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Question Number 148249 by mnjuly1970 last updated on 26/Jul/21
Commented by Kamel last updated on 26/Jul/21
Ω_n =∫_0 ^(+∞) ((cos(πx))/((x^2 +1^2 )(x^2 +2^2 )....(x^2 +n^2 )))dx (1/((x^2 +1^2 )(x^2 +2^2 )....(x^2 +n^2 )))=Σ_(k=1) ^n (a_k /((x^2 +k^2 )))...(E) X=x^2 , (E)×(x^2 +k^2 ), X→−k^2 a_k =(1/((1^2 −k^2 )(2^2 −k^2 )...((k−1)^2 −k^2 )((k+1)^2 −k^2 )((k+2)^2 −k^2 )...(n^2 −k^2 ))) =(((−1)^(k−1) k!(2k))/((k−1)!(2k−1)!(n−k)!(2k+1)(2k+2)(2k+3)...(n+k))) =((2(−1)^(k−1) k^2 )/((n+k)!(n−k)!)) Ω_n =2Σ_(k=1) ^n ((k^2 (−1)^(k−1) )/((n+k)!(n−k)!))∫_0 ^(+∞) ((cos(πx))/(x^2 +k^2 ))dx ∫_0 ^(+∞) ((cos(πx))/(x^2 +k^2 ))dx=(π/(2k))e^(−kπ) ∴ Ω_n =πΣ_(k=1) ^n (((−1)^(k−1) ke^(−kπ) )/((n+k)!(n−k)!)) ∴ ∫_0 ^(+∞) ((cos(πx))/((x^2 +1^2 )(x^2 +2^2 )....(x^2 +n^2 )))dx=πΣ_(k=1) ^n (((−1)^(k−1) ke^(−kπ) )/((n+k)!(n−k)!))
$$ \\ $$$$\Omega_{{n}} =\int_{\mathrm{0}} ^{+\infty} \frac{{cos}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \right)….\left({x}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)}{dx} \\ $$$$\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \right)….\left({x}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)}=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{a}_{{k}} }{\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}…\left({E}\right) \\ $$$${X}={x}^{\mathrm{2}} ,\:\left({E}\right)×\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right),\:{X}\rightarrow−{k}^{\mathrm{2}} \\ $$$${a}_{{k}} =\frac{\mathrm{1}}{\left(\mathrm{1}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)\left(\mathrm{2}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)…\left(\left({k}−\mathrm{1}\right)^{\mathrm{2}} −{k}^{\mathrm{2}} \right)\left(\left({k}+\mathrm{1}\right)^{\mathrm{2}} −{k}^{\mathrm{2}} \right)\left(\left({k}+\mathrm{2}\right)^{\mathrm{2}} −{k}^{\mathrm{2}} \right)…\left({n}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)} \\ $$$$\:\:\:\:\:=\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} {k}!\left(\mathrm{2}{k}\right)}{\left({k}−\mathrm{1}\right)!\left(\mathrm{2}{k}−\mathrm{1}\right)!\left({n}−{k}\right)!\left(\mathrm{2}{k}+\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{2}\right)\left(\mathrm{2}{k}+\mathrm{3}\right)…\left({n}+{k}\right)} \\ $$$$\:\:\:\:\:=\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} {k}^{\mathrm{2}} }{\left({n}+{k}\right)!\left({n}−{k}\right)!} \\ $$$$\Omega_{{n}} =\mathrm{2}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}^{\mathrm{2}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{\left({n}+{k}\right)!\left({n}−{k}\right)!