let-r-0-1-and-x-R-and-x-r-arctan-rsinx-1-r-cosx-1-prove-that-x-x-r-n-1-r-n-cos-nx-2-prove-that-x-r-n-1-r-n-sin-nx-n-

Question Number 35609 by abdo mathsup 649 cc last updated on 21/May/18

let r ∈[0,1[ and x∈ R and ϕ(x,r) = arctan( ((rsinx)/(1−r cosx))) 1) prove that (∂ϕ/∂x)(x,r) =Σ_(n=1) ^∞ r^n cos(nx) 2)prove that ϕ(x,r) = Σ_(n=1) ^∞ r^n ((sin(nx))/n)

$${let}\:{r}\:\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\in\:{R}\:\:{and}\:\right.\right. \\ $$$$\varphi\left({x},{r}\right)\:=\:{arctan}\left(\:\frac{{rsinx}}{\mathrm{1}−{r}\:{cosx}}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\frac{\partial\varphi}{\partial{x}}\left({x},{r}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:{cos}\left({nx}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\varphi\left({x},{r}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:\:\frac{{sin}\left({nx}\right)}{{n}} \\ $$$$ \\ $$

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let-r-0-1-and-x-R-and-x-r-arctan-rsinx-1-r-cosx-1-prove-that-x-x-r-n-1-r-n-cos-nx-2-prove-that-x-r-n-1-r-n-sin-nx-n-

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