let r∈[0,1[ and x from R F(x,r) = (1/(2π)) ∫_0 ^(2π) (((1−r^2 )f(t))/(1−2r cos(t−x) +r^2 ))dt with f ∈ C^0 (R) 2π periodic and ∣∣f∣∣=sup_(t∈R) ∣f(t)∣ prove that F(x,r)= (a_0 /2) + Σ_(n=1) ^∞ r^n (a_n cos(nx) +b_n sin(nx)) with a_n = (1/π) ∫_0 ^(2π) f(t) cos(nt)dt and b_n = (1/π) ∫


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Question Number 35608 by abdo mathsup 649 cc last updated on 21/May/18

$l e t r \in [0, 1 [a n d x f r o m R$ $F (x, r) = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - r^{2}) f (t)}{1 - 2 r c o s (t - x) + r^{2}} d t w i t h$ $f \in C^{0} (R) 2 π p e r i o d i c a n d ∣∣ f ∣∣= {s u p}_{t \in R} ∣ f (t) ∣$ $p r o v e t h a t F (x, r) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} r^{n} (a_{n} c o s (n x) + b_{n} s i n (n x))$ $w i t h a_{n} = \frac{1}{π} \int_{0}^{2 π} f (t) c o s (n t) d t a n d$ $b_{n} = \frac{1}{π} \int_{0}^{2 π} f (t) s i n (n t) d t$
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