There is Euler's formula: Gamma(z) Gamma(1-z) = pi/sin(pi*z) and Gauss's formula: sum{k=0 to n-1} Gamma(z + k/n) = (2*pi)^((n-1)/2) * n^(1/2 - n*z) * Gamma(n*z) for n a positive integer Apply the latter to z=1/10, z=3/10 to get Gamma(1/10) = sqrt(2) * sqrt(pi) * 2^(3/10) * Gamma(1/5)/ Gamma(3/5) and Gamma(3/10) = sqrt(2) * sqrt(pi) * 2^(-1/10) * Gamma(3/5) / Gamma(4/5) to get Gamma(1/10) * Gamma(3/10) = 2 * pi * 2^(1/10) * Gamma(1/5)/ Gamma(4/5) Plug that into the left hand side of Bill's formula to get (Gamma(1/5) * Gamma(4/5))^2 * sqrt(5)/(4*pi^2) Using Euler's formula, this becomes sqrt(5)/(4 * (sin(pi/5))^2). Finishing it off is now easy. Victor On Sun, Nov 5, 2017 at 7:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Out[344]= (2^(2/5) Sqrt[5] Gamma[1/5]^4)/(Gamma[1/10]^2 Gamma[3/10]^2) == GoldenRatio
In[345]:= N[%, 11111]
Out[345]= True --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun