Show that \int_0^\infty \frac{x\cos ax}{\sinh x}dx\frac{\pi^2}{4}\text{sech}^2(\frac{a\pi}{2})

Arthur Pratt

Arthur Pratt

Answered question

2022-01-03

Show that
0xcosaxsinhxdxπ24sech2(aπ2)

Answer & Explanation

puhnut1m

puhnut1m

Beginner2022-01-04Added 33 answers

Heres
Chanell Sanborn

Chanell Sanborn

Beginner2022-01-05Added 41 answers

We have that,
1sinh x=2k=2n+1, nNexp(kx)
So, 0sinaxsinh xdx=20sinaxk=0exp((2k+1)x)dx
=2a1a2+(2k+1)2=πcothπaπ2cothπa2
Differentiating with respect to a we get the answer.
Vasquez

Vasquez

Expert2022-01-09Added 669 answers

0xcos(ax)sinh xdx=dda0sin(ax)sinh (x)dx2ddaI0eiax1exexdx=2ddaI0e(1ia)xex1e2xdx=ddaI0e(1/2ia/2)xex/21e2xdx=ddaI0[exex/21exdx0exe(1/2ia/2)x1exdx]=ddaI[ψ(12)ψ(12a2i)]=dda[ψ(1/2)ai/2)+ψ(1/2+ai/2)2i]=i2dda[πcot(π[12a2i])]=πii2dda[tan(πa2i)]=π2dda[tanh(πa2)]=π24sech2(πa2)

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