The Fourier Transform of the Complex Exponential

The complex exponential is actually a complex sinusoidal function. Recall Euler's identity:
Recall from the previous page on the dirac-delta impulse that the Fourier Transform of
the shifted impulse is the complex exponential:
If we know the above is true, then the inverse Fourier Transform of the complex exponential must be the impulse:
What we are interested in is the Fourier Transform of the complex exponential in equation [1], or:

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

Next: Sinusoidal Functions - Sine and Cosine

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