nice-calculus-prove-that-cosh-px-cosh-x-pi-cos-pip-2-

Question Number 133334 by mnjuly1970 last updated on 21/Feb/21

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

p r o v e t h a t ::

ϕ = \int_{- \infty}^{+ \infty} \frac{c o s h (p x)}{c o s h (x)} = \frac{π}{c o s (\frac{π p}{2})}

Answered by mnjuly1970 last updated on 21/Feb/21

Commented by Dwaipayan Shikari last updated on 21/Feb/21

{H a v e n}^{'} t t h o u g h t o f . G r e a t w a y s i r!

Commented by mnjuly1970 last updated on 21/Feb/21

s i n c e r e l y y o u r s . .

g r a t e f u l s i r p a y a n \dots

Answered by mathmax by abdo last updated on 22/Feb/21

Φ = \int_{- \infty}^{+ \infty} \frac{ch (px)}{ch (x)} dx \Rightarrow Φ = \int_{- \infty}^{+ \infty} \frac{e^{px} + e^{- px}}{e^{x} + e^{- x}} dx

=_{e^{x} = t} \int_{0}^{\infty} \frac{t^{p} + t^{- p}}{t + t^{- 1}} \frac{dt}{t} = \int_{0}^{\infty} \frac{t^{p} + t^{- p}}{t^{2} + 1} dt = \int_{0}^{\infty} \frac{t^{p}}{1 + t^{2}} dt + \int_{0}^{\infty} \frac{t^{- p}}{1 + t^{2}} dt

we have \int_{0}^{\infty} \frac{t^{p}}{1 + t^{2}} dt =_{t = z^{\frac{1}{2}}} \frac{1}{2} \int_{0}^{\infty} \frac{z^{\frac{p}{2}}}{1 + z} z^{- \frac{1}{2}} dz

= \frac{1}{2} \int_{0}^{\infty} \frac{z^{\frac{p + 1}{2} - 1}}{1 + z} dz = \frac{1}{2} \frac{π}{\sin (π (\frac{p + 1}{2}))} = \frac{π}{2 \cos (\frac{p π}{2})} also

\int_{0}^{\infty} \frac{t^{- p}}{1 + t^{2}} dt = \frac{π}{2 \cos (\frac{- p π}{2})} = \frac{π}{2 \cos (\frac{p π}{2})} \Rightarrow

Φ = \frac{π}{2 \cos (\frac{p π}{2})} + \frac{π}{2 \cos (\frac{p π}{2})} = \frac{π}{\cos (\frac{p π}{2})}