0-xsinh-pix-e-x-2-1-x-2-dx-

Question Number 139092 by mathdanisur last updated on 22/Apr/21

Ω = \underset{0}{\int^{\infty}} \frac{x s i n h (π x) e^{- x^{2}}}{1 + x^{2}} d x

Answered by mathmax by abdo last updated on 23/Apr/21

i give this soution but not sure!

Φ = \int_{0}^{\infty} \frac{xsh (π x) e^{- x^{2}}}{x^{2} + 1} dx \Rightarrow Φ = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{xsh (π x)}{x^{2} + 1} e^{- x^{2}} dx

w (z) = \frac{zsh (π z) e^{- z^{2}}}{z^{2} + 1} \Rightarrow w (z) = \frac{zsh (π z) e^{- z^{2}}}{(z - i) (z + i)}

\int_{- \infty}^{+ \infty} w (z) dz = 2 i π Res (w, i) [= 2 i π \times \frac{ish (π i) e^{- (- 1)}}{2 i} = i π sh (π i) e

but sh (π i) = \frac{e^{i (π i)} - e^{- i (π i)}}{2 i} = \frac{e^{- π} - e^{π}}{2 i} \Rightarrow

\int_{- \infty}^{+ \infty} w (z) dz = i π (\frac{e^{- π} - e^{π}}{2 i}) e = π (e^{1 - π} - e^{1 + π}) \Rightarrow

Φ = \frac{π}{2} (e^{1 - π} - e^{1 + π})

Commented by mathdanisur last updated on 23/Apr/21

t h a k s t h a n k s S i r

Commented by mathdanisur last updated on 23/Apr/21

S i r, s i n h (i π) = 0 ?

Commented by mathmax by abdo last updated on 24/Apr/21

no sir its not like sinx \dots!

Commented by mathdanisur last updated on 24/Apr/21

S i r, o m e g a m u s t b e > 0 .

Y o u r a n s w e r i s a n e g a t i v e n u m b e r

Commented by mathdanisur last updated on 24/Apr/21

S i r, {i t}^{'} s n o t l i k e s i n, s i n (i x) / i = s i n h ?