let-r-0-1-and-x-from-R-F-x-r-1-2pi-0-2pi-1-r-2-f-t-1-2r-cos-t-x-r-2-dt-with-f-C-0-R-2pi-periodic-and-f-sup-t-R-f-t-prove-that-F-x-r-a-0-2-n-1-

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