Question Number 222937 by Nicholas666 last updated on 11/Jul/25
![Prove that the following identity holds : Σ_(λ∈Z[i]) ((1/((1 + 3λ)^3 ))) = ((π^(9/2) (√(1 + (√(3 )))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) Where Z[i] = {a + bi : a,b ∈ Z} denotes gaussian integers !](https://www.tinkutara.com/question/Q222937.png)
$$ \\ $$$$\:\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{identity}\:\mathrm{holds}\::\:\:\:\: \\ $$$$\:\:\:\underset{\lambda\in\mathbb{Z}\left[{i}\right]} {\sum}\:\left(\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:\mathrm{3}\lambda\right)^{\mathrm{3}} }\right)\:=\:\frac{\pi^{\mathrm{9}/\mathrm{2}} \:\sqrt{\mathrm{1}\:+\:\sqrt{\mathrm{3}\:}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \:\:\mathrm{3}^{\mathrm{27}/\mathrm{8}} \:\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \:}\:\:\:\:\: \\ $$$$\:\:\:\:\mathrm{Where}\:\mathbb{Z}\left[{i}\right]\:=\:\left\{{a}\:+\:{bi}\::\:{a},{b}\:\in\:\mathbb{Z}\right\}\:\mathrm{denotes}\:\mathrm{gaussian}\:\mathrm{integers}\:!\:\:\:\:\:\:\:\:\: \\ $$$$\: \\ $$
Answered by MrGaster last updated on 12/Jul/25
![Σ_(λ∈Z[i]) (1/((1 + 3λ)^3 ))=(1/(27))Σ_(λ∈Z[i]) (1/((λ+(1/3))^2 )) Σ_(λ∈Z[i]) (1/((λ+(1/3))^3 ))=−(1/2)℘′(−(1/3);Z[i]) ℘′(−(1/3);Z[i])=−℘′((1/3);Z[i]) ℘′((1/3);Z[i])=(√(4℘((1/3))^3 −g2℘((1/3))))(g_3 =0) g_2 =60G_4 (Z[i])=60 Σ_(λ∈Z[i]\{0}) (1/λ^4 )=((15Γ((1/4))^8 )/(16π^2 )) ℘((1/3))=((Γ((1/4))^4 )/(8(√2)π^(3/2) ))(√(1+(√3))) 4℘((1/3))^3 −g_2 ℘((1/3))=4(((Γ((1/4))^4 )/(8(√2)π^(3/2) ))(√(1+(√3))))^3 −(((15((1/4))^8 )/(16π)))(((Γ((1/4))^4 )/(8(√2)π^(3/2) ))(√(1+(√3)))) =((Γ((1/4))^(12) (√(1+(√3))))/(2048(√2)π^(7/2) ))(((4(1+(√3)))/π)−15) ((4(1+(√3)))/π)−15=((4+4(√3)−15π)/π) ℘′((1/3))=(√(((Γ((1/4))^(12) (√(1+(√3))))/(2048(√2)π^(7/2) ))∙((4+3(√3)−15π)/π)))=((Γ((1/6))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(23(2)^(1/4) π^(9/4) (√π))) Σ_(λ∈Z[i]) (1/((1+3λ)^3 ))=(1/(54))∙((Γ((1/4))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(32(2)^(1/3) π^(9/4) (√π)))=((Γ((1/4))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(1728(2)^(1/4) π^(11/4) )) Γ((3/4))=(((√2)π)/(Γ((1/4))))⇒Γ((3/4))^6 =((8π^6 )/(Γ((1/4))^6 )) ((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) ∙((8π^6 )/(Γ((1/4))^6 ))))=((Γ((1/4))^6 (√(1+(√3)))π^(9/2−6) )/(8∙2^(5/4) 3^(27/8) ))=((Γ((1/4))^6 (√(1+(√3))))/(8∙2^(5/4) 3^(27/8) π^(3/2) )) 8∙2^(5/4) =2^3 ∙2^(5/4) =2^(17/4) ((Γ((1/4))^6 (√(1+(√3))))/(2^(17/4) 3^(27/8) π^(3/2) ))=((Γ((1/4))^6 (√(1+(√3))))/(432∙2^(1/4) 3^(3/8) π^(3/2) )) ((Γ((1/4))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(1728(2)^(1/4) π^(11/4) ))=((Γ((1/4))^6 (√(1+(√3)))(√(4+4(√3)−15π)))/(1728(2)^(1/4) π^(11/4) (1+(√3))^(1/4) )) (√(4+4(√3)−15π))=(√(4(1+(√3))−15π)) 4(1+(√3))−15π=4+4(√3)−15π (√(4+4(√3)−15π))=(√(4(1+(√3))−15π))2(√(1+(√3)−((15π)/4))) ((Γ((1/4))^6 (√(1+(√3)))∙2(√(1+(√3)−((15π)/4))))/(1728(2)^(1/4) π^(11/4) (1+(√3))^(1/4) ))=((2Γ((1/4))^6 (√(1+(√3)))(√(1+(√3)−((15π)/4))))/(1728(2)^(1/4) π^(11/4) (1+(√3))^(1/4) )) =((2Γ((1/4))^6 (1+(√3))^(1/2) (√(1+(√3)−((15π)/4))))/(1728(2)^(1/4) π^(11/4) (1+(√3))^(1/4) ))=((2Γ((1/4))^6 (1+(√3))^(1/4) (√(1+(√3)−((15π)/4))))/(1728(2)^(1/4) π^(11/4) )) 1+(√3)−((15π)/4)=((4+4(√3)−15π)/4) (√(1+(√3)−((15π)/4)))=(1/2)(√(4+4(√3)−15π)) ((2Γ((1/4))^6 (1+(√3))^(1/4) ∙(1/2)(√(4+4(√3)−15π)))/(1728(2)^(1/4) π^(11/4) ))=((Γ((1/4))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(1728(2)^(1/4) π^(11/4) )) (2)^(1/4) =2^(1/4) ((Γ((1/4))^6 (1+(√3))^(1/4) (√(4+4(√3)−15π)))/(1728∙2^(1/4) π^(11/4) ))=((Γ((1/4))^6 (√(1+(√3)))(√(4+4(√3)−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(1/4) )) =((Γ((1/4))^6 (√(4+4(√3)−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(1/4) (√(1+(√3))))) (√(4+4(√3)−15π))=(√(4(1+(√3))−15π)) (1+(√3))^(1/4) (√(1+(√3)))=(1+(√3))^(3/4) ((Γ((1/4))^6 (√(4(1+(√3))−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) ))=((2Γ((1/4))^6 (√(1+(√3)((15π)/4))))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) )) 1+(√3)−((15π)/4)=((4+4(√3)−15π)/4) (√(1+(√3)−((15π)/4)))=(1/4)(√(4+4(√3)−15π)) ((2Γ((1/4))^6 ∙(1/2)(√(4+4(√3)−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) ))=((Γ((1/4))^6 (√(4+4(√3)−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) )) =((Γ((1/4))^6 (√(4(1+(√3))−15π)))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) ))=((2Γ((1/4))^6 (√(1+(√3)−((15π)/4))))/(1728∙2^(1/4) π^(11/4) (1+(√3))^(3/4) )) Γ((3/4))=(((√2)π)/(Γ((1/4))))⇒Γ((1/4))=(((√2)π)/(Γ((3/4)))) Γ((1/4))^6 =((8(√2)π^6 )/(Γ((3/4))^6 )) ((Γ((1/4))^6 (√(1+(√3))))/(2^(17/4) 3^(27/8) π^(3/2) ))=((8(√2)π^6 (√(1+(√3))))/(Γ((3/4))^6 2^(17/4) 3^(27/8) π^(3/2) ))=((8(√2)π^(6−3/2) (√(1+(√3))))/(Γ((3/4))^6 2^(17/4) 3^(27/8) )) 8=2^3 ,(√2)=2^(1/2) ,2^3 ∙2^(1/2) =2^(7/2) =2^(14/4) ((2^(14/4) π^(9/2) (√(1+(√3))))/(Γ((3/4))^6 2^(17/4) 3^(27/8) ))=((π^(9/2) (√(1+(√3))))/(Γ((3/4))^6 2^(3/4) 3^(27/8) )) 2^(3/4) =2^(6/8) =(2^6 )^(1/8) =64^(1/8) ,3^(27/8) =(3^(27) )^(1/8) =7625597484987^(1/8) ((π^(9/2) (√(1+(√3))))/(Γ((3/4))^6 2^(3/4) 3^(27/8) ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(2^(5/4) /2^(3/4) )=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙2^(1/2) (((√2)π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) (√2)(√(1+(√3)))=(√(2(1+(√3)))) (((√(2(1+(√3))))π^(9/2) )/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) 2(1+(√3))=2+2(√3) (√(2+2(√3)))=(√2)(√(1+(√3))) (((√2)(√(1+(√3)))π^(9/2) )/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4−1/2) 3^(27−8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(3/4) 3^(27/8) Γ((3/4))^6 )) 2^(3/4) =(2^3 )^(1/4) =8^(1/4) ,3^(27/8) =(3^(27) )^(1/8) ((π^(9/2) (√(1+(√3))))/(8^(1/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(2^(5/4) /8^(1/4) )=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(2^(5/4) /8^(1/4) )=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙2^(5/4−3/4) =((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙2^(1/2) =(((√2)π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) (√2)(√(1+(√3)))=(√(2+2(√3))) 2+2(√3)=2(1+(√3)) (√(2(1+(√3))))=(√2)(1+(√3)) (((√2)−(√(1+(√3)))π^(9/2) )/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4−1/2) 3^(27/8) Γ((3/2))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(3/4) 3^(27/8) Γ((3/4))^6 )) 2^(3/4) =2^(6/8) =64^(1/8) ((π^(9/2) (√(1+(√3))))/(64^(1/8) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/((64∙3^(27) )^(1/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/((64×7625597484987)^(1/8) Γ((3/4))^6 )) 64×7625597484987=488183418798368 ((π^(9/2) (√(1+(√3))))/(488183418798368^(1/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙((2^(5/4) 3^(27/8) )/(488183418798368^(1/8) )) 2^(5/4) 3^(27/8) =2^(10/8) 3^(27/8) =(2^(10) ∙3^(27) )=(1024×7625597484987)^(1/8) =7809032495822848^(1/8) 488183418798368^(1/8) =(488183418798368)^(1/8) 809032495822848/488183418798368=16 7809032495822848^(1/8) /488183418798368^(1/8) =16^(1/8) =2^(1/2) =(√2) ((2^(5/4) 3^(27/8) )/(488183418798368^(1/8) ))=(√2) ((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(√2)=(((√2)π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) (√2)(√(1+(√3)))=(√(2+2(√3))) (((√(2+2(√3)))π^(9/2) )/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(2(1+(√3)))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) (√(2(1+(√3))))=(√2)(√(1+(√3))) (((√2)π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4−1/2) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(3/4) 3^(27/8) Γ((3/4))^6 )) 2^(3/4) =(2^3 )^(1/4) =8^(1/2) ((π^(9/2) (√(1+(√3))))/(2^(3/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(2^(5/4) /8^(1/4) )=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙2^(5/4−3/4) =((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/6))^6 ))∙2^(1/2) =(((√2)π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 )) (√2)(√(1+(√3)))=(√(2+2(√3))) 2+2(√3)=2(1+(√3)) (√(2(1+(√3))))=(√2)(√(1+(√3))) (((√2)π^(9/2) (√(1−(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(√2)=((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) Γ((3/4))^6 ))∙(√2) Γ((3/4))=(((√2)π)/(Γ((1/4))))⇒Γ((3/4))^6 =((8π^6 )/(Γ((1/4))^6 )) ((π^(9/2) (√(1+(√3))))/(2^(5/4) 3^(27/8) ∙((8π^6 )/(Γ((1/4))^6 ))))=((Γ((1/4))^6 