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Question Number 144597 by mathmax by abdo last updated on 26/Jun/21
let ϕ(x)=(1/(3+cosx)) developp f at fourier serie
$$\mathrm{let}\:\varphi\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cosx}} \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$
Answered by Olaf_Thorendsen last updated on 26/Jun/21
a_0 = (1/T)∫_(−(T/2)) ^(+(T/2)) f(x)dx a_0 = (1/(2π))∫_(−π) ^(+π) (dx/(3+cosx)) a_0 = (1/(2π))∫_(−π) ^(+π) (dx/(3+cosx)) Let t = tan(x/2) a_0 = (1/(2π))∫_(−∞) ^(+∞) (1/(3+((1−t^2 )/(1+t^2 )))).((2dt)/(1+t^2 )) a_0 = (1/π)∫_(−∞) ^(+∞) (dt/(3(1+t^2 )+(1−t^2 ))) a_0 = (1/π)∫_(−∞) ^(+∞) (dt/(2t^2 +4)) a_0 = (1/(2π))∫_(−∞) ^(+∞) (dt/(t^2 +2)) a_0 = (1/(2π))[(1/( (√2)))arctan(t/( (√2)))]_(−∞) ^(+∞) a_0 = (1/(2(√2)π))((π/2)) = (1/(4(√2))) a_n = (2/T)∫_(−(T/2)) ^(+(T/2)) f(x)cos(((2πnx)/T))dx a_n = (1/π)∫_(−π) ^(+π) (1/(3+cosx))cos(nx)dx a_(n+2) +a_n = (1/π)∫_(−π) ^(+π) ((cos((n+2)x)+cos(nx))/(3+cosx)) dx a_(n+2) +a_n = (1/π)∫_(−π) ^(+π) ((2cos((n+1)x).cosx)/(3+cosx)) dx a_(n+2) +a_n = (2/π)∫_(−π) ^(+π) cos((n+1)x) dx − (2/π)∫_(−π) ^(+π) ((3cos((n+1)x))/(3+cosx)) dx a_(n+2) +a_n = (2/π)[((sin((n+1)x))/(n+1))]_(−π) ^(+π) −6a_(n+1) a_(n+2) +6a_(n+1) +a_n = 0 (1) a_1 = (1/π)∫_(−π) ^(+π) ((cosx)/(3+cosx)) dx a_1 = (1/π)∫_(−π) ^(+π) (1−(3/(3+cosx))) dx a_1 = (1/π)∫_(−π) ^(+π) dx−6a_0 a_1 = 2−(6/(4(√2))) = 2−(3/(2(√2))) (1) : r^2 +6r+1 = 0 r = ((−6±(√(36−4)))/2) = −3±2(√2) a_n = λr_1 ^n +μr_2 ^n a_0 = λ+μ = (1/(4(√2))) a_1 = λr_1 +μr_2 = 2−(3/(2(√2))) λ = (((r_2 /(4(√2)))−(2−(3/(2(√2)))))/(r_2 −r_1 )) λ = ((((−3+2(√2))/(4(√2)))−(2−(3/(2(√2)))))/((−3+2(√2))−(−3−2(√(2))))) λ = (((3/(4(√2)))−(3/2))/( 4(√2))) = (3/(32))(1−2(√2)) μ = (((2−(3/(2(√2))))−(r_1 /(4(√2))))/(r_2 −r_1 )) μ = (((2−(3/(2(√2))))−((−3−2(√2))/(4(√2))))/((−3+2(√2))−(−3−2(√2)))) μ = (((5/2)−(3/( 4(√2))))/( 4(√2))) = (1/(32))(10(√2)−3) a_n = λr_1 ^n +μr_2 ^n a_n = (3/(32))(1−2(√2))(−3+2(√2))^n +(1/(32))(10(√2)−3)(−3+2(√2))^n b_n = 0 ∀n (f is even) f(x) = a_0 +Σ_(n=1) ^∞ a_n cos(((2πnx)/T)) f(x) = (1/(4(√2)))+Σ_(n=1) ^∞ a_n cos(nx) a_n = (3/(32))(1−2(√2))(−3+2(√2))^n +(1/(32))(10(√2)−3)(−3+2(√2))^n
$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{T}}\int_{−\frac{\mathrm{T}}{\mathrm{2}}} ^{+\frac{\mathrm{T}}{\mathrm{2}}} {f}\left({x}\right){dx} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\pi} ^{+\pi} \frac{{dx}}{\mathrm{3}+\mathrm{cos}{x}} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\pi} ^{+\pi} \frac{{dx}}{\mathrm{3}+\mathrm{cos}{x}} \\ $$$$\mathrm{Let}\:{t}\:=\:\mathrm{tan}\frac{{x}}{\mathrm{2}} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\infty} ^{+\infty} \frac{\mathrm{1}}{\mathrm{3}+\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}.\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\infty} ^{+\infty} \frac{{dt}}{\mathrm{3}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)+\left(\mathrm{1}−{t}^{\mathrm{2}} \right)} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\infty} ^{+\infty} \frac{{dt}}{\mathrm{2}{t}^{\mathrm{2}} +\mathrm{4}} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{−\infty} ^{+\infty} \frac{{dt}}{{t}^{\mathrm{2}} +\mathrm{2}} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\left[\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{arctan}\frac{{t}}{\:\sqrt{\mathrm{2}}}\right]_{−\infty} ^{+\infty} \\ $$$${a}_{\mathrm{0}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}\pi}\left(\frac{\pi}{\mathrm{2}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}} \\ $$$$ \\ $$$${a}_{{n}} \:=\:\frac{\mathrm{2}}{\mathrm{T}}\int_{−\frac{\mathrm{T}}{\mathrm{2}}} ^{+\frac{\mathrm{T}}{\mathrm{2}}} {f}\left({x}\right)\mathrm{cos}\left(\frac{\mathrm{2}\pi{nx}}{\mathrm{T}}\right){dx} \\ $$$${a}_{{n}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} \frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}{x}}\mathrm{cos}\left({nx}\right){dx} \\ $$$${a}_{{n}+\mathrm{2}} +{a}_{{n}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} \frac{\mathrm{cos}\left(\left({n}+\mathrm{2}\right){x}\right)+\mathrm{cos}\left({nx}\right)}{\mathrm{3}+\mathrm{cos}{x}}\:{dx} \\ $$$${a}_{{n}+\mathrm{2}} +{a}_{{n}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} \frac{\mathrm{2cos}\left(\left({n}+\mathrm{1}\right){x}\right).\mathrm{cos}{x}}{\mathrm{3}+\mathrm{cos}{x}}\:{dx} \\ $$$${a}_{{n}+\mathrm{2}} +{a}_{{n}} \:=\:\frac{\mathrm{2}}{\pi}\int_{−\pi} ^{+\pi} \mathrm{cos}\left(\left({n}+\mathrm{1}\right){x}\right)\:{dx} \\ $$$$−\:\frac{\mathrm{2}}{\pi}\int_{−\pi} ^{+\pi} \frac{\mathrm{3cos}\left(\left({n}+\mathrm{1}\right){x}\right)}{\mathrm{3}+\mathrm{cos}{x}}\:{dx} \\ $$$${a}_{{n}+\mathrm{2}} +{a}_{{n}} \:=\:\frac{\mathrm{2}}{\pi}\left[\frac{\mathrm{sin}\left(\left({n}+\mathrm{1}\right){x}\right)}{{n}+\mathrm{1}}\right]_{−\pi} ^{+\pi} −\mathrm{6}{a}_{{n}+\mathrm{1}} \\ $$$${a}_{{n}+\mathrm{2}} +\mathrm{6}{a}_{{n}+\mathrm{1}} +{a}_{{n}} \:=\:\mathrm{0}\:\:\:\left(\mathrm{1}\right) \\ $$$${a}_{\mathrm{1}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} \frac{\mathrm{cos}{x}}{\mathrm{3}+\mathrm{cos}{x}}\:{dx} \\ $$$${a}_{\mathrm{1}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} \left(\mathrm{1}−\frac{\mathrm{3}}{\mathrm{3}+\mathrm{cos}{x}}\right)\:{dx} \\ $$$${a}_{\mathrm{1}} \:=\:\frac{\mathrm{1}}{\pi}\int_{−\pi} ^{+\pi} {dx}−\mathrm{6}{a}_{\mathrm{0}} \\ $$$${a}_{\mathrm{1}} \:=\:\mathrm{2}−\frac{\mathrm{6}}{\mathrm{4}\sqrt{\mathrm{2}}}\:=\:\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$$$\left(\mathrm{1}\right)\::\:{r}^{\mathrm{2}} +\mathrm{6}{r}+\mathrm{1}\:=\:\mathrm{0} \\ $$$${r}\:=\:\frac{−\mathrm{6}\pm\sqrt{\mathrm{36}−\mathrm{4}}}{\mathrm{2}}\:=\:−\mathrm{3}\pm\mathrm{2}\sqrt{\mathrm{2}} \\ $$$${a}_{{n}} \:=\:\lambda{r}_{\mathrm{1}} ^{{n}} +\mu{r}_{\mathrm{2}} ^{{n}} \\ $$$${a}_{\mathrm{0}} \:=\:\lambda+\mu\:=\:\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}} \\ $$$${a}_{\mathrm{1}} \:=\:\lambda{r}_{\mathrm{1}} +\mu{r}_{\mathrm{2}} \:=\:\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$$$\lambda\:\:=\:\frac{\frac{{r}_{\mathrm{2}} }{\mathrm{4}\sqrt{\mathrm{2}}}−\left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}}\right)}{{r}_{\mathrm{2}} −{r}_{\mathrm{1}} } \\ $$$$\lambda\:\:=\:\frac{\frac{−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{4}\sqrt{\mathrm{2}}}−\left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}}\right)}{\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)−\left(−\mathrm{3}−\mathrm{2}\sqrt{\left.\mathrm{2}\right)}\right.