prove-1-1-e-x-2-dx-pi-1-2-1-2-

Question Number 129763 by Eric002 last updated on 18/Jan/21

p r o v e

\int_{- \infty}^{+ \infty} \frac{1}{1 + e^{x^{2}}} d x = \sqrt{π} (1 - \sqrt{2}) ξ (\frac{1}{2})

Answered by Dwaipayan Shikari last updated on 18/Jan/21

\int_{- \infty}^{\infty} \frac{1}{1 + e^{x^{2}}} d x

= \underset{n = 1}{\sum^{\infty}} {(- 1)}^{n + 1} \int_{- \infty}^{\infty} e^{- {n x}^{2}} d x

= \sqrt{π} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{\sqrt{n}} = \sqrt{π} η (\frac{1}{2}) = \sqrt{π} ζ (\frac{1}{2}) (1 - \frac{1}{2^{- \frac{1}{2}}})

= \sqrt{π} ζ (\frac{1}{2}) (1 - \sqrt{2})

Commented by Eric002 last updated on 18/Jan/21

w e l l d o n e

Answered by mnjuly1970 last updated on 18/Jan/21

ϕ = 2 \int_{0}^{\infty} \frac{1}{1 + e^{x^{2}}} d x \overset{x^{2} = t}{=} \int_{0}^{\infty} \frac{d t}{\sqrt{t} (1 + e^{t})}

= \int_{0}^{\infty} \frac{t^{\frac{1}{2} - 1}}{1 + e^{t}} = Γ (s) . η (s)

w h e r e Γ a n d η a r e E u l e r g a m m a a n d

D r i c h l e t e t a f u n c t i o n s r e s p e c t i v e l y .

∴ ϕ = Γ (\frac{1}{2}) η (\frac{1}{2})

= Γ (\frac{1}{2}) (1 - 2^{1 - \frac{1}{2}}) ζ (\frac{1}{2})

= \sqrt{π} (1 - \sqrt{2}) ζ (\frac{1}{2}) \dots