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Question Number 29844 by abdo imad last updated on 12/Feb/18
![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 ))) .](Q29844.png)
$${let}\:{give}\:{f}_{\alpha} \left({t}\right)={cos}\left(\alpha{t}\right)\:\:\mathrm{2}\pi\:{periodic}\:{with}\:{t}\:\in\left[−\pi,\pi\right]{and} \\ $$$$\alpha\in\:{R}−{Z} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}_{\alpha} \:\:{at}\:{fourier}\:{serie}\:{and}\:{prove}\:{that} \\ $$$${cotan}\left(\alpha\pi\right)=\:\frac{\mathrm{1}}{\alpha\pi}\:\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{2}\alpha}{\pi\left(\alpha^{\mathrm{2}} −{n}^{\mathrm{2}} \right)} \\ $$$$\left.\mathrm{2}\left.\right)\left.{let}\:{x}\in\right]\mathrm{0},\pi\left[\:{ant}\:{g}\left({t}\right)={cotant}\:−\frac{\mathrm{1}}{{t}}\:\:{if}\:{t}\in\right]\mathrm{0},{x}\right]{andg}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:{g}\:{is}\:{continue}\:{in}\left[\mathrm{0},{x}\right]\:{and}\:{find}\:\int_{\mathrm{0}} ^{{x}} {g}\left({t}\right){dt} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall\:{t}\in\left[\mathrm{0},{x}\right]\:{g}\left({t}\right)=\mathrm{2}{t}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{t}^{\mathrm{2}} −{n}^{\mathrm{2}} \pi^{\mathrm{24}} } \\ $$$$\left.\mathrm{4}\left.\right)\:{chow}\:{that}\:\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right)=\:\frac{{sinx}}{{x}}\:\:{and}\:{for}\:{x}\in\right]−\pi,\pi\left[\right. \\ $$$${sinx}={x}\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right)\:\:. \\ $$