The exponential form is a common format used to express complex numbers. The derivation and meaning of this format can be explained through the Maclaurin series of mathematical functions. Therefore, we will begin by discussing the Maclaurin series.

3.6.1. Maclaurin Series

Maclaurin series are infinite series constructed based on the derivatives of a function at x = 0. These series express functions in terms of increasing powers of x and are commonly used to approximate values of functions. The Maclaurin series for the functions ex, sin x, and cos x are as follows:

• ex = 1 + x + x2/2! + x3/3! + x4/4! + ...
• sin x = x - x3/3! + x5/5! - ...
• cos x = 1 - x2/2! + x4/4! - ...

These series are valid for any real or complex value of x.

3.6.2. Exponential Form of Complex Numbers and Euler’s Formula

For complex numbers, the exponential form is directly related to Euler’s formula. Euler’s formula establishes a connection between the exponential function eiθ and trigonometric functions, and it is expressed as:

• eiθ = cos θ + i sin θ

This relationship holds when θ is replaced with any angle value and allows complex numbers to be expressed in exponential form in polar coordinates.

Let’s explain what we discussed with an example. Suppose we have two complex numbers, z1 and z2, defined as follows:

• z1 = 3eiπ/4 and z2 = 2eiπ/6

From the two complex numbers expressed in exponential form above, we can extract the following information:

• r1 = 3 and r2 = 2
• θ1 = π/4 and θ2 = π/6

General Formula for Multiplication:

z1 × z2 = (r1 × r2)ei(θ1 + θ2) = 6ei5π/12

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NOTE!

Let us assume that two complex numbers are given in the following form:

• z1 = r1eiθ1
• z2 = r2eiθ2

When performing multiplication and division of these numbers, we use the following formulas:

• z1 × z2 = (r1 × r2)ei(θ1 + θ2)
• z1 / z2 = (r1 / r2)ei(θ1 - θ2)

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