let-z-r-e-i-fins-f-z-z-x-2-dx-

Question Number 36916 by maxmathsup by imad last updated on 07/Jun/18

l e t z = r e^{i θ} f i n s f (z) = \int_{- \infty}^{+ \infty} z^{- x^{2}} d x

Commented by math khazana by abdo last updated on 09/Jun/18

f (z) = \int_{- \infty}^{+ \infty} {(r e^{i θ})}^{- x^{2}} d x

= \int_{- \infty}^{+ \infty} {e^{l n (r) + i θ}}^{- x^{2}} d x

= \int_{- \infty}^{+ \infty} e^{- (l n (r) + i θ) x^{2}} d x b u t c h a n g e m e n t

\sqrt{l n (r) + i θ} x = t g i v e

f (z) = \frac{1}{\sqrt{l n (r) + i θ}} \int_{- \infty}^{+ \infty} e^{- t^{2}} d t

= \frac{\sqrt{π}}{\sqrt{l n (r) + i θ}} .