let-r-0-1-and-R-x-R-prove-that-1-1-2-n-1-r-n-cos-1-r-2-1-2r-cos-r-2-2-1-1-2pi-0-2pi-1-r-2-1-2rcos-t-x-r-2-dt-

Question Number 35605 by abdo mathsup 649 cc last updated on 21/May/18

let r∈[0,1[ and θ ∈ R,x∈ R prove that 1) 1+ 2 Σ_(n=1) ^(+∞) r^n cosθ = ((1−r^2 )/(1−2r cosθ +r^2 )) 2)1 =(1/(2π)) ∫_0 ^(2π) (((1−r^2 ))/(1−2rcos(t−x) +r^2 ))dt

$${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:\theta\:\in\:{R},{x}\in\:{R}\:{prove}\:{that}\right.\right. \\ $$$$\left.\mathrm{1}\right)\:\mathrm{1}+\:\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:{r}^{{n}} {cos}\theta\:=\:\frac{\mathrm{1}−{r}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{r}\:{cos}\theta\:+{r}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\mathrm{1}\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{2}{rcos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}} }{dt} \\ $$

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