prove-that-1-2-e-4x-6-12x-5-15x-4-10x-3-4x-2-x-dx-e-1-8-3-1-6-1-2-2-2-1-3-1F2-1-3-2-3-1-6-1-69-2-5-6-128-4-1-3-1F2-4-3-5

Question Number 99120 by M±th+et+s last updated on 18/Jun/20

p r o v e t h a t :

\int_{- \frac{1}{2}}^{\infty} e^{- (4 x^{6} + 12 x^{5} + 15 x^{4} + 10 x^{3} + 4 x^{2} + x)} d x

= \frac{\sqrt[8]{e}}{3} [\frac{Γ {(\frac{1}{6})}^{\frac{- 1}{2}}}{2 \sqrt[3]{2}} 1 F 2 (_{1 / 3, 2 / 3}^{1 / 6} ∣ \frac{- 1}{69 / 2}) + \frac{Γ (5 / 6)}{128 \sqrt[3]{4}} 1 F 2 (_{4 / 3, 5 / 3}^{5 / 6} ∣ \frac{- 1}{69 / 2}) - \frac{\sqrt{π}}{16} 12 (_{2 / 3, 4 / 3}^{1 / 2} ∣ \frac{- 1}{69 / 2})

Commented by M±th+et+s last updated on 18/Jun/20

I = \frac{\sqrt[8]{e} . π \sqrt{π}}{2 \sqrt[3]{4} \sqrt[6]{3}} [{(Ai (\frac{- 1}{8 \sqrt[3]{6}}))}^{2} + {(Bi (\frac{- 1}{8 \sqrt[3]{6}}))}^{2}]

Commented by M±th+et+s last updated on 18/Jun/20

Ai (z) : A i r y f u n c t i o n

Bi (z) : A i r y Bi f u n c t i o n

1 F 2 : i s h y p e r g e o m e t r i c f u n c t i o n

Γ (s) : g a m m a f u n c t i o n