Let be the integral equation satisfied by Pi(x)(adsbygoogle = window.adsbygoogle || []).push({});

LnR(s)/s=Int(0,8)K(x,y)Pi(x) with K(x,s)=1/x(x^2-1) then we split the integral into 2 ones

Int(0,8=Infinite)=Int(0,r)+Int(r,8) in the second one we make the change x=1/t, we will chose r so we have that after using Gaussian integration

LnR(s)/s=Sum(j)K(xj,s)Pi(xj)+K(xk,s)Pi(a) we have chosen r so this happen

then we take (sk) so we have the system of equations:

LnR(sk)/sk=Sum(j)K(xj,sk)Pi(xj)+CkK(a,s)Pi(a) now from this system we have only to solve the value of Pi(a) and this is valid whatever a is

that is whatever the integral is,using gaussian integration we can always obtain Int(a,b)f(x)=f(c)ck+sum(j)cjf(xj) only have to choose a d so

Int(a,b)=Int(a,d)+Int(d,b) in the firs integral after making the change of variable x=(d-a)xk+(d+a)/2=c so d=2c-(1+xk)a/1+xk

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# A trick to calculate Pi(a) for any a?

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