Question Number 38204 by prof Abdo imad last updated on 22/Jun/18
$${if}\:\:\frac{\mathrm{1}}{{cosx}}\:=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {cos}\left({nx}\right) \\ $$$${calculate}\:{a}_{\mathrm{0}} \:{and}\:{a}_{{n}} \\ $$
Commented by math khazana by abdo last updated on 26/Jun/18
$${let}?{developp}\:{at}\:{fourier}?{serie}\:{tbe}?{function} \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}}\:\:\left(\mathrm{2}\pi\:{periodic}\:{even}\right) \\ $$$${f}\left({x}\right)=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} {a}_{{n}} \:{cos}\left({nx}\right) \\ $$$${a}_{{n}} =\frac{\mathrm{2}}{{T}}\:\int_{\left[{T}\right]} {f}\left({x}\right){cos}\left({nx}\right){dx}=\frac{\mathrm{2}}{\mathrm{2}\pi}\:\int_{−\pi} ^{\pi} \:\frac{{cos}\left({nx}\right)}{{cosx}}{dx}\:\Rightarrow \\ $$$$\pi\:{a}_{{n}} =\:\int_{−\pi} ^{\pi} \:\:\frac{{cos}\left({nx}\right)}{{cosx}}{dx}={Re}\left(\:\int_{−\pi} ^{\pi} \:\frac{{e}^{{inx}} }{{cosx}}{dx}\right) \\ $$$${changement}\:{e}^{{ix}} ={z}\:{give} \\ $$$$\int_{−\pi} ^{\pi} \:\:\frac{{e}^{{inx}} }{{cosx}}{dx}\:=\int_{\mid{z}\mid=\mathrm{1}} \:\:\:\mathrm{2}\frac{{z}^{{n}} }{{z}+{z}^{−\mathrm{1}} }\:\frac{{dz}}{{iz}} \\ $$$$=\:\int_{\mid{z}\mid=\mathrm{1}} \:\frac{−\mathrm{2}{iz}^{{n}} }{{z}^{\mathrm{2}} \:+\mathrm{1}}{dz}\:{let}\:\varphi\left({z}\right)=\:\:\frac{−\mathrm{2}{iz}^{{n}} }{{z}^{\mathrm{2}} \:+\mathrm{1}}\:{the}\:{poles}\:{of} \\ $$$$\varphi\:{are}\:{i}\:{and}\:−{i}\:,{residus}\:{theorem}?{give} \\ $$$$\int_{\mid{z}\mid=\mathrm{1}} \:\:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:{Res}\left(\varphi,{i}\right) \\ $$$$=\mathrm{2}{i}\pi\:\:\frac{−\mathrm{2}{i}.{i}^{{n}} }{\mathrm{2}{i}}\:=−\mathrm{2}{i}\pi{i}^{{n}} \:=−\mathrm{2}\pi\:{i}^{{n}+\mathrm{1}} \:\Rightarrow \\ $$$$\pi\:{a}_{\mathrm{2}{n}} \:={Re}\left(\:−\mathrm{2}\pi\:{i}^{\mathrm{2}{n}+\mathrm{1}} \right)={Re}\left(\:−\mathrm{2}\pi{i}\left(−\mathrm{1}\right)^{{n}} \right)=\mathrm{0} \\ $$$$\pi\:{a}_{\mathrm{2}{n}+\mathrm{1}} ={Re}\left(−\mathrm{2}\pi{i}^{\mathrm{2}{n}+\mathrm{2}} \right)={Re}\left(\:\mathrm{2}\pi\left(−\mathrm{1}\right)^{{n}} \right) \\ $$$$=\mathrm{2}\pi\left(−\mathrm{1}\right)^{{n}} \:\Rightarrow\:{a}_{\mathrm{2}{n}+\mathrm{1}} =\:\mathrm{2}\left(−\mathrm{1}\right)^{{n}} \:\Rightarrow{a}_{\mathrm{0}} =\mathrm{0}\:{and} \\ $$$$\frac{\mathrm{1}}{{cosx}}\:=\:\mathrm{2}\sum_{{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{{n}} \:{cos}\left(\mathrm{2}{n}+\mathrm{1}\right){x} \\ $$$$ \\ $$$$ \\ $$