find-a-integral-form-of-L-e-x-2-L-f-means-laplace-transform-of-f-

Question Number 26057 by abdo imad last updated on 18/Dec/17

f i n d a i n t e g r a l f o r m o f L (e^{- x^{2}})

L (f) m e a n s l a p l a c e t r a n s f o r m o f f .

Commented by abdo imad last updated on 21/Dec/17

w e d e f i n e L (f (x)) = \int_{0}^{\infty} f (t) e^{- x t} d t w i t h x ⩾ 0 w i t h \int_{0}^{\infty} / f (t) / d t c o n v e r g e n t

L (e^{- x^{2}}) = \int_{0}^{\infty} e^{- t^{2}} . e^{- x t} d t = \int_{0}^{\infty} e^{- (t^{2} + x t)} d t = \int_{0}^{\infty} e^{- (t^{2} + 2 \frac{x}{2} t + \frac{x^{2}}{4} - \frac{x^{2}}{4})} d t

= e^{\frac{x^{2}}{4}} \int_{0}^{\infty} e^{- {(t + \frac{x}{2})}^{2}} d t a n d b y t h e c h a n g e m e n t α = t + \frac{x}{2}

L (e^{- x^{2}}) = e^{\frac{x^{2}}{4}} \int_{\frac{x}{2}}^{\propto} e^{- α^{2}} d α = e^{\frac{x^{2}}{4}} (\int_{\frac{x}{2}}^{0} e^{- α^{2}} d α + \int_{0}^{\propto} e^{- α^{2}} d α)

b u t \int_{0}^{\propto} e^{- α^{2}} d α = \frac{\sqrt{π}}{2}

\Rightarrow L (e^{- x^{2}}) = e^{\frac{x^{2}}{4}} (\frac{\sqrt{π}}{2} - \int_{0}^{\frac{x}{2}} e^{- α^{2}} d α)