Proof Using Complex Variables

To show by means of phasor analysis that Eq. (A.1) always has a solution, we can express each component sinusoid as

$$A_i\cos(\omega t + \phi_i) =$$   re$$\left\{A_i e^{j(\omega t + \phi_i)}\right\}$$

Equation (A.1) therefore becomes

$$\begin{eqnarray*} \mbox{re}\left\{A e^{j(\omega t + \phi)}\right\} &=& \sum_{i=1... ...}\right\}\\ &=& \mbox{re}\left\{A e^{j(\omega t+\phi)}\right\}. \end{eqnarray*}$$

Thus, equality holds when we define

$$A e^{j\phi} \isdef \sum_{i=1}^N A_i e^{j\phi_i}. \protect$$ (A.4)

Since $A e^{j\phi}$ is just the polar representation of a complex number, there is always some value of $A\geq 0$ and $\phi\in[-\pi,\pi)$ such that $A e^{j\phi}$ equals whatever complex number results on the right-hand side of Eq. (A.4).

As is often the case, we see that the use of Euler's identity and complex analysis gives a simplified algebraic proof which replaces a proof based on trigonometric identities.