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Question Number 35678 by abdo imad last updated on 21/May/18

l e t f (t) = \int_{0}^{\infty} \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} d x w i t h t > 0

1) s t u d y t h e e x i s t e n c t e o f f (t)

2) c a l c u l a t e f^{^{'}} (t)

3) f i n d a s i m p l e f o r m o f f (t) .

Commented by prof Abdo imad last updated on 23/May/18

1) f (t) = \int_{0}^{1} \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} d x + \int_{1}^{+ \infty} \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} d x

= I + J b u t w e h a v e \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} \sim e^{- {t x}^{2}} \Rightarrow

{l i m}_{x \to 0} \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} = 1 s o I c o n v e r g e s

a l s o w e h a v e {l i m}_{x \to + \infty} x^{2} \frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}} = 0

t h e c o m v e r g e n c e o f J i s a s s u r e d s o f (t) ? e x i s t s

f o r t > 0

2) f^{^{'}} (t) = \int_{0}^{\infty} \frac{\partial}{\partial t} {\frac{e^{- {t x}^{2}} a r c t a n (x^{2})}{x^{2}}} d x

= - \int_{0}^{\infty} e^{- {t x}^{2}} a r c t a n (x^{2}) d x .