} \\ $$$$\lambda\:\:=\:\frac{\frac{\mathrm{3}}{\mathrm{4}\sqrt{\mathrm{2}}}−\frac{\mathrm{3}}{\mathrm{2}}}{\:\mathrm{4}\sqrt{\mathrm{2}}}\:=\:\frac{\mathrm{3}}{\mathrm{32}}\left(\mathrm{1}−\mathrm{2}\sqrt{\mathrm{2}}\right) \\ $$$$\mu\:\:=\:\frac{\left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}}\right)−\frac{{r}_{\mathrm{1}} }{\mathrm{4}\sqrt{\mathrm{2}}}}{{r}_{\mathrm{2}} −{r}_{\mathrm{1}} } \\ $$$$\mu\:\:=\:\frac{\left(\mathrm{2}−\frac{\mathrm{3}}{\mathrm{2}\sqrt{\mathrm{2}}}\right)−\frac{−\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{4}\sqrt{\mathrm{2}}}}{\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)−\left(−\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)} \\ $$$$\mu\:\:=\:\frac{\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{3}}{\:\mathrm{4}\sqrt{\mathrm{2}}}}{\:\mathrm{4}\sqrt{\mathrm{2}}}\:=\:\frac{\mathrm{1}}{\mathrm{32}}\left(\mathrm{10}\sqrt{\mathrm{2}}−\mathrm{3}\right) \\ $$$${a}_{{n}} \:=\:\lambda{r}_{\mathrm{1}} ^{{n}} +\mu{r}_{\mathrm{2}} ^{{n}} \\ $$$${a}_{{n}} \:=\:\frac{\mathrm{3}}{\mathrm{32}}\left(\mathrm{1}−\mathrm{2}\sqrt{\mathrm{2}}\right)\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} +\frac{\mathrm{1}}{\mathrm{32}}\left(\mathrm{10}\sqrt{\mathrm{2}}−\mathrm{3}\right)\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} \\ $$$$ \\ $$$${b}_{{n}} \:=\:\mathrm{0}\:\forall{n}\:\left({f}\:\mathrm{is}\:\mathrm{even}\right) \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:{a}_{\mathrm{0}} +\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} \mathrm{cos}\left(\frac{\mathrm{2}\pi{nx}}{\mathrm{T}}\right) \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}}+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} \mathrm{cos}\left({nx}\right) \\ $$$${a}_{{n}} \:=\:\frac{\mathrm{3}}{\mathrm{32}}\left(\mathrm{1}−\mathrm{2}\sqrt{\mathrm{2}}\right)\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} +\frac{\mathrm{1}}{\mathrm{32}}\left(\mathrm{10}\sqrt{\mathrm{2}}−\mathrm{3}\right)\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} \\ $$$$ \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 27/Jun/21
thank you sir
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by mathmax by abdo last updated on 27/Jun/21
ϕ(x)=(1/(3+cosx))=(1/(3+((e^(ix) +e^(−ix) )/2)))=(2/(6+e^(ix) +e^(−ix) )) =_(e^(ix) =z) (2/(6+z+z^(−1) ))=((2z)/(6z+z^2 +1))=((2z)/(z^2 +6z +1))=w(z) Δ^′ =3^2 −1=8 ⇒z_1 =−3+2(√2) and z_2 =−3−2(√2) ⇒w(z)=((2z)/((z−z_1 )(z−z_2 )))=2z((1/(z−z_1 ))−(1/(z−z_2 ))).(1/(4(√2))) =(1/(2(√2)))((z/(z−z_1 ))−(z/(z−z_2 ))) ∣(z/z_1 )∣−1=(1/(3−2(√2)))−1=((1−3+2(√2))/(3−2(√2)))=((2(√2)−2)/(3−2(√2)))>0 ⇒∣(z/z_1 )∣>1 ∣(z/z_2 )∣−1=(1/(3+2(√2)))−1=((1−3−2(√2))/(3+2(√2)))<0 ⇒∣(z/z_2 )∣<1 ⇒ w(z)=(1/(2(√2)))((1/(1−(z_1 /z)))−(z/z_2 )(1/((z/z_2 )−1))) =(1/(2(√2)))(Σ_(n=0) ^∞ (z_1 ^n /z^n )+(z/z_2 )Σ_(n=0) ^∞ (z^n /z_2 ^n )) =(1/(2(√2)))(Σ_(n=0) ^∞ (−3+2(√2))^n e^(−inx) +(z/z_2 )Σ_(n=0) ^∞ (1/((−3−2(√2))^n ))e^(inx) ) =(1/(2(√2)))(Σ_(n=0) ^∞ (−1)^n (3−2(√2))^n e^(−inx) +Σ_(n=0) ^∞ (−1)^(n+1) (3−2(√2))^(n+1) e^(i(n+1)x) ) =(1/(2(√2)))(Σ_(n=0) ^∞ (−1)^n (3−2(√2))^n e^(−inx) +Σ_(n=1) ^∞ (−1)^n (3−2(√2))^n e^(inx) ) =(1/(2(√2)))(1+Σ_(n=1) ^∞ (−1)^n (3−2(√2))^n (2cos(nx))) w(z)=(1/(2(√2))) +(1/( (√2)))Σ_(n=1) ^∞ (−1)^n (3−2(√2))^n cos(nx)=ϕ(x)
