calculste-f-a-1-x-2-x-2-a-2-dx-with-a-gt-0-

Question Number 69379 by mathmax by abdo last updated on 22/Sep/19

c a l c u l s t e f (a) = \int_{- \infty}^{+ \infty} \frac{{(- 1)}^{x^{2}}}{x^{2} + a^{2}} d x w i t h a > 0

Commented by mathmax by abdo last updated on 23/Sep/19

w e h a v e f (a) = \int_{- \infty}^{+ \infty} \frac{e^{i π x^{2}}}{x^{2} + a^{2}} d x l e t φ (z) = \frac{e^{i π z^{2}}}{x^{2} + a^{2}}

φ (z) = \frac{e^{i π z^{2}}}{(z - i a) (z + i a)} r e s i d u s t h e o r e m g i v e

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i a) = 2 i π \frac{e^{i π {(i a)}^{2}}}{2 i a} = \frac{π}{a} e^{- i π a^{2}}

= \frac{π}{a} {(- 1)}^{a^{2}} \Rightarrow f (a) = \frac{π}{a} {(- 1)}^{a^{2}}