Product of Complex Numbers in Exponential Form
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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.
Then:
- $z_1 z_2 = r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }$
Proof
\(\ds z_1 z_2\) | \(=\) | \(\ds r_1 e^{i \theta_1} r_2 e^{i \theta_2}\) | Definition of Exponential Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 \paren {\cos \theta_1 + i \sin \theta_1 } r_2 \paren {\cos \theta_2 + i \sin \theta_2}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }\) | Product of Complex Numbers in Polar Form | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }\) | Euler's Formula |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Operations with Complex Numbers in Polar Form: $7.25$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $19$
- 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers