Polar Coordinates and Complex Numbers

A complex number has the form $z=a+bi$ where $a$ is the real part, $b$ is the imaginary part, and $i^2=-1$.

• Add / subtract: combine real parts and imaginary parts separately.
• Multiply: FOIL, then replace $i^2=-1$.
• Conjugate: $\bar z=a-bi$. To divide, multiply top and bottom by the conjugate of the denominator.
• Complex plane: plot $z=a+bi$ as the point $(a,b)$; the horizontal axis is real (Re), the vertical axis is imaginary (Im).

For $z=a+bi$:

The trigonometric (polar) form is

where $r$ is the modulus and $\theta$ is the argument. To go back to rectangular form, evaluate $a=r\cos\theta$ and $b=r\sin\theta$.

Let $z_1=r_1\,\text{cis}\,\theta_1$ and $z_2=r_2\,\text{cis}\,\theta_2$.

DeMoivre's Theorem: for any positive integer $n$,

Every nonzero complex number $z=r\,\text{cis}\,\theta$ has exactly $n$ distinct $n$th roots:

All $n$ roots lie on a circle of radius $\sqrt[n]{r}$, equally spaced $\dfrac{360^\circ}{n}$ apart.

A polar point $(r,\theta)$ is located by rotating $\theta$ from the positive $x$-axis, then moving $r$ along that ray (if $r<0$, move backward, i.e. $180^\circ$ opposite).

Multiple representations: $(r,\theta)=(r,\theta+360^\circ)=(-r,\theta+180^\circ)$ all name the same point.

Use $x=r\cos\theta$, $y=r\sin\theta$, and $r^2=x^2+y^2$.

• Polar $\to$ Rect: multiply through by $r$ to create $r^2$, $r\cos\theta$, or $r\sin\theta$, then substitute.
• Rect $\to$ Polar: replace $x,y$; solve for $r$. Useful: $r\cos\theta=x$, $r\sin\theta=y$, so a vertical line $x=c$ becomes $r=c\sec\theta$ and a horizontal line $y=c$ becomes $r=c\csc\theta$.

• Circle: $r=a$ (center origin), $r=a\cos\theta$ or $r=a\sin\theta$ (through origin).
• Cardioid: $r=a\pm a\cos\theta$ or $r=a\pm a\sin\theta$ (heart shape; $a=b$).
• Lima con: $r=a\pm b\cos\theta$ or $r=a\pm b\sin\theta$. Ratio $\tfrac{a}{b}$: $<1$ inner loop, $=1$ cardioid, $1<\tfrac{a}{b}<2$ dimpled, $\ge 2$ convex.
• Rose: $r=a\cos(n\theta)$ or $r=a\sin(n\theta)$. Petal length $|a|$; $n$ odd $\Rightarrow n$ petals, $n$ even $\Rightarrow 2n$ petals.
• Lemniscate: $r^2=a^2\cos2\theta$ or $r^2=a^2\sin2\theta$ (figure-eight).

The $n$th roots of unity are the solutions of $z^n=1$. Writing $\omega=\text{cis}\dfrac{360^\circ}{n}$, they are the $n$ equally spaced points

Two identities fall out of the geometry:

• Sum is zero: $\displaystyle\sum_{k=0}^{n-1}\omega^k=1+\omega+\dots+\omega^{\,n-1}=0$ for $n\ge 2$ (finite geometric series with ratio $\omega\ne1$ and $\omega^n=1$). The roots balance about the origin.
• Product identity: $\displaystyle\prod_{k=1}^{n-1}\bigl(1-\omega^k\bigr)=n$. (Factor $z^n-1=\prod_{k=0}^{n-1}(z-\omega^k)$, divide by $z-1$, then let $z\to1$.)

If $z=\text{cis}\,\theta$ lies on the unit circle then $\dfrac1z=z^{-1}=\text{cis}(-\theta)=\cos\theta-i\sin\theta$. By DeMoivre, $z^n=\text{cis}\,n\theta$ and $z^{-n}=\text{cis}(-n\theta)$, so adding and subtracting gives

The special case $z+\dfrac1z=2\cos\theta$ is the workhorse: raising it to a power and expanding lets you rewrite $\cos^m\theta$ as a sum of terms $\cos k\theta$ (Fourier/power-reduction identities).

Treat $z=x+iy$ as the point $(x,y)$ and read $|z-a|$ as the distance from $z$ to the point $a$.

• $|z-a|=k$ is a circle of radius $k$ centered at $a$.
• $|z-a|=|z-b|$ is the perpendicular bisector of the segment $ab$.
• $|z-a|=k|z-b|$ with $k\ne1$ is the Apollonius circle: the set of points whose distances to $a$ and $b$ are in a fixed ratio $k$.

To find it, square both sides and expand with $|z-a|^2=(x-a_1)^2+(y-a_2)^2$; the $x^2$ and $y^2$ terms survive with equal coefficients, so the result is always a circle.

A thin wedge of angle $d\theta$ at radius $r$ is a circular sector of area $\tfrac12 r^2\,d\theta$. Summing wedges, the area swept by $r=f(\theta)$ from $\theta=\alpha$ to $\theta=\beta$ is

Choosing limits: for one rose petal, integrate between the two consecutive $\theta$ where $r=0$; for a full cardioid, use $0$ to $360^\circ$ once around. For the area between two curves, subtract: $A=\tfrac12\int(r_{\text{out}}^2-r_{\text{in}}^2)\,d\theta$.