We derived the frequency response above using trig identities in order
to minimize the mathematical level involved. However, it turns out it
is actually easier, though more advanced, to use *complex
numbers* for this purpose. To do this, we need *Euler's
identity*:

where is the imaginary unit for complex numbers, and is a transcendental constant approximately equal to . Euler's identity is fully derived in [84]; here we will simply use it ``on faith.'' It can be proved by computing the Taylor series expansion of each side of Eq. (1.8) and showing equality term by term [84,14].

- Complex Sinusoids
- Complex Amplitude
- Phasor Notation
- Complex Sinusoids as Circular Motion
- Rederiving the Frequency Response

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