# Example 1. Fourier transform of sin(t)

F:=unapply(inttrans[fourier](sin(t),t,w),w);F(1); # singularity at freq=1


# Example 2. Fourier transform of a sinusoid pulse

f:=t->4*sin(100*2*Pi*t)*(Heaviside(t)-Heaviside(t-2*Pi));
F:=unapply(inttrans[fourier](f(t),t,w),w);assume(w,real);simplify(Re(F(w)));
# Singularity at zero of the denominator indicates Dirac answer


# Example 3. A mixed signal with exponential and sinusoidal terms

f:=t->(4*exp(-t)+sin(4*t))*(Heaviside(t)-Heaviside(t-2*Pi));assume(w,real);simplify(Re(F(w)));
F:=unapply(inttrans[fourier](f(t),t,w),w);

Re(F(2)); Im(F(2)); abs(F(2)); evalf(abs(F(2)));evalf(arctan(Im(F(2))/Re(F(2))));