let f(x) =∫_0 ^∞ ((cos(xt))/(x^2 +t^2 )) dt with x>0 1) find f(x) 2) find the values of ∫_0 ^∞ ((cos(t))/(1+t^2 ))dt and ∫_0 ^∞ ((cos(2t))/(4+t^2 ))dt 3) let U_n =∫_0 ^∞ ((cos(nt))/(n^2 +t^2 ))dt find lim_(n→+∞) U_n and study the convergenge of Σ U_n and Σ U


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Question Number 56311 by maxmathsup by imad last updated on 13/Mar/19

$l e t f (x) = \int_{0}^{\infty} \frac{c o s (x t)}{x^{2} + t^{2}} d t w i t h x > 0$ $1) f i n d f (x)$ $2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{c o s (t)}{1 + t^{2}} d t a n d \int_{0}^{\infty} \frac{c o s (2 t)}{4 + t^{2}} d t$ $3) l e t U_{n} = \int_{0}^{\infty} \frac{c o s (n t)}{n^{2} + t^{2}} d t f i n d {l i m}_{n \to + \infty} U_{n} a n d s t u d y t h e c o n v e r g e n g e o f$ $Σ U_{n} a n d Σ U_{n}^{2}$
Commented by maxmathsup by imad last updated on 15/Mar/19

$1) w e h a v e 2 f (x) = \int_{- \infty}^{+ \infty} \frac{c o s (x t)}{t^{2} + x^{2}} d t = R e (\int_{- \infty}^{+ \infty} \frac{e^{i x t}}{t^{2} + x^{2}} d t) l e t c o n s i d e r t h e$ $c o m p l e x f u n c t i o n φ (z) = \frac{e^{i x z}}{z^{2} + x^{2}} \Rightarrow φ (z) = \frac{e^{i x z}}{(z - i x) (z + i x)} s o t h e p o l e s o f φ a r e$ $\bar{+} i x r e s i d u s t h e o r e m g i v e \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i x)$ $R e s (φ, i x) = {l i m}_{z \to i x} (z - i x) φ (z) = \frac{e^{i x (i x)}}{2 i x} = \frac{e^{- x^{2}}}{2 i x} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{e^{- x^{2}}}{2 i x}$ $= \frac{π}{x} e^{- x^{2}} \Rightarrow ★ f (x) = \frac{π}{2 x} e^{- x^{2}} ★$ $2) \int_{0}^{\infty} \frac{c o s (t)}{1 + t^{2}} d t = f (1) = \frac{π}{2 e}$ $\int_{0}^{\infty} \frac{c o s (2 t)}{4 + t^{2}} = f (2) = \frac{π}{4} e^{- 4}$ $3) w e h a v e U_{n} = f (n) = \frac{π}{2 n} e^{- n^{2}} \Rightarrow {l i m}_{n \to + \infty} U_{n} = 0$ $w e h a v e U_{n} > 0 a n d U_{n} ⩽ \frac{π}{2} e^{- n^{2}} ⩽ \frac{π}{2} e^{- n} (n > 0)$ $\Rightarrow \sum_{n = 1}^{\infty} U_{n} ⩽ \frac{π}{2} \sum_{n = 1}^{\infty} e^{- n} a n d t h i s s e r i e c o n v e r g e s \Rightarrow Σ U_{n} i s c o n v e r g e n t e$ $w e h a v e U_{n}^{2} = \frac{π^{2}}{4 n^{2}} e^{- 2 n^{2}} \Rightarrow U_{n}^{2} ⩽ \frac{π^{2}}{4} e^{- 2 n} \Rightarrow \sum_{n = 1}^{\infty} U_{n}^{2} ⩽ \frac{π^{2}}{4} \sum_{n = 1}^{\infty} e^{- 2 n} a n d t h i s$ $s e r i e c o n v e r g e s \Rightarrow Σ U_{n}^{2} c o n v e r g e s .$
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