Complex Numbers

Algebraic form: a complex number $z$ is written as $z=a+bi$ where $a,b\in\mathbb{R}$ are respectively the real part $(\Re(z)=a)$ and the imaginary part $(\Im(z)=b)$, and $i$ is the imaginary unit with $i^2=-1$.

Equality: $a+bi = c+di$ iff $a=c$ and $b=d$.

Sum: $(a+bi)+(c+di) = (a+c)+(b+d)i$.

Product: $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$ (use distributivity and $i^2=-1$).

Complex conjugate: $$\overline{z} = a-bi.$$ The conjugate reflects the point across the real axis. Properties: $$\overline{z_1+z_2}=\overline{z_1}+\overline{z_2},\qquad \overline{z_1 z_2}=\overline{z_1}\cdot\overline{z_2},$$ and $z\cdot\overline{z}=a^2+b^2$ (real and non-negative).

Modulus: $|z| = \sqrt{a^2+b^2}$. Geometrically, it is the distance from the origin to the point $(a,b)$ in the complex plane. Properties: $|z_1z_2|=|z_1||z_2|$, $|z_1+z_2|\leq|z_1|+|z_2|$ (triangle inequality).

Division: to compute $z_1/z_2$ (with $z_2\neq0$), multiply numerator and denominator by $\overline{z_2}$:

$$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}.$$

Argand–Gauss plane: complex numbers are represented as points $(a,b)$ in the Cartesian plane. The horizontal axis is the real axis, the vertical axis is the imaginary axis.

Trigonometric (polar) form: a complex number $z=a+bi$ can be written as:

$$z = r(\cos\theta + i\sin\theta),$$

where $r=|z|=\sqrt{a^2+b^2}$ is the modulus and $\theta = \arg(z)$ is the argument (angle with the positive real axis). The argument is defined modulo $2\pi$; the principal argument $\mbox{Arg}(z)\in(-\pi,\pi]$.

Conversion formulas: $a = r\cos\theta$, $b = r\sin\theta$; $\tan\theta = b/a$ (with quadrant adjustments).

Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$. This is one of the most beautiful formulas in mathematics, connecting exponentials and trigonometry. It gives the exponential form of a complex number:

$$z = r e^{i\theta}.$$

Euler's identity for $\theta=\pi$: $e^{i\pi}+1=0$, which relates five fundamental constants ($e$, $i$, $\pi$, $1$, $0$).

Working in exponential form greatly simplifies algebraic operations.

Product: $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$ — moduli multiply, arguments add.

Quotient: $$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$$ (provided $r_2\neq0$) — moduli divide, arguments subtract.

Power (De Moivre formula): for any integer $n$,

$$(r e^{i\theta})^n = r^n e^{in\theta} = r^n(\cos n\theta + i\sin n\theta).$$

De Moivre's theorem is extremely useful for computing high powers of complex numbers without expanding binomials.

Example: compute $(1+i)^{10}$. Write $1+i = \sqrt{2}e^{i\pi/4}$, then $(1+i)^{10} = (\sqrt{2})^{10}e^{i10\pi/4} = 2^5 e^{i5\pi/2} = 32 e^{i\pi/2} = 32i$.

The $n$-th roots of a complex number $z = r e^{i\theta}$ are given by:

$$z_k = r^{1/n} e^{i(\theta+2\pi k)/n}, \qquad k = 0,1,\dots,n-1.$$

There are exactly $n$ distinct $n$-th roots, equally spaced on a circle of radius $r^{1/n}$ in the complex plane. The angle between consecutive roots is $2\pi/n$.

Roots of unity: when $z=1$ ($r=1,\theta=0$), the $n$-th roots of unity are:

$$\omega_k = e^{2\pi i k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k=0,1,\dots,n-1.$$

Geometrically, they are the vertices of a regular $n$-gon inscribed in the unit circle, with one vertex at $(1,0)$.

Properties: $$\sum_{k=0}^{n-1} \omega_k = 0$$ (the centroid is at the origin); $(\omega_1)^n = 1$; the roots form a cyclic group under multiplication.

The complex exponential is defined by extending the real exponential to the complex plane:

$$e^z = e^{x+iy} = e^x(\cos y + i\sin y).$$

This definition uses Euler's formula and preserves all analytic properties: $(e^z)' = e^z$, and $e^{z_1+z_2}=e^{z_1}e^{z_2}$.

Properties:

• $|e^z| = e^x$ (the modulus depends only on the real part)
• $\arg(e^z) = y \pmod{2\pi}$ (the argument is the imaginary part)
• $e^z$ is periodic with period $2\pi i$: $e^{z+2\pi i}=e^z$

Trigonometric functions in complex form:

$$\cos z = \frac{e^{iz}+e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz}-e^{-iz}}{2i}.$$

Application — AC circuits: in electrical engineering, sinusoidal voltages and currents are represented as $V(t) = V_0 e^{i(\omega t+\phi)}$ (phasor notation). The real part $V_0\cos(\omega t+\phi)$ is the physical signal. Using complex exponentials simplifies the analysis of RLC circuits because differentiation and integration become algebraic operations on phasors.

Solving equations in $\mathbb{C}$ means finding all $z\in\mathbb{C}$ that satisfy a given equation. The fundamental theorem of algebra (Gauss, 1799) states that every polynomial of degree $n\geq1$ with complex coefficients has exactly $n$ roots in $\mathbb{C}$ (counted with multiplicity).

Polynomial equations: to solve $P(z)=0$ of degree $n$:

• If the polynomial is real (real coefficients), complex roots always occur in conjugate pairs.
• For quadratic equations $az^2+bz+c=0$, the quadratic formula $z = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ holds in $\mathbb{C}$ too, with $\sqrt{\Delta}$ denoting a complex square root.
• For higher degrees: factorization, search for elementary roots, substitutions (e.g. $w=z^n$ for trinomial equations).

Factorization: every polynomial $P(z)$ can be split into linear factors: $P(z)=a_n(z-z_1)(z-z_2)\cdots(z-z_n)$, where $z_k$ are the roots (not necessarily distinct).