1-calculate-f-x-y-0-e-xt-cos-yt-t-dt-and-g-x-y-0-e-xt-sin-yt-t-dt-with-x-gt-0-and-y-gt-0-2-find-the-values-of-0-e-2t-cos-t-t-dt-and-0-

Question Number 62335 by maxmathsup by imad last updated on 19/Jun/19

1) c a l c u l a t e f (x, y) = \int_{0}^{\infty} \frac{e^{- x t} c o s (y t)}{\sqrt{t}} d t a n d g (x, y) = \int_{0}^{\infty} \frac{e^{- x t} s i n (y t)}{\sqrt{t}} d t

w i t h x > 0 a n d y > 0

2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{e^{- 2 t} c o s (t)}{\sqrt{t}} d t a n d \int_{0}^{\infty} \frac{e^{- t} c o s (2 t)}{\sqrt{t}} d t

Commented by maxmathsup by imad last updated on 20/Jun/19

1) w e h a v e f (x, y) + i g (x, y) = \int_{0}^{\infty} \frac{e^{- x t}}{\sqrt{t}} e^{i y t} d t = \int_{0}^{\infty} \frac{e^{- (x + i y) t}}{\sqrt{t}} d t

c h a n g e m e n t \sqrt{x + i y} \sqrt{t} = u g i v e f (x, y) + i g (x, y) = \sqrt{x + i y} \int_{0}^{\infty} \frac{e^{- u^{2}}}{u}

\sqrt{t} = \frac{u}{\sqrt{x + i y}} \Rightarrow t = \frac{u^{2}}{x + i y} \Rightarrow d t = \frac{2 u d u}{x + i y} \Rightarrow f (x, y) + i g (x, y) = \sqrt{x + i y} \int_{0}^{\infty} \frac{e^{- u^{2}}}{u} \frac{2 u d u}{x + i y}

= \frac{2}{\sqrt{x + i y}} \int_{0}^{\infty} e^{- u^{2}} d u = \frac{2}{\sqrt{x + i y}} \frac{\sqrt{π}}{2} = \frac{\sqrt{π}}{\sqrt{x + i y}}

w e h a v e x + i y = \sqrt{x^{2} + y^{2}} {\frac{x}{\sqrt{x^{2} + y^{2}}} + i \frac{y}{\sqrt{x^{2} + y^{2}}}} = r e^{i θ} \Rightarrow r = \sqrt{x^{2} + y^{2}}

c o s θ = \frac{x}{\sqrt{x^{2} + y^{2}}} a n d s i n θ = \frac{y}{\sqrt{x^{2} + y^{2}}} \Rightarrow t a n θ = \frac{y}{x} \Rightarrow

θ = a r c t a n (\frac{y}{x}) \Rightarrow x + i y = {(x^{2} + y^{2})}^{\frac{1}{2}} e^{i a r c t a n (\frac{y}{x})} \Rightarrow \sqrt{x + i y} = {(x^{2} + y^{2})}^{\frac{1}{4}} e^{\frac{i}{2} a r c t a n (\frac{y}{x})} \Rightarrow

f (x, y) = {(x^{2} + y^{2})}^{\frac{1}{4}} c o s (\frac{1}{2} a r c t a n (\frac{y}{x})) a n d g (x, y) = {(x^{2} + y^{2})}^{\frac{1}{4}} s i n (\frac{1}{2} a r c t a n (\frac{y}{x}))

Commented by maxmathsup by imad last updated on 20/Jun/19

2) \int_{0}^{\infty} \frac{e^{- 2 t} c o s (t)}{\sqrt{t}} d t = f (2, 1) =^{4} \sqrt{5} c o s (\frac{1}{2} a r c t a n (\frac{1}{2}))

\int_{0}^{\infty} \frac{e^{- t} c o s (2 t)}{\sqrt{t}} d t = f (1, 2) =^{4} \sqrt{5} c o s (\frac{1}{2} a r c t a n (2))