}\int_{\mathrm{0}} ^{+\infty} \frac{{cos}\left(\pi{x}\right)}{{x}^{\mathrm{2}} +{k}^{\mathrm{2}} }{dx} \\ $$$$\int_{\mathrm{0}} ^{+\infty} \frac{{cos}\left(\pi{x}\right)}{{x}^{\mathrm{2}} +{k}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}{k}}{e}^{−{k}\pi} \\ $$$$\therefore\:\Omega_{{n}} =\pi\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} {ke}^{−{k}\pi} }{\left({n}+{k}\right)!\left({n}−{k}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\int_{\mathrm{0}} ^{+\infty} \frac{{cos}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \right)….\left({x}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)}{dx}=\pi\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} {ke}^{−{k}\pi} }{\left({n}+{k}\right)!\left({n}−{k}\right)!} \\ $$
Answered by Olaf_Thorendsen last updated on 26/Jul/21
Ω = ∫_0 ^∞ ((cos(πx))/(Π_(k=1) ^n (x^2 +k^2 ))) dx Ω = (1/2)∫_(−∞) ^∞ ((cos(πx))/(Π_(k=1) ^n (x^2 +k^2 ))) dx Let f(x) = ((cos(πx))/(Π_(k=1) ^n (x^2 +k^2 ))) dx The function f is even : Ω = (1/2)Re∫_(−∞) ^∞ (e^(iπx) /(Π_(k=1) ^n (x^2 +k^2 ))) dx = (1/2)∫_(−∞) ^∞ (e^(iπx) /(Π_(k=1) ^n (x^2 +k^2 ))) dx Ω = 2iπΣ_(k=1) ^n Res(f(x),ik) Res(f(x),ik) = lim_(x→ik) ((e^(iπx) /((x+ik)Π_(p=1_(p≠k) ) ^n (x^2 +p^2 )))) Res(f(x),ik) = (e^(−kπ) /((2ik)Π_(p=1_(p≠k) ) ^n (p^2 −k^2 ))) Ω = πΣ_(k=1) ^n (e^(−kπ) /(kΠ_(p=1_(p≠k) ) ^n (p^2 −k^2 ))) ...to be continued...
$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\left(\pi{x}\right)}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\Omega\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\infty} \frac{\mathrm{cos}\left(\pi{x}\right)}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{cos}\left(\pi{x}\right)}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\mathrm{The}\:\mathrm{function}\:{f}\:\mathrm{is}\:\mathrm{even}\:: \\ $$$$\Omega\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{Re}\int_{−\infty} ^{\infty} \frac{{e}^{{i}\pi{x}} }{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\:{dx}\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\infty} \frac{{e}^{{i}\pi{x}} }{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\Omega\:=\:\mathrm{2}{i}\pi\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{Res}\left({f}\left({x}\right),{ik}\right) \\ $$$$\mathrm{Res}\left({f}\left({x}\right),{ik}\right)\:=\:\underset{{x}\rightarrow\mathrm{i}{k}} {\mathrm{lim}}\left(\frac{{e}^{{i}\pi{x}} }{\left({x}+{ik}\right)\underset{\underset{{p}\neq{k}} {{p}=\mathrm{1}}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +{p}^{\mathrm{2}} \right)}\right) \\ $$$$\mathrm{Res}\left({f}\left({x}\right),{ik}\right)\:=\:\frac{{e}^{−{k}\pi} }{\left(\mathrm{2}{ik}\right)\underset{\underset{{p}\neq{k}} {{p}=\mathrm{1}}} {\overset{{n}} {\prod}}\left({p}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)} \\ $$$$\Omega\:=\:\:\pi\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{e}^{−{k}\pi} }{{k}\underset{\underset{{p}\neq{k}} {{p}=\mathrm{1}}} {\overset{{n}} {\prod}}\left({p}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)} \\ $$$$…{to}\:{be}\:{continued}… \\ $$
