nice-calculus-calculate-0-tanh-pix-x-1-x-2-dx-

Question Number 126476 by mnjuly1970 last updated on 20/Dec/20

\dots n i c e c a l c u l u s \dots

c a l c u l a t e ::

Ω = \int_{0}^{\infty} \frac{t a n h (π x)}{x (1 + x^{2})} d x = ?

Answered by mathmax by abdo last updated on 21/Dec/20

Ψ = \int_{0}^{\infty} \frac{th (π x)}{x (1 + x^{2})} dx \Rightarrow Ψ =_{π x = t} \int_{0}^{\infty} \frac{th (t)}{\frac{t}{π} (1 + \frac{t^{2}}{π^{2}})} \frac{dt}{π}

= π^{2} \int_{0}^{\infty} \frac{th (t)}{t (t^{2} + π^{2})} dt = \frac{π^{2}}{2} \int_{- \infty}^{+ \infty} \frac{th (t)}{t (t^{2} + π^{2})} dt let φ (z) = \frac{th (z)}{z (z^{2} + π^{2})}

residus give \int_{- \infty}^{+ \infty} φ (z) dz = 2 i π {Res (φ, o) + Res (φ, i π)}

Res (φ, o) = 0

Res (φ, i π) = \frac{th (i π)}{i π (2 i π)} = - \frac{1}{2 π^{2}} th (i π) and

th (i π) = \frac{sh (i π)}{ch (i π)} = \frac{e^{i π} - e^{- i π}}{e^{i π} + e^{- i π}} = \frac{isin (π)}{\cos (π)} = 0 \Rightarrow Ψ = 0

Answered by mathmax by abdo last updated on 21/Dec/20

if you have another way post it sir mnj