Exponential Form of Complex Numbers

October 2, 2024

This is a deduction of the exponential form of a complex number.

To deduce the exponential form of a complex number, we will consider the following relationships:

(1) (cos⁡(θ)+isin⁡(θ))⋅(cos⁡(θ′)+isin⁡(θ′))=cos⁡(θ+θ′)+isin⁡(θ+θ′)(\cos(\theta) + i \sin(\theta)) \cdot (\cos(\theta ') + i \sin(\theta ')) = \cos(\theta + \theta ') + i \sin(\theta + \theta ')

(2) (cos⁡(θ)+isin⁡(θ))n=cos⁡(nθ)+isin⁡(nθ)(\cos(\theta) + i \sin(\theta))^n = \cos(n \theta) + i \sin(n \theta)

We can deduce that there is an application ϕ:R→C\phi: \mathbb{R} \rightarrow \mathbb{C} such that:

ϕ(θ)=cos⁡(θ)+isin⁡(θ)\phi(\theta) = \cos(\theta) + i \sin(\theta)

Comparing the relationships (1) and (2) for the application ϕ\phi, we can deduce that these relationships are quite similar to the properties of the exponential function ϕ:R→R\phi: \mathbb{R} \rightarrow \mathbb{R}, as we can see in the following relationships:

(3) ϕ(θ+θ′)=ϕ(θ)⋅ϕ(θ′)\phi(\theta + \theta ') = \phi(\theta) \cdot \phi(\theta ')

(4) ϕ(nθ)=ϕ(θ)n\phi(n \theta) = \phi(\theta)^n

Written in another way:

(3) e(θ+θ′)=eθ⋅eθ′e^{(\theta + \theta ')} = e^{\theta} \cdot e^{\theta '}

(4) enθ=(eθ)ne^{n \theta} = (e^{\theta})^n

Thus, we can use the following form to represent a complex number:

eiθ=cos⁡(θ)+isin⁡(θ)∀θ∈Re^{i \theta} = \cos(\theta) + i \sin(\theta) \quad \forall \theta \in \mathbb{R}

Finally, if z≠0z \neq 0 is a complex number, we can write it in exponential form as:

z=r⋅eiθz = r \cdot e^{i \theta}

Where r=∣z∣r = |z| and θ=arg⁡(z)\theta = \arg(z).

From the following formulas:

eiθ=cos⁡(θ)+isin⁡(θ)e^{i \theta} = \cos(\theta) + i \sin(\theta)

e−iθ=cos⁡(−θ)+isin⁡(−θ)=cos⁡(θ)−isin⁡(θ)e^{-i \theta} = \cos(-\theta) + i \sin(-\theta) = \cos(\theta) - i \sin(\theta)

Summing and subtracting these formulas, we can deduce Euler’s formulas:

eiθ+e−iθ2=cos⁡(θ)eiθ−e−iθ2i=sin⁡(θ)\begingroup \begin{array}{l} \frac{e^{i \theta} + e^{-i \theta}}{2} = \cos(\theta) \\ \\ \frac{e^{i \theta} - e^{-i \theta}}{2i} = \sin(\theta) \end{array} \endgroup