let give f_α (t)=cos(αt) 2π periodic with t ∈[−π,π]and
α∈ R−Z
1) developp f_α at fourier serie and prove that
cotan(απ)= (1/(απ)) +Σ_(n=1) ^∞ ((2α)/(π(α^2 −n^2 )))
2)let x∈]0,π[ ant g(t)=cotant −(1/t) if t∈]0,x]andg(0)=0
prove that g is continue in[0,x] and find ∫_0 ^x g(t)dt
3)prove that ∀ t∈[0,x] g(t)=2t Σ_(n=1) ^∞ (1/(t^2 −n^2 π^(24) ))
4) chow that Π_(n=1) ^∞ (1−(x^2 /(n^2 π^2 )))= ((sinx)/x) and for x∈]−π,π[
sinx=x Π_(n=1) ^∞ (1−(x^2 /(n^2 π^2 ))) .
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