Answered by mathmax by abdo last updated on 26/Jul/21
Ψ_n =∫_0 ^∞ ((cos(πx))/((x^2 +1^2 )(x^2 +2^2 ).....(x^2 +n^2 )))dx ⇒ 2Ψ_n =Re(∫_(−∞) ^(+∞) (e^(iπx) /((x^2 +1^2 )(x^2 +2^2 )....(x^2 +n^2 )))dx) let ϕ(z)=(e^(iπz) /((z^2 +1^2 )(z^2 +2^2 )....(z^2 +n^2 ))) ⇒ ϕ(z)=(e^(iπz) /((z−i)(z+i)(z+2i)(z−2i)....(z−ni)(z+ni))) the poles of ϕ are +^− ki rdsidus theorem give ∫_(−∞) ^(+∞) ϕ(z)dz =2iπΣ_(k=1) ^n Res(ϕ,ki) ϕ(z)=(e^(iπz) /((z−i)(z−2i)....(z−ni)(z+i)(z+2i)...(z+ni))) =(e^(iπz) /((z−i)(z−2i)....(z−(k−1)i)(z−ki)(z−(k+1)i)...(z−ni)Π_(k=1) ^n (z+ki))) ⇒Res(ϕ,ki)=(e^(iπ(ki)) /((ki−i)(ki−2i)....(i)(−i)...(ki−ni)Π_(k=1) ^n (2ki))) =(e^(−πk) /(i^(k−1) (k−1)!(−i)^(n−k) (n−k)!(2i)^n n!)) =(e^(−kπ) /((−1)^(n−k) i^(n−1) (i)^n 2^n (k−1)!(n−k)!n!)) =i(e^(−kπ) /((−1)^k 2^n (k−1)!(n−k)!n!)) ⇒ ∫_(−∞) ^(+∞) ϕ(z)dz =((2iπ)/(n!×2^n ))iΣ_(k=1) ^n (((−1)^k e^(−kπ) )/((k−1)!(n−k)!)) =−(π/(n! 2^(n−1) ))Σ_(k=1) ^n (((−1)^k e^(−kπ) )/((k−1)!(n−k)!)) =(π/(n!2^(n−1) ))Σ_(k=1) ^n (((−1)^(k−1) e^(−kπ) )/((k−1)!(n−k)!))=2Ψ_n ⇒ Ψ_n =(π/(2^n n!))Σ_(k=1) ^n (((−1)^(k−1) e^(−kπ) )/((k−1)!(n−k)!))
$$\Psi_{\mathrm{n}} \:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{cos}\left(\pi\mathrm{x}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \right)…..\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} \right)}\mathrm{dx}\:\Rightarrow \\ $$$$\mathrm{2}\Psi_{\mathrm{n}} =\mathrm{Re}\left(\int_{−\infty} ^{+\infty} \:\frac{\mathrm{e}^{\mathrm{i}\pi\mathrm{x}} }{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \right)….\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} \right)}\mathrm{dx}\right) \\ $$$$\mathrm{let}\:\varphi\left(\mathrm{z}\right)=\frac{\mathrm{e}^{\mathrm{i}\pi\mathrm{z}} }{\left(\mathrm{z}^{\mathrm{2}} \:+\mathrm{1}^{\mathrm{2}} \right)\left(\mathrm{z}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \right)….\left(\mathrm{z}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)}\:\Rightarrow \\ $$$$\varphi\left(\mathrm{z}\right)=\frac{\mathrm{e}^{\mathrm{i}\pi\mathrm{z}} }{\left(\mathrm{z}−\mathrm{i}\right)\left(\mathrm{z}+\mathrm{i}\right)\left(\mathrm{z}+\mathrm{2i}\right)\left(\mathrm{z}−\mathrm{2i}\right)….\left(\mathrm{z}−\mathrm{ni}\right)\left(\mathrm{z}+\mathrm{ni}\right)} \\ $$$$\mathrm{the}\:\mathrm{poles}\:\mathrm{of}\:\varphi\:\mathrm{are}\:\overset{−} {+}\mathrm{ki}\:\:\:\:\mathrm{rdsidus}\:\mathrm{theorem}\:\mathrm{give} \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left(\mathrm{z}\right)\mathrm{dz}\:=\mathrm{2i}\pi\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{Res}\left(\varphi,\mathrm{ki}\right) \\ $$$$\varphi\left(\mathrm{z}\right)=\frac{\mathrm{e}^{\mathrm{i}\pi\mathrm{z}} }{\left(\mathrm{z}−\mathrm{i}\right)\left(\mathrm{z}−\mathrm{2i}\right)….