$$\varphi\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cosx}}=\frac{\mathrm{1}}{\mathrm{3}+\frac{\mathrm{e}^{\mathrm{ix}} \:+\mathrm{e}^{−\mathrm{ix}} }{\mathrm{2}}}=\frac{\mathrm{2}}{\mathrm{6}+\mathrm{e}^{\mathrm{ix}} \:+\mathrm{e}^{−\mathrm{ix}} } \\ $$$$=_{\mathrm{e}^{\mathrm{ix}} \:=\mathrm{z}} \:\:\:\frac{\mathrm{2}}{\mathrm{6}+\mathrm{z}+\mathrm{z}^{−\mathrm{1}} }=\frac{\mathrm{2z}}{\mathrm{6z}+\mathrm{z}^{\mathrm{2}} \:+\mathrm{1}}=\frac{\mathrm{2z}}{\mathrm{z}^{\mathrm{2}} \:+\mathrm{6z}\:+\mathrm{1}}=\mathrm{w}\left(\mathrm{z}\right) \\ $$$$\Delta^{'} \:=\mathrm{3}^{\mathrm{2}} −\mathrm{1}=\mathrm{8}\:\Rightarrow\mathrm{z}_{\mathrm{1}} =−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\:\mathrm{and}\:\mathrm{z}_{\mathrm{2}} =−\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{w}\left(\mathrm{z}\right)=\frac{\mathrm{2z}}{\left(\mathrm{z}−\mathrm{z}_{\mathrm{1}} \right)\left(\mathrm{z}−\mathrm{z}_{\mathrm{2}} \right)}=\mathrm{2z}\left(\frac{\mathrm{1}}{\mathrm{z}−\mathrm{z}_{\mathrm{1}} }−\frac{\mathrm{1}}{\mathrm{z}−\mathrm{z}_{\mathrm{2}} }\right).\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\frac{\mathrm{z}}{\mathrm{z}−\mathrm{z}_{\mathrm{1}} }−\frac{\mathrm{z}}{\mathrm{z}−\mathrm{z}_{\mathrm{2}} }\right) \\ $$$$\mid\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{1}} }\mid−\mathrm{1}=\frac{\mathrm{1}}{\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}−\mathrm{1}=\frac{\mathrm{1}−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}=\frac{\mathrm{2}\sqrt{\mathrm{2}}−\mathrm{2}}{\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}>\mathrm{0}\:\Rightarrow\mid\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{1}} }\mid>\mathrm{1} \\ $$$$\mid\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }\mid−\mathrm{1}=\frac{\mathrm{1}}{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}−\mathrm{1}=\frac{\mathrm{1}−\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}<\mathrm{0}\:\Rightarrow\mid\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }\mid<\mathrm{1}\:\Rightarrow \\ $$$$\mathrm{w}\left(\mathrm{z}\right)=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{z}_{\mathrm{1}} }{\mathrm{z}}}−\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }\frac{\mathrm{1}}{\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }−\mathrm{1}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{z}_{\mathrm{1}} ^{\mathrm{n}} }{\mathrm{z}^{\mathrm{n}} }+\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{z}^{\mathrm{n}} }{\mathrm{z}_{\mathrm{2}} ^{\mathrm{n}} }\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \mathrm{e}^{−\mathrm{inx}} +\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{2}} }\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left(−\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} }\mathrm{e}^{\mathrm{inx}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{inx}} \:+\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} \left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}+\mathrm{1}} \mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{inx}} \:+\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \:\mathrm{e}^{\mathrm{inx}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\mathrm{1}+\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \left(\mathrm{2cos}\left(\mathrm{nx}\right)\right)\right) \\ $$$$\mathrm{w}\left(\mathrm{z}\right)=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{n}} \:\mathrm{cos}\left(\mathrm{nx}\right)=\varphi\left(\mathrm{x}\right) \\ $$

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