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Euler's formula

Leonhard Euler
Leonhard Euler (1707--1783) was born in Basel, Swiss Confederacy, but in 1727 he left his motherland for Saint Petersburg, Russian Empire and never returned back. In 1748, Euler’s landmark book Introductio in analysin infinitorum (In latin) was published in Lausanne, Switzerland, by Marc-Michel Bousquet. Euler (promounced as "oiler" not uler) systematically defined e = 2.7182818284590452353602… as the unique number for which the area under the curve y = 1/x from 1 to e equals 1. He established e as the natural base of logarithms and the exponential function, giving it a central role in analysis. Euler also introduced the notation f(x) for functions and unified trigonometric and exponential functions through series expansions.

Leonhard Euler contributed to many science areas as he published more than 800 works. He was pivotal in popularizing and integrating the unit imaginary vector into mainstream mathematics, particularly through Euler's formula

Theorem 1 (Euler, 1748): For any real number θ, one has \begin{equation} \label{EqEuler.1} e^{{\bf j}\theta} = \cos\theta + {\bf j}\,\sin\theta . \end{equation}

   
Example 1:    ■
End of Example 1

Polar representation

Let xOy be an orthogonal axes system on a plane. We will identify the set ℝ², the direct product of two real lines, with the plane xy. Since ℝ² and ℂ are isomorphic as vector spaces over ℝ, it follows that we could identify the complex set ℂ with the plane xy, such that x ∈ ℝ (abscissa) and y is its ordinate. Thus, the axis x will be called the real axis, and the ordinate will be called the imaginary axis.

To every point from ℝ², it corresponds one and only one complex number from ℂ, and vice versa. Thus, there exists a bijective correspondence between the sets ℝ² and ℂ; i. e., (x, y) ←→ x + ⅉy, where ⅉ = [0, 1] is the unit vector of the ordinate. Besides rectangular coordinates, it will be convenient to use polar coordinate system. Polar coordinates describe a point's location using a distance from a central point (the pole, which is usually the origin) and an angle from a reference line (the polar axis, a ray drawn from the pole), represented as (r, θ), instead of the familiar (x, y). 'r' is the radial distance, and 'θ' (theta) is the angle, measured counter-clockwise (which is considered also as positive direction) from the positive x-axis (polar axis), making it ideal for circular or rotational systems like radar and navigation. Remember that the pole is a singular point for polar coordinate system because it corresponds to r = 0, but no angle can be associated with.

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Figure 1: Polar system of coordinates

Letting z ∈ ℂ \ {(0, 0)}, z = x + ⅉy, from the relation

\[ \frac{z}{|z|} = \frac{x}{\sqrt{x^2 + y^2}} + {\bf j}\,\frac{y}{\sqrt{x^2 + y^2}} , \]
combined with elementary knowledge of trigonometry, it follows that the system
\[ \begin{cases} \cos\theta &= \frac{x}{\sqrt{x^2 + y^2}} , \\ \sin\theta &= \frac{y}{\sqrt{x^2 + y^2}} , \end{cases} \]
is compatible, with respect to the θ unknown. This fact allows us to give the following definition.    
Example 2: the reciprocal of z    ■
End of Example 2
Let z ∈ ℂ ∖ {0}. Every solution θ of the equation \[ \cos\theta + {\bf j}\,\sin\theta = \frac{z}{|z|} \] is called the argument of the complex number z. The equation has a unique solution in the interval (−π, π], and it is called the principal value of argument of the complex number z, and is denoted by arg z. If we denote by Arg z the set of all the arguments of the number z, then \[ \mbox{Arg} z = \left\{ \arg z + 2k\pi \ : \quad k \in \mathbb{Z} \right\} . \]
The correspondence z → Arg z is a multivalued function from ℂ ∖ {0} to ℝ.
Theorem 2: For any complex numbers z₁, z₂ ∈ ℂ ∖ {0}, we have
  1. Arg(z₁ z₂) = Arg(z₁) + Arg(z₂) .
  2. Arg(z₁/z₂) = Arg(z₁) − Arg(z₂) .

