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Thursday, September 29, 2016

Entry 27

Here are some Ramanujan-inspired formulas for pi using the complete elliptic integral of the first kind K(kd) and the golden ratio ϕ=1+521π=12K(k5)n=0(2n)!3n!61(26ϕ6)n 1π=12K(k15)n=0(2n)!3n!61(212ϕ8)n 1π=12K(k25)n=0(2n)!3n!61(26ϕ24)n The denominators having different powers of ϕ are the exact values of (η(τ)η(2τ))24 for τ=1+52,1+152,1+252, respectively. (Using d=35 would already need a sextic.) The last implies eπ2526ϕ2424.00004 And K(kn) can be expressed as a product of gamma functions as given in the link above. For example K(k5)=ϕ3/4Γ(120)Γ(320)Γ(720)Γ(920)410π=1.576390 which can also be numerically evaluated by Wolfram Alpha.

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