ^∞ ∫_0 ^∞ te^(βˆ’2Ο€nt+it) +te^(βˆ’2Ο€ntβˆ’it) dt =Ξ£_(n=0) ^∞ (1/((2Ο€nβˆ’i)^2 ))+(1/((2Ο€n+i)^2 ))=(1/(4Ο€^2 ))(ψ^1 (βˆ’(i/(2Ο€)))+ψ^1 ((i/(2Ο€)))) =(1/(4Ο€^2 ))(Ο€^2 csc^2 Ο€((i/(2Ο€))))+1=1+(1/4)(((2i)/(e^((βˆ’1)/2) βˆ’e^(1/2) )))^2 =1βˆ’(e/((eβˆ’1)^2 ))

$${x}={t}^{\mathrm{2}} \\ $$$$\mathrm{2}\int_{\mathrm{0}} ^{\infty} {t}\frac{{cos}\left({t}\right)}{{e}^{\mathrm{2}\pi{t}} βˆ’\mathrm{1}}{dt}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\infty} {te}^{βˆ’\mathrm{2}\pi{nt}+{it}} +{te}^{βˆ’\mathrm{2}\pi{nt}βˆ’{it}} {dt} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}\pi{n}βˆ’{i}\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{2}\pi{n}+{i}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}\pi^{\mathrm{2}} }\left(\psi^{\mathrm{1}} \left(βˆ’\frac{{i}}{\mathrm{2}\pi}\right)+\psi^{\mathrm{1}} \left(\frac{{i}}{\mathrm{2}\pi}\right)\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}\pi^{\mathrm{2}} }\left(\pi^{\mathrm{2}} {csc}^{\mathrm{2}} \pi\left(\frac{{i}}{\mathrm{2}\pi}\right)\right)+\mathrm{1}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{2}{i}}{{e}^{\frac{βˆ’\mathrm{1}}{\mathrm{2}}} βˆ’{e}^{\frac{\mathrm{1}}{\mathrm{2}}} }\right)^{\mathrm{2}} =\mathrm{1}βˆ’\frac{{e}}{\left({e}βˆ’\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Commented by mnjuly1970 last updated on 27/Feb/21

bravo bravo mr payan ...excellent...

$${bravo}\:{bravo} \\ $$$${mr}\:{payan}\:...{excellent}... \\ $$

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