A.1  Euler’s Equation

where $x$ is given in radians. When $x\; =\; \pi$, it leads to the famous identity $e^{j\pi }=\; -1$. The value $e^{jx}$ can be interpreted as a complex number with magnitude one and angle $x$ rad. Hence, Eq. (A.1) represents the conversion of this complex number from the polar to the Cartesian form $cos(x)\; +\; jsin(x)$.

Using the fact that cosine and sine are even and odd functions, respectively, one can write $e^{-jx}=\; cos(x)\; -\; jsin(x)$ and using Eq. (A.1) obtain

and