An Easier Way

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:

$$\zbox {e^{j\theta} = \cos(\theta) + j \sin(\theta)}\qquad \hbox{(Euler's Identity)} \protect$$ (2.8)

where $j\isdef \sqrt{-1}$ is the imaginary unit for complex numbers, and $e$ is a transcendental constant approximately equal to $2.718\ldots$. Euler's identity is fully derived in [83]; 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 [83,14].