let-b-C-and-Re-b-gt-0-prove-that-e-iax-x-ib-dx-2ipi-e-ab-and-e-iax-x-ib-dx-0-

Question Number 37356 by math khazana by abdo last updated on 12/Jun/18

l e t b \in C a n d R e (b) > 0 p r o v e t h a t

\int_{- \infty}^{+ \infty} \frac{e^{i a x}}{x - i b} d x = 2 i π e^{- a b} a n d

\int_{- \infty}^{+ \infty} \frac{e^{i a x}}{x + i b} d x = 0

Commented by math khazana by abdo last updated on 12/Jun/18

a > 0

Commented by math khazana by abdo last updated on 13/Jun/18

l e t φ (z) = \frac{e^{i a z}}{z - i b} s o i b i s a s i m p l e p o l e o f φ

w i t h R e (b) > 0 \Rightarrow I m (i b) > 0 \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i b)

= 2 i π e^{i a (i b)} = 2 i π e^{- a b}

l e t ψ (z) = \frac{e^{i a z}}{x + i b} t h e p o l e o f ψ i s - i b b u t I m (- i b) < 0

s o \int_{- \infty}^{+ \infty} ψ (z) d z = 0