Complex Number Formula Handbook

A Comprehensive Reference for Students and Engineers

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1 Basic Definitions

A complex number is a number of the form \(z = a + ib\), where \(a, b \in \mathbb{R}\) and \(i = \sqrt{-1}\) is the imaginary unit.

Real part: \(\text{Re}(z) = a\)
Imaginary part: \(\text{Im}(z) = b\)
Imaginary unit: \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\)
Purely real: \(z\) is real if \(b = 0\)
Purely imaginary: \(z\) is purely imaginary if \(a = 0\) and \(b \neq 0\)

2 Algebraic Operations

Addition

\[(a+ib) + (c+id) = (a+c) + i(b+d)\]

Subtraction

\[(a+ib) - (c+id) = (a-c) + i(b-d)\]

Multiplication

\[(a+ib)(c+id) = (ac-bd) + i(ad+bc)\]

Division

\[\frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{(c+id)(c-id)} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}\]

Note: At least one of \(c\) and \(d\) must be non-zero.

3 Complex Conjugate

The conjugate of \(z = a + ib\) is denoted \(\bar{z}\) or \(z^*\):

\[\bar{z} = a - ib\]

Properties of Conjugate

\(\overline{\bar{z}} = z\)
\(\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}\)
\(\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}\)
\(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}\)
\(z \cdot \bar{z} = |z|^2 = a^2 + b^2\)
\(z + \bar{z} = 2\text{Re}(z)\)
\(z - \bar{z} = 2i\text{Im}(z)\)
\(\overline{z^n} = (\bar{z})^n\)
\(z\) is real if and only if \(z = \bar{z}\)
\(z\) is purely imaginary if and only if \(z = -\bar{z}\)

4 Modulus (Absolute Value)

The modulus of \(z = a + ib\) is:

\[|z| = \sqrt{a^2 + b^2} = \sqrt{z \cdot \bar{z}}\]

Properties of Modulus

\(|z| \geq 0\) and \(|z| = 0\) if and only if \(z = 0\)
\(|z| = |\bar{z}| = |-z|\)
\(|\text{Re}(z)| \leq |z|\) and \(|\text{Im}(z)| \leq |z|\)
\(-|z| \leq \text{Re}(z) \leq |z|\)
\(-|z| \leq \text{Im}(z) \leq |z|\)
\(|z^n| = |z|^n\) for \(n \in \mathbb{N}\)
\(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)
\(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) (for \(z_2 \neq 0\))
Triangle Inequality: \(|z_1 + z_2| \leq |z_1| + |z_2|\)
Reverse Triangle Inequality: \(|z_1 - z_2| \geq ||z_1| - |z_2||\)
\(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2\text{Re}(z_1\bar{z_2})\)
\(|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2\text{Re}(z_1\bar{z_2})\)
Parallelogram Law: \(|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)\)

5 Argument (Phase Angle)

The argument of \(z = a + ib\) is the angle \(\theta\) that the line from origin to \(z\) makes with the positive real axis:

\[\arg(z) = \theta = \arctan\left(\frac{b}{a}\right)\]

Important: Consider the quadrant when finding argument:

Quadrant I: \(\theta = \arctan(b/a)\)
Quadrant II: \(\theta = \pi - \arctan(|b|/|a|)\)
Quadrant III: \(\theta = -\pi + \arctan(|b|/|a|)\)
Quadrant IV: \(\theta = -\arctan(|b|/|a|)\)

Principal Argument

The principal value of argument, denoted \(\text{Arg}(z)\), satisfies \(-\pi < \text{Arg}(z) \leq \pi\).

\[\arg(z) = \text{Arg}(z) + 2n\pi, \quad n \in \mathbb{Z}\]

Properties of Argument

\(\arg(z_1 \cdot z_2) = \arg(z_1) + \arg(z_2) + 2n\pi\)
\(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) + 2n\pi\)
\(\arg(z^n) = n\arg(z) + 2k\pi\)
\(\arg(\bar{z}) = -\arg(z)\)
\(\arg(-z) = \arg(z) \pm \pi\)

6 Different Representations

Cartesian (Rectangular) Form

\[z = a + ib\]

where \(a = \text{Re}(z)\) and \(b = \text{Im}(z)\)

Polar Form

\[z = r(\cos\theta + i\sin\theta) = r\text{cis}(\theta) = r\angle\theta\]

where \(r = |z|\) and \(\theta = \arg(z)\)

Exponential (Euler) Form

\[z = re^{i\theta}\]

Euler's Formula: \(e^{i\theta} = \cos\theta + i\sin\theta\)

Conversion Formulas

From	To	Formula
Cartesian	Polar	\(r = \sqrt{a^2 + b^2}\), \(\theta = \arctan(b/a)\)
Polar	Cartesian	\(a = r\cos\theta\), \(b = r\sin\theta\)

7 Powers and Roots

Powers of Complex Numbers

\[z^n = r^n(\cos(n\theta) + i\sin(n\theta)) = r^ne^{in\theta}\]

