We can use the phase shift formula above to find the Fourier transform of
any complex sinusoid with frequency , simply by setting
in the formula and using the Fourier transform for DC:

where is the Dirichlet kernel and is an ugly phase term:

If the sinusoid's frequency is an integer multiple of the fundamental frequency , the Dirichlet kernel is shifted to the left or right by an integer. In this case the zero crossings of the Dirichlet kernel line up with integer values of , so that only one partial is nonzero. This is pictured in Figure 9.3 (part a).

Part (b) shows the result when the frequency falls halfway between two integers. The partials have amplitudes falling off roughly as in both directions, measured from the actual frequency . That the energy should be spread over many partials, when after all we started with a single sinusoid, might seem surprising at first. However, as shown in Figure 9.4, the signal repeats at a period which disagrees with the frequency of the sinusoid. As a result there is a discontinuity at the beginning of each period, and energy is flung over a wide range of frequencies.