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Question-221583

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Question Number 221583 by MrGaster last updated on 08/Jun/25
Answered by MrGaster last updated on 08/Jun/25
=∫_0 ^1 t^(1/11−1) (1−t)^(9/11−1) β((( 1)/(11)),(9/(11)))Γ((1/(11)))Γ((9/(11)))(∫_0 ^1 s^(3/11−1) (1−s)^(5/11−1) β((3/(11)),(5/(11)))Γ((3/(11)))Γ((5/(11)))∫_0 ^1 u^4 (1−u)^0 β((4/(11)),1)Γ((4/(11)))du ds dt) =β((( 1)/(11)),(9/(11)))β((3/(11)),(5/(11)))β((4/(11)),1)Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11))) β((1/(11)),(9/(11)))=((Γ((1/(11)))Γ((9/(11))))/(Γ(((10)/(11))))),β((3/(11)),(5/(11)))=((Γ((3/(11)))Γ((5/(11))))/(Γ((8/(11))))),β((4/(11)),1)=((Γ((4/(11)))Γ(1))/(Γ(((15)/(11))))) Γ((5/(11)))=Γ(1+(4/(11)))=(4/(11))Γ((4/(11))),Γ(1)=1 β((4/(11)),1)=((Γ((4/(11))))/((4/(11))Γ((1/(11)))))=((11)/4) Π_(k=1) ^(10) Γ((k/(11)))=(((2π)^5 )/( (√(11)))) Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11)))=((Γ((1/(11)))((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11)))Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11))))/(Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11)))))=((((2π)^5 )/( (√(11))))/(Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11))))) Γ((2/(11)))=Γ(1−(9/(11)))=π/(sin(((9π)/(11)))Γ((9/(11)))),Γ((6/(11)))=Γ(1−(5/(11)))=π/(sin(((5π)/(11)))Γ((5/(11)))),Γ((7/(11)))=Γ(1−(4/(11)))=π/(sin(((4π)/(11)))Γ((4/(11)))),Γ((8/(11)))=Γ(1−(3/(11)))=π/(sin(((3π)/(11)))Γ((3/(11)))),Γ(((10)/(11)))=Γ(1−(1/(11)))=π/(sin((π/(11)))Γ((1/(11)))) Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11)))=(π^5 /(Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11)))sin((π/(11)))sin(((3π)/(11)))sin(((4π)/(11)))sin(((5π)/(11)))sin(((9π)/(11))))) sin(((9π)/(11)))=sin(π−((2π)/(11)))=sin(((2π)/(11))),sin(((5π)/(11)))=sin(π−((6π)/(11)))=sin(((6π)/(11)))sin(((7π)/(11)))sin(((8π)/(11)))sin((π/(11)))=((11)/2^5 ) Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11)))=(((2π)^5 )/( (√(11))))∙((Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ( ((9/(11)))sin((π/(11)))sin(((3π)/(11)))sin)(((4π)/(11)))sin(((5π)/(11)))sin(((9π)/(11))))/)∙((Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11))))/(Γ((2/(11)))Γ((6/(11)))Γ((7/(11)))Γ((8/(11)))Γ(((10)/(11))))) =(((2π)^5 )/( (√(11))))∙((11)/(2^5 π^6 ))=(((2π)^5 )/( (√(11))))∙((11)/(32π^5 ))=((2^5 π^5 )/( (√(11))))∙((11)/(32π^5 ))=((11)/( (√(11∙32))))∙32(√(11))=(√(11)) =(1/(6(√(2π^2 ))))Γ((1/(11)))Γ((3/(11)))Γ((4/(11)))Γ((5/(11)))Γ((9/(11)))
$$=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\mathrm{1}/\mathrm{11}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{\mathrm{9}/\mathrm{11}−\mathrm{1}} \beta\left(\frac{\:\mathrm{1}}{\mathrm{11}},\frac{\mathrm{9}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\left(\int_{\mathrm{0}} ^{\mathrm{1}} {s}^{\mathrm{3}/\mathrm{11}−\mathrm{1}} \left(\mathrm{1}−{s}\right)^{\mathrm{5}/\mathrm{11}−\mathrm{1}} \beta\left(\frac{\mathrm{3}}{\mathrm{11}},\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{\mathrm{4}} \left(\mathrm{1}−{u}\right)^{\mathrm{0}} \beta\left(\frac{\mathrm{4}}{\mathrm{11}},\mathrm{1}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right){du}\:{ds}\:{dt}\right) \\ $$$$=\beta\left(\frac{\:\mathrm{1}}{\mathrm{11}},\frac{\mathrm{9}}{\mathrm{11}}\right)\beta\left(\frac{\mathrm{3}}{\mathrm{11}},\frac{\mathrm{5}}{\mathrm{11}}\right)\beta\left(\frac{\mathrm{4}}{\mathrm{11}},\mathrm{1}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right) \\ $$$$\beta\left(\frac{\mathrm{1}}{\mathrm{11}},\frac{\mathrm{9}}{\mathrm{11}}\right)=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)}{\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)},\beta\left(\frac{\mathrm{3}}{\mathrm{11}},\frac{\mathrm{5}}{\mathrm{11}}\right)=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)}{\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)},\beta\left(\frac{\mathrm{4}}{\mathrm{11}},\mathrm{1}\right)=\frac{\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\mathrm{1}\right)}{\Gamma\left(\frac{\mathrm{15}}{\mathrm{11}}\right)} \\ $$$$\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{11}}\right)=\frac{\mathrm{4}}{\mathrm{11}}\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right),\Gamma\left(\mathrm{1}\right)=\mathrm{1} \\ $$$$\beta\left(\frac{\mathrm{4}}{\mathrm{11}},\mathrm{1}\right)=\frac{\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)}{\frac{\mathrm{4}}{\mathrm{11}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)}=\frac{\mathrm{11}}{\mathrm{4}} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{10}} {\prod}}\Gamma\left(\frac{{k}}{\mathrm{11}}\right)=\frac{\left(\mathrm{2}\pi\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}} \\ $$$$\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)}{\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)}=\frac{\frac{\left(\mathrm{2}\pi\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}}}{\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)} \\ $$$$\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}−\frac{\mathrm{9}}{\mathrm{11}}\right)=\pi/\left(\mathrm{sin}\left(\frac{\mathrm{9}\pi}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\right),\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}−\frac{\mathrm{5}}{\mathrm{11}}\right)=\pi/\left(\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\right),\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}−\frac{\mathrm{4}}{\mathrm{11}}\right)=\pi/\left(\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\right),\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}−\frac{\mathrm{3}}{\mathrm{11}}\right)=\pi/\left(\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\right),\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)=\Gamma\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{11}}\right)=\pi/\left(\mathrm{sin}\left(\frac{\pi}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\right) \\ $$$$\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)=\frac{\pi^{\mathrm{5}} }{\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{9}\pi}{\mathrm{11}}\right)} \\ $$$$\mathrm{sin}\left(\frac{\mathrm{9}\pi}{\mathrm{11}}\right)=\mathrm{sin}\left(\pi−\frac{\mathrm{2}\pi}{\mathrm{11}}\right)=\mathrm{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{11}}\right),\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{11}}\right)=\mathrm{sin}\left(\pi−\frac{\mathrm{6}\pi}{\mathrm{11}}\right)=\mathrm{sin}\left(\frac{\mathrm{6}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{7}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{8}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{11}}\right)=\frac{\mathrm{11}}{\mathrm{2}^{\mathrm{5}} } \\ $$$$\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)=\frac{\left(\mathrm{2}\pi\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}}\centerdot\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\:\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{11}}\right)\mathrm{sin}\right)\left(\frac{\mathrm{4}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{11}}\right)\mathrm{sin}\left(\frac{\mathrm{9}\pi}{\mathrm{11}}\right)}{}\centerdot\frac{\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)}{\Gamma\left(\frac{\mathrm{2}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{6}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{8}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{10}}{\mathrm{11}}\right)} \\ $$$$=\frac{\left(\mathrm{2}\pi\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}}\centerdot\frac{\mathrm{11}}{\mathrm{2}^{\mathrm{5}} \pi^{\mathrm{6}} }=\frac{\left(\mathrm{2}\pi\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}}\centerdot\frac{\mathrm{11}}{\mathrm{32}\pi^{\mathrm{5}} }=\frac{\mathrm{2}^{\mathrm{5}} \pi^{\mathrm{5}} }{\:\sqrt{\mathrm{11}}}\centerdot\frac{\mathrm{11}}{\mathrm{32}\pi^{\mathrm{5}} }=\frac{\mathrm{11}}{\:\sqrt{\mathrm{11}\centerdot\mathrm{32}}}\centerdot\mathrm{32}\sqrt{\mathrm{11}}=\sqrt{\mathrm{11}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}\sqrt{\mathrm{2}\pi^{\mathrm{2}} }}\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{4}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right) \\ $$

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