π^(9/2) (√(1+(√3))))/(8∙2^(5/4) 3^(27/8) π^6 ))=((Γ((1/4))^6 (√(1−(√3))))/(8∙2^(5/4) 3^(27/8) π^(3/2) )) 8∙2^(5/4) =2^3 ∙5^(5/4) =2^(17/4) ((Γ((1/4))^6 (√(1+(√3))))/(2^(17/4) 3^(27/8) π^(3/2) ))=((Γ((1/4))^6 (√(1+(√3))))/(8∙2^(5/4) 3^(27/8) π^(3/2) )) 8∙2^(5/4) =2^3 ∙2^(5/4) =2^(17/4) ((Γ((1/4))^6 (√(1−(√3))))/(2^(17/4) 3^(27/8) π^(3/2) ))=((Γ((1/4))^6 (√(1+(√3))))/(432∙2^(1/4) 3^(3/8) π^(3/2) )) 3^(3/8) =(3^3 )^(1/8) =27^(1/8) ((Γ((1/4))^6 (√(1+(√3))))/(432∙2^(1/4) 27^(1/8) π^(3/2) ))=((Γ((1/4))^6 (√(1+(√3))))/(432∙2^(1/4) ∙3^(3/8) π^(3/2) )) ((π^(9/2) (√(1+(√3))))/(2^(5/4) ∙3^((27)/8) Γ((3/4))^6 ))=((Γ((1/6))^6 (√(1+(√3))))/(432∙2^(1/4) 3^(3/8) π^(3×2) ))](https://www.tinkutara.com/question/Q222966.png)
$$\:\:\:\underset{\lambda\in\mathbb{Z}\left[{i}\right]} {\sum}\:\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:\mathrm{3}\lambda\right)^{\mathrm{3}} }=\frac{\mathrm{1}}{\mathrm{27}}\underset{\lambda\in\mathbb{Z}\left[{i}\right]} {\sum}\frac{\mathrm{1}}{\left(\lambda+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} } \\ $$$$\underset{\lambda\in\mathbb{Z}\left[{i}\right]} {\sum}\frac{\mathrm{1}}{\left(\lambda+\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} }=−\frac{\mathrm{1}}{\mathrm{2}}\wp'\left(−\frac{\mathrm{1}}{\mathrm{3}};\mathbb{Z}\left[{i}\right]\right) \\ $$$$\wp'\left(−\frac{\mathrm{1}}{\mathrm{3}};\mathbb{Z}\left[{i}\right]\right)=−\wp'\left(\frac{\mathrm{1}}{\mathrm{3}};\mathbb{Z}\left[{i}\right]\right) \\ $$$$\wp'\left(\frac{\mathrm{1}}{\mathrm{3}};\mathbb{Z}\left[{i}\right]\right)=\sqrt{\mathrm{4}\wp\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} −{g}\mathrm{2}\wp\left(\frac{\mathrm{1}}{\mathrm{3}}\right)}\left({g}_{\mathrm{3}} =\mathrm{0}\right) \\ $$$${g}_{\mathrm{2}} =\mathrm{60}{G}_{\mathrm{4}} \left(\mathbb{Z}\left[{i}\right]\right)=\mathrm{60}\:\underset{\lambda\in\mathbb{Z}\left[{i}\right]\backslash\left\{\mathrm{0}\right\}} {\sum}\frac{\mathrm{1}}{\lambda^{\mathrm{4}} }=\frac{\mathrm{15}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{8}} }{\mathrm{16}\pi^{\mathrm{2}} } \\ $$$$\wp\left(\frac{\mathrm{1}}{\mathrm{3}}\right)=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} }{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{3}/\mathrm{2}} }\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}} \\ $$$$\mathrm{4}\wp\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} −{g}_{\mathrm{2}} \wp\left(\frac{\mathrm{1}}{\mathrm{3}}\right)=\mathrm{4}\left(\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} }{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{3}/\mathrm{2}} }\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\right)^{\mathrm{3}} −\left(\frac{\mathrm{15}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{8}} }{\mathrm{16}\pi}\right)\left(\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} }{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{3}/\mathrm{2}} }\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\right) \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{12}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2048}\sqrt{\mathrm{2}}\pi^{\mathrm{7}/\mathrm{2}} }\left(\frac{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}{\pi}−\mathrm{15}\right) \\ $$$$\frac{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}{\pi}−\mathrm{15}=\frac{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}{\pi} \\ $$$$\wp'\left(\frac{\mathrm{1}}{\mathrm{3}}\right)=\sqrt{\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{12}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2048}\sqrt{\mathrm{2}}\pi^{\mathrm{7}/\mathrm{2}} }\centerdot\frac{\mathrm{4}+\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{15}\pi}{\pi}}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{23}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{4}} \sqrt{\pi}} \\ $$$$\underset{\lambda\in\mathbb{Z}\left[{i}\right]} {\sum}\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{3}\lambda\right)^{\mathrm{3}} }=\frac{\mathrm{1}}{\mathrm{54}}\centerdot\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{32}\sqrt[{\mathrm{3}}]{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{4}} \sqrt{\pi}}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} } \\ $$$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)=\frac{\sqrt{\mathrm{2}}\pi}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\Rightarrow\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} =\frac{\mathrm{8}\pi^{\mathrm{6}} }{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \centerdot\frac{\mathrm{8}\pi^{\mathrm{6}} }{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} }}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\pi^{\mathrm{9}/\mathrm{2}−\mathrm{6}} }{\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{17}/\mathrm{4}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{432}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \mathrm{3}^{\mathrm{3}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} } \\ $$$$\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}=\sqrt{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi} \\ $$$$\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi=\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi \\ $$$$\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}=\sqrt{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi}\mathrm{2}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\centerdot\mathrm{2}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} }=\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} } \\ $$$$=\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} }=\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} } \\ $$$$\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}=\frac{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}{\mathrm{4}} \\ $$$$\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi} \\ $$$$\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \centerdot\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\sqrt[{\mathrm{4}}]{\mathrm{2}}\pi^{\mathrm{11}/\mathrm{4}} } \\ $$$$\sqrt[{\mathrm{4}}]{\mathrm{2}}=\mathrm{2}^{\mathrm{1}/\mathrm{4}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} } \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}} \\ $$$$\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}=\sqrt{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi} \\ $$$$\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{4}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}=\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} }=\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} } \\ $$$$\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}=\frac{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}{\mathrm{4}} \\ $$$$\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi} \\ $$$$\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \centerdot\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} } \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)−\mathrm{15}\pi}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} }=\frac{\mathrm{2}\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}−\frac{\mathrm{15}\pi}{\mathrm{4}}}}{\mathrm{1728}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \pi^{\mathrm{11}/\mathrm{4}} \left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{3}/\mathrm{4}} } \\ $$$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)=\frac{\sqrt{\mathrm{2}}\pi}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\Rightarrow\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)=\frac{\sqrt{\mathrm{2}}\pi}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)} \\ $$$$\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} =\frac{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{6}} }{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\mathrm{8}\sqrt{\mathrm{2}}\pi^{\mathrm{6}−\mathrm{3}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} } \\ $$$$\mathrm{8}=\mathrm{2}^{\mathrm{3}} ,\sqrt{\mathrm{2}}=\mathrm{2}^{\mathrm{1}/\mathrm{2}} ,\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{2}^{\mathrm{1}/\mathrm{2}} =\mathrm{2}^{\mathrm{7}/\mathrm{2}} =\mathrm{2}^{\mathrm{14}/\mathrm{4}} \\ $$$$\frac{\mathrm{2}^{\mathrm{14}/\mathrm{4}} \pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} } \\ $$$$\mathrm{2}^{\mathrm{3}/\mathrm{4}} =\mathrm{2}^{\mathrm{6}/\mathrm{8}} =\left(\mathrm{2}^{\mathrm{6}} \right)^{\mathrm{1}/\mathrm{8}} =\mathrm{64}^{\mathrm{1}/\mathrm{8}} ,\mathrm{3}^{\mathrm{27}/\mathrm{8}} =\left(\mathrm{3}^{\mathrm{27}} \right)^{\mathrm{1}/\mathrm{8}} =\mathrm{7625597484987}^{\mathrm{1}/\mathrm{8}} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} \mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} }{\mathrm{2}^{\mathrm{3}/\mathrm{4}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{2}} \frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}=\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)} \\ $$$$\frac{\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}\pi^{\mathrm{9}/\mathrm{2}} }{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)=\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}} \\ $$$$\sqrt{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}}=\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}} \\ $$$$\frac{\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\pi^{\mathrm{9}/\mathrm{2}} }{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}−\mathrm{1}/\mathrm{2}} \mathrm{3}^{\mathrm{27}−\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\mathrm{2}^{\mathrm{3}/\mathrm{4}} =\left(\mathrm{2}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{4}} =\mathrm{8}^{\mathrm{1}/\mathrm{4}} ,\mathrm{3}^{\mathrm{27}/\mathrm{8}} =\left(\mathrm{3}^{\mathrm{27}} \right)^{\mathrm{1}/\mathrm{8}} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{8}^{\mathrm{1}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} }{\mathrm{8}^{\mathrm{1}/\mathrm{4}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} }{\mathrm{8}^{\mathrm{1}/\mathrm{4}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}−\mathrm{3}/\mathrm{4}} =\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{2}} =\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}=\sqrt{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}=\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}=\sqrt{\mathrm{2}}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\frac{\sqrt{\mathrm{2}}−\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}\pi^{\mathrm{9}/\mathrm{2}} }{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}−\mathrm{1}/\mathrm{2}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\mathrm{2}^{\mathrm{3}/\mathrm{4}} =\mathrm{2}^{\mathrm{6}/\mathrm{8}} =\mathrm{64}^{\mathrm{1}/\mathrm{8}} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{64}^{\mathrm{1}/\mathrm{8}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\left(\mathrm{64}\centerdot\mathrm{3}^{\mathrm{27}} \right)^{\mathrm{1}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\left(\mathrm{64}×\mathrm{7625597484987}\right)^{\mathrm{1}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\mathrm{64}×\mathrm{7625597484987}=\mathrm{488183418798368} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{488183418798368}^{\mathrm{1}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} }{\mathrm{488183418798368}^{\mathrm{1}/\mathrm{8}} } \\ $$$$\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} =\mathrm{2}^{\mathrm{10}/\mathrm{8}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} =\left(\mathrm{2}^{\mathrm{10}} \centerdot\mathrm{3}^{\mathrm{27}} \right)=\left(\mathrm{1024}×\mathrm{7625597484987}\right)^{\mathrm{1}/\mathrm{8}} =\mathrm{7809032495822848}^{\mathrm{1}/\mathrm{8}} \\ $$$$\mathrm{488183418798368}^{\mathrm{1}/\mathrm{8}} =\left(\mathrm{488183418798368}\right)^{\mathrm{1}/\mathrm{8}} \\ $$$$\mathrm{809032495822848}/\mathrm{488183418798368}=\mathrm{16} \\ $$$$\mathrm{7809032495822848}^{\mathrm{1}/\mathrm{8}} /\mathrm{488183418798368}^{\mathrm{1}/\mathrm{8}} =\mathrm{16}^{\mathrm{1}/\mathrm{8}} =\mathrm{2}^{\mathrm{1}/\mathrm{2}} =\sqrt{\mathrm{2}} \\ $$$$\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} }{\mathrm{488183418798368}^{\mathrm{1}/\mathrm{8}} }=\sqrt{\mathrm{2}} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\sqrt{\mathrm{2}}=\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}=\sqrt{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\frac{\sqrt{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}}\pi^{\mathrm{9}/\mathrm{2}} }{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}=\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}} \\ $$$$\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}−\mathrm{1}/\mathrm{2}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\mathrm{2}^{\mathrm{3}/\mathrm{4}} =\left(\mathrm{2}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{4}} =\mathrm{8}^{\mathrm{1}/\mathrm{2}} \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{3}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\frac{\mathrm{2}^{\mathrm{5}/\mathrm{4}} }{\mathrm{8}^{\mathrm{1}/\mathrm{4}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}−\mathrm{3}/\mathrm{4}} =\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{6}}\right)^{\mathrm{6}} }\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{2}} =\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}=\sqrt{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\mathrm{2}+\mathrm{2}\sqrt{\mathrm{3}}=\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\sqrt{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)}=\sqrt{\mathrm{2}}\sqrt{\mathrm{1}+\sqrt{\mathrm{3}}} \\ $$$$\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}−\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\sqrt{\mathrm{2}}=\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }\centerdot\sqrt{\mathrm{2}} \\ $$$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)=\frac{\sqrt{\mathrm{2}}\pi}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\Rightarrow\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} =\frac{\mathrm{8}\pi^{\mathrm{6}} }{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} } \\ $$$$\frac{\pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \centerdot\frac{\mathrm{8}\pi^{\mathrm{6}} }{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} }}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \pi^{\mathrm{9}/\mathrm{2}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{6}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}−\sqrt{\mathrm{3}}}}{\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{5}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{17}/\mathrm{4}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\mathrm{8}\centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{2}^{\mathrm{5}/\mathrm{4}} =\mathrm{2}^{\mathrm{17}/\mathrm{4}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}−\sqrt{\mathrm{3}}}}{\mathrm{2}^{\mathrm{17}/\mathrm{4}} \mathrm{3}^{\mathrm{27}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{432}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \mathrm{3}^{\mathrm{3}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\mathrm{3}^{\mathrm{3}/\mathrm{8}} =\left(\mathrm{3}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{8}} =\mathrm{27}^{\mathrm{1}/\mathrm{8}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{432}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \mathrm{27}^{\mathrm{1}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{432}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \centerdot\mathrm{3}^{\mathrm{3}/\mathrm{8}} \pi^{\mathrm{3}/\mathrm{2}} } \\ $$$$\frac{\pi^{\frac{\mathrm{9}}{\mathrm{2}}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{2}^{\frac{\mathrm{5}}{\mathrm{4}}} \centerdot\mathrm{3}^{\frac{\mathrm{27}}{\mathrm{8}}} \Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{6}} }=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)^{\mathrm{6}} \sqrt{\mathrm{1}+\sqrt{\mathrm{3}}}}{\mathrm{432}\centerdot\mathrm{2}^{\mathrm{1}/\mathrm{4}} \mathrm{3}^{\mathrm{3}/\mathrm{8}} \pi^{\mathrm{3}×\mathrm{2}} } \\ $$
Commented by gabthemathguy25 last updated on 12/Jul/25
$${Bravo}! \\ $$