Let θ₁ ∈ Arg(z₁) and θ₂ ∈ Arg(z₂), then \[ \cos\theta_i + {\bf j}\,\sin\theta_i = \frac{z_i}{| z_i |} , \qquad i = 1,2 . \] Using simple trigonometric properties, we obtain that \[ \frac{z_1}{| z_1 |}\,\frac{z_2}{| z_2 |} = \left( \cos\theta_1 + {\bf j}\,\sin\theta_1 \right) \left( \cos\theta_2 + {\bf j}\,\sin\theta_2 \right) , \] and \[ \frac{z_1}{| z_1 |}\,\frac{z_2}{| z_2 |} = \frac{z_1 z_2}{| z_1 z_2 |} \quad \Longrightarrow \quad \theta_1 + \theta_2 \in \mbox{Arg}(z_1 z_2 ) , \] hence \[ \mbox{Arg} z_1 + \mbox{Arg} z_2 \, \subset \mbox{Arg} \left( z_1 z_2 \right) . \] Similarly, we deduce that \begin{align*} \frac{\frac{z_1}{z_2}}{\frac{|z_1 |}{| z_2 |}} &= \frac{\frac{z_1}{|z_1 |}}{\frac{z_2}{|z_2 |}} = \frac{\cos\theta_1 +{\bf j}\,\sin\theta_1}{\cos\theta_2 + {\bf j}\,\sin\theta_2} \\ &= \frac{\left( \cos\theta_1 + {\bf j}\,\sin\theta_1 \right) \left( \cos\theta_2 - {\bf j}\,\sin\theta_2 \right)}{\cos^2 \theta_2 + \sin^2 \theta_2} \\ &= \cos\left( \theta_1 - \theta_2 \right) + {\bf j}\,\sin \left( \theta_1 - \theta_2 \right) . \end{align*} So θ₁ − θ₂ ∈ Arg(z₁/z₂), or \[ \mbox{Arg}(z_1 ) - \mbox{Arg}(z_2 ) \subset \mbox{Arg}\left( \frac{z_1}{z_2} \right) . \] Let θ ∈ Arg(z₁z₂) be arbitrary, and consider an arbitrary angle θ₁ ∈ Arg(z₁). From above relation, we have \[ \theta - \theta_1 \in \mbox{arg}\left( \frac{z_1 z_2}{z_1} \right) = \mbox{Arg}(z_1 ) , \] and hence \[ \theta \in \mbox{Arg} z_2 + \theta_1 \subset \mbox{Arg}(z_1 ) + \mbox{Arg}(z_2 ) . \] From these relations follows the equality of the first point.

Since \[ \mbox{Arg}(z_1 ) = \mbox{Arg}\left( \frac{z_1}{z_2}\,z_2 \right) = \mbox{Arg}\left( \frac{z_1}{z_2} \right) + \mbox{Arg}(z_2 ) , \] we get \[ \forall \theta \in \mbox{Arg} \left( \frac{z_1}{z_2} \right) , \quad \exists \theta_1 ,\quad \exists \theta_2 \in \mbox{Arg} z_2 \] such that θ₁ = θ + θ₂. Therefore, \[ \theta = \theta_1 - \theta_2 \in \mbox{Arg} z_1 - \mbox{Arg} z_2 \] and thus \[ \mbox{Arg} \left( \frac{z_1}{z_2} \right) \subset \mbox{Arg} z_1 - \mbox{Arg} z_2 . \] From these relations, it follows the equality of the second point.

   
Example 5:    ■
End of Example 5
If z ∈ ℂ∖{(0,0)}, then \[ z = | z |\left( \cos\theta + {\bf j}\,\sin\theta \right) , \qquad \mbox{where} \quad \theta \in \mbox{Arg}(z) , \] is called the trigonometric form of the complex number z.
Since complex numbers are ordered pairs of real numbers, they can be represented on the complex plane as a vector started at the origin. The abscissa axis is .{(x, 0) ∈ ℂ} and it is called the real axis. The ordinate axis is .{(0, y) ∈ ℂ}, and it is called the imaginary axis. The use of polar coordinates gives the polar (or trigonometric) representation of complex numbers, z = [x, y] = r(cosθ, sinθ), where \( \displaystyle \quad |z| = r = +\sqrt{x 2 + y 2} \quad \) is called the modulus3 or absolute value of z = x + ⅉy, and θ is called an argument of z. So, argument of z ≠ 0 is defined up to addition of integer multiples of 2π; if the argument of z is in interval ] − π, π], it is called the principal value of argument of z. With the real arctan function whose range is ] − π/2 , π/2[, we have
\[ \arg z = \begin{cases} \arctan \left(\frac{\mbox{Im}z}{\mbox{Re}z}\right) + \pi , & \quad \mbox{if} \quad \mbox{Re} z < 0, \ \mbox{Im} z \ge 0 , \\ + \frac{\pi}{2} , & \quad \mbox{if} \quad \mbox{Re} z = 0, \ \mbox{Im}z > 0 , \\ \arctan \left(\frac{\mbox{Im}z}{\mbox{Re}z}\right) , & \quad \mbox{if} \quad \mbox{Re} z > 0, \\ - \frac{\pi}{2} , & \quad \mbox{if} \quad \mbox{Re} z = 0, \ \mbox{Im}z < 0 , \\ \arctan \left(\frac{\mbox{Im}z}{\mbox{Re}z}\right) - \pi , & \quad \mbox{if} \quad \mbox{Re} z < 0, \ \mbox{Im} z < 0 . \end{cases} \]
A succinct version of this formula is
\[ \mbox{Arg} z = \begin{cases} \arctan \left(\frac{\mbox{Im}z}{\mbox{Re}z}\right) + 2k\pi &\quad (I \mbox{ and } IV \mbox{ quadrants}), \quad k \in \mathbb{Z} , \\ \arctan \left(\frac{\mbox{Im}z}{\mbox{Re}z}\right) + \left( 2k + 1 \right)\pi &\quad (II \mbox{ and } III \mbox{ quadrants}) , \quad k \in \mathbb{Z} . \end{cases} \]
Here arctan means the principle value of artangent, which belongs to the interval (−π/2, π/2]. Recall the following facts from the previous section.

 

 

  1. Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: ‎ 978-0471000075.