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# Polar representation and complex exponential

## Polar coordinates

Polar coordinates use a fixed point called the pole, and a fixed direction, the initial line to describe the position of other points. In the Cartesian plane, the pole is the origin and the initial line is the positive $x$-axis.

The polar coordinate of a point $A$ is written $(r, \theta)$. It consists of the radius $r\ge 0$, which is distance from the pole $O$, and the angle $\theta$ that the ray $OA$ makes with the initial line.

The polar coordinates $(r, \theta)$ is equivalent to the Cartesian coordinates $$(r\cos \theta, r\sin \theta).$$

The $r$-coordinate is unique, but the $\theta$-coordinate can only be given up to adding a multiple of $2\pi$. Turning through $2\pi$ keeps a point unchanged.

$(r,\theta)$ and $(r, \theta-2\pi)$ are the same point, A

## Polar representation of a complex number

Every complex number can be written as $$z = r(\cos\theta +i \sin\theta)$$ with $r\ge0$. $r$ is the modulus and $\theta$ is the argument. This is the polar representation of $z$, and is sometimes called the modulus-argument form. It corresponds to the polar coordinates $(r, \theta)$, and is an alternative to the Cartesian representation $z=x+iy$.

The polar angle $\theta$ is the argument of $z$. It is characterised by $$\cos\theta = \frac{x}{\sqrt{x^2+y^2}},\quad \sin\theta = \frac{y}{\sqrt{x^2+y^2}}.$$

The argument, denoted $\arg z$, is unique up to multiples of $2\pi$. The principal argument, denoted $\Arg z$, is the argument in the interval $(-\pi, \pi ]$.

Polar representation of a complex number

## Complex exponential

We often use the exponential notation for the polar representation of complex numbers. If we have a number $z$ with modulus $r$ and argument $\theta$, we can write $$z = r e^{i\theta}$$

Exponential representations of some complex numbers are: $$1 = e^{i0},\; i = e^{i\pi/2},\; -1 = e^{i\pi},\; -i = e^{-i\pi/2},\; 1+i = \sqrt{2}e^{i\pi/4}$$

In particular, $e^{i\pi} + 1 = 0$ is known as Euler identity.

Polar representation of a complex number

## Algebraic operations in polar coordinates

Conjugation, and multiplicative operations (including inverse, division, root extraction, etc.) are simple using the polar representation.

The complex exponential $e^{i\theta} = (\cos\theta +i \sin\theta)$ has the usual properties of the exponential. For instance, $$e^{i(\theta_1+\theta_2)} = e^{i\theta_1} e^{i\theta_2}.$$ This arises from the trigonometric identities \begin{gather*} \cos(\theta_1+\theta_2) = \cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2,\\ \sin(\theta_1+\theta_2) = \cos\theta_1\sin\theta_2+\sin\theta_1\cos\theta_2. \end{gather*}

If $z=re^{i\theta}$, using the properties of the exponential, we find $$\zo = r e^{-i\theta},\quad z^{-1}= r^{-1}e^{-i\theta},\quad z_1z_2 = r_1r_2 e^{i(\theta_1+\theta_2)}.$$

This is equivalent to \begin{gather*} \vert\zo\vert = \vert z\vert,\; \arg\zo = -\arg z, \; \vert z^{-1}\vert = \vert z\vert^{-1},\; \arg z^{-1} = -\arg z\\ \vert z_1z_2\vert = \vert z_1\vert\,\vert z_2\vert,\; \arg(z_1z_2) = \arg z_1 + \arg z_2 \end{gather*}

Multiplication of complex numbers

## De Moivre's formula

De Moivre's formula states that for all $\theta\in\R$ and all integers $n$, we have $$\cos(n\theta)+i\sin(n\theta) = (\cos\theta + i \sin\theta)^n.$$

The formula can be seen using the property of the exponential $$e^{i n\theta} = (e^{i\theta})^n.$$

We can also use proof by induction and trigonometric formulae.

Sometimes the expression $\cos\theta + i \sin\theta$ is abbreviated to cis.

## De Moivre's formula and trigonometry

De Moivre's formula and the binomial formula are often used to express polynomials in $\cos\th$ or $\sin\th$ of degree $n$ as a linear combination of $\cos(k\th)$ and $\sin(k\th)$ (for $0\le k\le n$) and vice versa.

To express $\cos(n\theta)$ and $\sin(n\theta)$ in powers of $\cos\theta$ and $\sin\theta$, we use \begin{gather*} \cos(n\th) = \Real(e^{in\th}) = \Real\big((\cos\th + i\sin\th)^n\big),\\ \sin(n\th) = \Imag(e^{in\th}) = \Imag\big((\cos\th + i\sin\th)^n\big). \end{gather*}

Writing $c=\cos\theta$ and $s=\sin\theta$, we can expand $\cos(4\th)$. \begin{gather*}\cos(4\theta) = \Real(c+is)^4 = c^4 + 6i^2c^2s^2+i^4s^4\\ =c^4-6c^2s^2+s^4. \end{gather*}

To express polynomials of $\sin\theta$ and $\cos\theta$ using sums of $\cos(n\theta)$ and $\sin(n\theta)$, we use $$\cos^n(\th) = \frac{(e^{i\th}+e^{-i\th})^n}{2^n} ,\quad \sin^n(\th) = \frac{(e^{i\th}-e^{-i\th})^n}{(2i)^n}$$

We can express $\sin^2\th$ in terms of $\cos(2\th)$. $$\sin^2\theta = -\frac{(e^{i\th}-e^{-i\th})^2}{4} =-\frac{e^{i2\th}-2 + e^{-i2\th}}{4} = \frac{1-\cos(2\th)}{2}.$$

## Complex roots of unity

The solutions of $z^n=1$, called the $n^{\text{th}}$ roots of unity, are given by $$z_k = e^{i 2k\pi/n},\quad k\in\{0,1,\dots,n-1\}.$$ To see this, we note that $(z_k)^n = e^{i 2kn\pi} = 1$. Since the equation has $n$ roots, we have them all.

The $n^{\text{th}}$ roots of unity sum to $0$. They are often denoted $\omega$. All except $1$ satisfy the equation $$\omega^{n-1} + \omega^{n-2} + \dots + \omega + 1 =\frac{\omega^n - 1}{\omega -1} = 0.$$

The roots of unity lie at the vertices of a regular polygon.

More generally, the solutions of $z^n = Z = R e^{i\Theta}$, called the $n^{\text{th}}$ roots of a complex number, are given by $$z_k = R^{1/n}e^{i(\Theta/n + 2k\pi/n)},\quad k\in\{0,1,\dots,n-1\}.$$

Third roots of a complex number