The Fourier Transform of the Complex Exponential

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The complex exponential function is common in applied mathematics. The basic form is written in Equation [1]:

complex exponential       [1]

The complex exponential is actually a complex sinusoidal function. Recall Euler's identity:

euler's identity       [2]

Recall from the previous page on the dirac-delta impulse that the Fourier Transform of the shifted impulse is the complex exponential:

      [3]

If we know the above is true, then the inverse Fourier Transform of the complex exponential must be the impulse:

      [4]

What we are interested in is the Fourier Transform of the complex exponential in equation [1], or:

fourier transform of complex exponential       [5]

Using the second line of Equation [4], equation [5] can be quickly solved:

fourier transform of complex exponential       [6]

The last equal sign equation follows directly from equation [4]. Hence, the Fourier Transform of the complex exponential given in equation [1] is the shifted impulse in the frequency domain. This should also make intuitive sense: since the Fourier Transform decomposes a waveform into its individual frequency components, and since g(t) is a single frequency component (see equation [2]), then the Fourier Transform should be zero everywhere except where f=a, where it has infinite energy.


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