Question Number 35608 by abdo mathsup 649 cc last updated on 21/May/18
$${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\:{from}\:{R}\right.\right. \\ $$$${F}\left({x},{r}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right){f}\left({t}\right)}{\mathrm{1}−\mathrm{2}{r}\:{cos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}} }{dt}\:\:{with} \\ $$$${f}\:\:\in\:{C}^{\mathrm{0}} \left({R}\right)\:\:\mathrm{2}\pi\:{periodic}\:\:{and}\:\:\mid\mid{f}\mid\mid={sup}_{{t}\in{R}} \mid{f}\left({t}\right)\mid \\ $$$$\:{prove}\:{that}\:{F}\left({x},{r}\right)=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\:\sum_{{n}=\mathrm{1}} ^{\infty} {r}^{{n}} \left({a}_{{n}} {cos}\left({nx}\right)\:+{b}_{{n}} {sin}\left({nx}\right)\right) \\ $$$${with}\:{a}_{{n}} =\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{f}\left({t}\right)\:{cos}\left({nt}\right){dt}\:{and} \\ $$$${b}_{{n}} =\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{f}\left({t}\right){sin}\left({nt}\right){dt} \\ $$