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#1/2[(int_-∞^∞e^(jω0t)e^(-jωt)dt)+(int_-∞^∞e^(-jω0t)e^(-jωt)dt)] #

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Answer 1 · 2018-05-06T17:29:36

#sqrt(2pi)/2delta(omega-omega_0)+ sqrt(2pi)/2delta(omega+omega_0)#

where #delta# is the Dirac delta function.

The integrals are what one does when attempting to compute the Fourier transform. We can replace the integrals with:

#1/2(\mathcalF{e^(jomega_0t)}+ \mathcalF{e^(j-omega_0t)})#

These transforms can be found in any table:

#1/2(sqrt(2pi)delta(omega-omega_0)+ sqrt(2pi)delta(omega+omega_0))#

Distribute the #1/2#:

#sqrt(2pi)/2delta(omega-omega_0)+ sqrt(2pi)/2delta(omega+omega_0)#

where #delta# is the Dirac delta function.