Sum of Complex Numbers in Exponential Form/General Result
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{>0}$ be a positive integer.
For all $k \in \set {1, 2, \dotsc, n}$, let:
- $z_k = r_k e^{i \theta_k}$
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
Let:
\(\ds r e^{i \theta}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n z_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z_1 + z_2 + \dotsb + z_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2} + \dotsb + r_n \paren {\cos \theta_n + i \sin \theta_n}\) | Definition of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n + i \paren {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n}\) | rerranging |
By the definition of the complex modulus, with $z = x + i y$, $r$ is defined as:
- $r = \sqrt {\map {\Re^2} z + \map {\Im^2} z}$
Hence
\(\ds r\) | \(=\) | \(\ds \sqrt {\map {\Re^2} z + \map {\Im^2} z}\) | ||||||||||||
\(\ds r\) | \(=\) | \(\ds \sqrt {\paren {r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n }^2 + \paren {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n}^2 }\) |
In the above we have two types of pairs of terms:
\(\text {(1)}: \quad\) | \(\ds 1 \le k \le n: \, \) | \(\ds {r_k}^2 \cos^2 {\theta_k}^2 + {r_k}^2 \sin^2 {\theta_k}^2\) | \(=\) | \(\ds {r_k}^2 \paren {\cos^2 {\theta_k}^2 + \sin^2 {\theta_k}^2}\) | ||||||||||
\(\ds \) | \(=\) | \(\ds {r_k}^2\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 1 \le j < k \le n: \, \) | \(\ds 2 r_j r_k \cos \theta_j \cos \theta_k + 2 {r_j} {r_k} \sin \theta_j \sin \theta_k\) | \(=\) | \(\ds 2 r_j r_k \paren {\cos \theta_j \cos \theta_k + \sin \theta_j \sin \theta_k}\) | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 r_j r_k \map \cos {\theta_j - \theta_k}\) | Cosine of Difference |
Hence:
- $\ds r = \sqrt {\sum_{k \mathop = 1}^n {r_k}^2 + \sum_{1 \mathop \le j \mathop < k \mathop \le n} 2 {r_j} {r_k} \map \cos {\theta_j - \theta_k} }$
Note that $r > 0$ since $r_k > 0$ for all $k$.
Hence we may safely assume that $r > 0$ when determining the argument below.
By definition of the argument of a complex number, with $z = x + i y$, $\theta$ is defined as any solution to the pair of equations:
- $(1): \quad \dfrac x {\cmod z} = \map \cos \theta$
- $(2): \quad \dfrac y {\cmod z} = \map \sin \theta$
where $\cmod z$ is the modulus of $z$.
As $r > 0$ we have that $\cmod z \ne 0$ by definition of modulus.
Hence we can divide $(2)$ by $(1)$, to get:
\(\ds \map \tan \theta\) | \(=\) | \(\ds \frac y x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Im z} {\map \Re z}\) |
Hence:
\(\ds \theta\) | \(=\) | \(\ds \map \arctan {\frac {\map \Im {r e^{i \theta} } } {\map \Re {r e^{i \theta} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n} {r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n} }\) |
$\blacksquare$
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: $88 \ \text{(b)}$