calculate-f-1-i-x-2-dx-

Question Number 36919 by maxmathsup by imad last updated on 07/Jun/18

c a l c u l a t e f (α) = \int_{- \infty}^{+ \infty} {(1 + α i)}^{- x^{2}} d x .

Commented by abdo.msup.com last updated on 10/Jun/18

w e h a v e 1 + α i = \sqrt{1 + α^{2}} (\frac{1}{\sqrt{1 + α^{2}}} + \frac{α}{\sqrt{1 + α^{2}}} i)

= r e^{i θ} \Rightarrow r = \sqrt{1 + α^{2}} a n d c o s θ = \frac{1}{\sqrt{1 + α^{2}}}

s i n θ = \frac{α}{\sqrt{1 + α^{2}}} \Rightarrow t a n θ = α \Rightarrow θ = a r c t a n (α)

1 + α i = r e^{i a r c t a n (α)} \Rightarrow

f (α) = \int_{- \infty}^{+ \infty} {(r e^{i a r c t a n (α)})}^{- x^{2}} d x

= \int_{- \infty}^{+ \infty} {(e^{l n (r) + i a r c t a n (α)})}^{- x^{2}} d x

= \int_{- \infty}^{+ \infty} e^{- (l n {(\sqrt{1 + α^{2}} + i a r c t a n (α))}^{} x^{2}} d x

c h a n g e m e n t \sqrt{l n (\sqrt{1 + α^{2})} + i a r t a n (α)} x = t

g i v e

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

= \frac{\sqrt{π}}{\sqrt{l n (\sqrt{1 + α^{2})} + i a r c t a n (α)}} .