[PDF] fourier transform of e^ t

How do you Fourier transform a simple exponential function?

The Fourier transform of a function of x gives a function of k, where k is the wavenumber. This function insures that our oscillator starts at time t = 0. If didn’t include, the amplitude would blow up as t? ??. So we can Fourier transform the simpler exponential function. Starting with the first term, we find ? f 1 ?

What is the modulus squared F () 2 of the Fourier transform?

The function and the modulus squared f ?(?) 2 of its Fourier transform are then: Figure 2. An underdamped oscillator and its power spectrum (modulus of its Fourier transform squared) for ? =2 and ?0=10. The widths of the peaks give how much each harmonic damps with time. The width at half maximum gives the damping factor ?.

Is the Fourier transform limited to a function of time?

The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. The Fourier inversion theorem provides a synthesis process that recreates the original function from its frequency domain representation.

Does a Fourier transform have imaginary parts?

This is a Fourier sine transform. Thus the imaginary part vanishes only if the function has no sine components which happens if and only if the function is even. For an odd function, the Fourier transform is purely imaginary. For a general real function, the Fourier transform will have both real and imaginary parts.