Sketch the Fourier transforms of the following functions assuming |omega_1|>|omega_2| a) f(t)=[cos(omega_1 t)+i sin(omega_1 t)]-[cos(omega_2 t)+i sin(omega_2 t)] b) f(t)=[cos(omega_0 t)]e^{-frac{t}{T_2}}

Step 1
Consider the equation
Step 2
$(a)f(t)=[\cos ({\omega }_{1}t)+i\sin ({\omega }_{1}t)]-[\cos ({\omega }_{2}t)+i\sin ({\omega }_{2}t)]$ F is a function of a real variable omega ,the function value $F(\omega )$ is a complex number.
$|F(t)|$ is called amplitude specturm of f
F=f(T) means F is the Fourier transform of f as for Laplace transform Laplace transform of f
$f(t)=[\cos ({\omega }_{1}t)+i\sin ({\omega }_{1}t)]-[\cos ({\omega }_{2}t)+i\sin ({\omega }_{2}t)]$
$=\cos ({\omega }_{1}t)+i\sin ({\omega }_{1}t)-\cos ({\omega }_{2}t)+i\sin ({\omega }_{2}t)$
(b) $f(t)=[\cos ({\omega }_{0}t)]e^{-\frac{t}{T_2}}$
F is a function of real variable t the function value F(t) is a complex number, F=f(t) means F is the Fourier transform of f as for Laplace transform
$f(t)=[\cos ({\omega }_{0}t)]e^{-\frac{t}{T_2}}$