\left(\mathrm{z}−\mathrm{ni}\right)\left(\mathrm{z}+\mathrm{i}\right)\left(\mathrm{z}+\mathrm{2i}\right)…\left(\mathrm{z}+\mathrm{ni}\right)} \\ $$$$=\frac{\mathrm{e}^{\mathrm{i}\pi\mathrm{z}} }{\left(\mathrm{z}−\mathrm{i}\right)\left(\mathrm{z}−\mathrm{2i}\right)….\left(\mathrm{z}−\left(\mathrm{k}−\mathrm{1}\right)\mathrm{i}\right)\left(\mathrm{z}−\mathrm{ki}\right)\left(\mathrm{z}−\left(\mathrm{k}+\mathrm{1}\right)\mathrm{i}\right)…\left(\mathrm{z}−\mathrm{ni}\right)\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \left(\mathrm{z}+\mathrm{ki}\right)} \\ $$$$\Rightarrow\mathrm{Res}\left(\varphi,\mathrm{ki}\right)=\frac{\mathrm{e}^{\mathrm{i}\pi\left(\mathrm{ki}\right)} }{\left(\mathrm{ki}−\mathrm{i}\right)\left(\mathrm{ki}−\mathrm{2i}\right)….\left(\mathrm{i}\right)\left(−\mathrm{i}\right)…\left(\mathrm{ki}−\mathrm{ni}\right)\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \left(\mathrm{2ki}\right)} \\ $$$$=\frac{\mathrm{e}^{−\pi\mathrm{k}} }{\mathrm{i}^{\mathrm{k}−\mathrm{1}} \left(\mathrm{k}−\mathrm{1}\right)!\left(−\mathrm{i}\right)^{\mathrm{n}−\mathrm{k}} \left(\mathrm{n}−\mathrm{k}\right)!\left(\mathrm{2i}\right)^{\mathrm{n}} \:\mathrm{n}!} \\ $$$$=\frac{\mathrm{e}^{−\mathrm{k}\pi} }{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{k}} \mathrm{i}^{\mathrm{n}−\mathrm{1}} \left(\mathrm{i}\right)^{\mathrm{n}} \:\mathrm{2}^{\mathrm{n}} \left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!\mathrm{n}!} \\ $$$$=\mathrm{i}\frac{\mathrm{e}^{−\mathrm{k}\pi} }{\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{2}^{\mathrm{n}} \left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!\mathrm{n}!}\:\Rightarrow \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left(\mathrm{z}\right)\mathrm{dz}\:=\frac{\mathrm{2i}\pi}{\mathrm{n}!×\mathrm{2}^{\mathrm{n}} }\mathrm{i}\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{e}^{−\mathrm{k}\pi} }{\left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!} \\ $$$$=−\frac{\pi}{\mathrm{n}!\:\mathrm{2}^{\mathrm{n}−\mathrm{1}} }\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{e}^{−\mathrm{k}\pi} }{\left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!} \\ $$$$=\frac{\pi}{\mathrm{n}!\mathrm{2}^{\mathrm{n}−\mathrm{1}} }\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \:\mathrm{e}^{−\mathrm{k}\pi} }{\left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!}=\mathrm{2}\Psi_{\mathrm{n}} \:\Rightarrow \\ $$$$\Psi_{\mathrm{n}} =\frac{\pi}{\mathrm{2}^{\mathrm{n}} \mathrm{n}!}\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \:\mathrm{e}^{−\mathrm{k}\pi} }{\left(\mathrm{k}−\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!} \\ $$

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