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


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

$l e t r \in [0, 1 [a n d x \in R a n d$ $φ (x, r) = a r c t a n (\frac{r s i n x}{1 - r c o s x})$ $1) p r o v e t h a t \frac{\partial φ}{\partial x} (x, r) = \sum_{n = 1}^{\infty} r^{n} c o s (n x)$ $2) p r o v e t h a t φ (x, r) = \sum_{n = 1}^{\infty} r^{n} \frac{s i n (n x)}{n}$
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