# Sum of Complex Numbers in Exponential Form/General Result

## Theorem

Let $n \in \Z_{>0}$ be a positive integer.

For all $k \in \set {1, 2, \dotsc, n}$, let:

- $z_k = r_k e^{i \theta_k}$

Let:

- $r e^{i \theta} = \displaystyle \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$

Then:

\(\displaystyle r\) | \(=\) | \(\displaystyle \sqrt { {r_1}^2 + {r_2}^2 + \dotsb + {r_n}^2 + \ldots}\) | |||||||||||

\(\displaystyle \theta\) | \(=\) | \(\displaystyle \map \arctan {\dfrac {r_1 \sin \theta_1 + r_1 \sin \theta_2 + \dotsb + r_n \sin \theta_2} {r_1 \cos \theta_1 + r_1 \cos \theta_2 + \dotsb + r_n \cos \theta_2} }\) |

## Proof

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $88 \ \text{(b)}$