Definition:Complex Number/Polar Form/Exponential Form

Definition

Let $z := \polar {r, \theta}$ be a complex number in polar form.

From Euler's Formula:

$e^{i \theta} = \cos \theta + i \sin \theta$

so $z$ can also be written in the form:

$z = r e^{i \theta}$

This form of presentation of a complex number is known as exponential form.

Also known as

Some sources refer to the form $z = r e^{i \theta}$ as polar form, and do not feel the need to treat it as a different representation from the $z = r \paren {\cos \theta + i \sin \theta}$ form.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Polar Form of Complex Numbers Expressed as an Exponential: $7.24$
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation
• 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): polar form of a complex number
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): polar form of a complex number