Sum of Complex Numbers in Exponential Form
Jump to navigation
Jump to search
Theorem
Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.
Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$.
Then:
- $r_3 = \sqrt { {r_1}^2 + {r_2}^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$
- $\theta_3 = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2} }$
General Result
Let $n \in \Z_{>0}$ be a positive integer.
For all $k \in \set {1, 2, \dotsc, n}$, let:
- $z_k = r_k e^{i \theta_k}$
be non-zero complex numbers in exponential form.
Let:
- $r e^{i \theta} = \ds \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$
Then:
\(\ds r\) | \(=\) | \(\ds \sqrt {\sum_{k \mathop = 1}^n {r_k}^2 + \sum_{1 \mathop \le j \mathop < k \mathop \le n} 2 {r_j} {r_k} \map \cos {\theta_j - \theta_k} }\) | ||||||||||||
\(\ds \theta\) | \(=\) | \(\ds \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n} {r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n} }\) |
Proof
We have:
\(\ds r_1 e^{i \theta_1} + r_2 e^{i \theta_2}\) | \(=\) | \(\ds r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2}\) | Definition of Polar Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r_1 \cos \theta_1 + r_2 \cos \theta_2} + i \paren {r_1 \sin \theta_1 + r_2 \sin \theta_2}\) |
Then:
\(\ds {r_3}^2\) | \(=\) | \(\ds {r_1}^2 + {r_2}^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2}\) | Complex Modulus of Sum of Complex Numbers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds r_3\) | \(=\) | \(\ds \sqrt { {r_1}^2 + {r_2}^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }\) |
and similarly:
- $\theta_3 = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2} }$
$\blacksquare$
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: $88 \ \text{(a)}$