De Moivre's Theorem

\[(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\]

This holds for all \(n \in \mathbb{R}\)

nth Roots of Complex Numbers

The \(n\) distinct \(n\)th roots of \(z = re^{i\theta}\) are:

\[z_k = r^{1/n}e^{i(\theta + 2\pi k)/n} = r^{1/n}\left[\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right]\]

where \(k = 0, 1, 2, \ldots, n-1\)

nth Roots of Unity

The \(n\) roots of the equation \(z^n = 1\) are:

\[\omega_k = e^{2\pi ik/n} = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right)\]

where \(k = 0, 1, 2, \ldots, n-1\)

Properties:

Sum of all \(n\)th roots of unity: \(\sum_{k=0}^{n-1}\omega_k = 0\)
Product of all \(n\)th roots of unity: \(\prod_{k=0}^{n-1}\omega_k = (-1)^{n+1}\)

8 Exponential and Logarithmic Functions

Exponential Function

\[e^z = e^{a+ib} = e^a(\cos b + i\sin b) = e^ae^{ib}\]

Important Exponential Identities

\(e^{i\pi} = -1\) (Euler's Identity)
\(e^{2\pi i} = 1\)
\(e^{i\pi/2} = i\)
\(|e^{iz}| = e^{-\text{Im}(z)}\) for any complex \(z\)

Logarithm of Complex Numbers

\[\ln(z) = \ln|z| + i\arg(z) = \ln r + i(\theta + 2n\pi)\]

The principal value is obtained when \(n = 0\):

\[\text{Log}(z) = \ln|z| + i\text{Arg}(z)\]

9 Trigonometric and Hyperbolic Functions

Trigonometric Functions

\[\cos z = \frac{e^{iz} + e^{-iz}}{2}\]

\[\sin z = \frac{e^{iz} - e^{-iz}}{2i}\]

\[\tan z = \frac{\sin z}{\cos z}\]

Hyperbolic Functions

\[\cosh z = \frac{e^z + e^{-z}}{2}\]

\[\sinh z = \frac{e^z - e^{-z}}{2}\]

\[\tanh z = \frac{\sinh z}{\cosh z}\]

Relationships

\(\cos(iz) = \cosh z\)
\(\sin(iz) = i\sinh z\)
\(\cosh(iz) = \cos z\)
\(\sinh(iz) = i\sin z\)

10 Geometric Interpretations

Distance Formula

The distance between two complex numbers \(z_1\) and \(z_2\) is:

\[d(z_1, z_2) = |z_1 - z_2|\]

Important Geometric Relations

Circle: \(|z - z_0| = r\) represents a circle with center \(z_0\) and radius \(r\)
Perpendicular bisector: \(|z - z_1| = |z - z_2|\) is the perpendicular bisector of the line segment joining \(z_1\) and \(z_2\)
Ellipse: \(|z - z_1| + |z - z_2| = k\) (where \(k > |z_1 - z_2|\))
Hyperbola: \(||z - z_1| - |z - z_2|| = k\) (where \(k < |z_1 - z_2|\))
Straight line: \(z = z_1 + t(z_2 - z_1)\), \(t \in \mathbb{R}\)

11 Important Inequalities

Triangle Inequality: \(|z_1 + z_2| \leq |z_1| + |z_2|\)
Generalized Triangle Inequality: \(|z_1 + z_2 + \cdots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|\)
Reverse Triangle Inequality: \(||z_1| - |z_2|| \leq |z_1 - z_2|\)
\(|z_1 - z_2| \leq |z_1| + |z_2|\)
\(|z_1 + z_2| \geq ||z_1| - |z_2||\)

12 Special Results and Theorems

Fundamental Theorem of Algebra

Every non-constant polynomial of degree \(n\) with complex coefficients has exactly \(n\) complex roots (counting multiplicities).

Square Root of Complex Numbers

If \(z = a + ib\), then \(\sqrt{z} = \pm(x + iy)\) where:

\[x = \sqrt{\frac{|z| + a}{2}}, \quad y = \text{sgn}(b)\sqrt{\frac{|z| - a}{2}}\]

Square Root of i

\[\sqrt{i} = \pm\frac{1+i}{\sqrt{2}}\]

\[\sqrt{-i} = \pm\frac{1-i}{\sqrt{2}}\]

Rotation Formula

Multiplying a complex number \(z\) by \(e^{i\alpha}\) rotates it by angle \(\alpha\) counterclockwise:

\[z' = ze^{i\alpha}\]

13 Useful Identities

\((1 + i)^2 = 2i\)
\((1 - i)^2 = -2i\)
\(i^n\) has period 4: \(i^{4k} = 1\), \(i^{4k+1} = i\), \(i^{4k+2} = -1\), \(i^{4k+3} = -i\)
\(\frac{1}{i} = -i\)
\(|z_1||z_2|\cdots|z_n| = |z_1 z_2 \cdots z_n|\)
If \(|z| = 1\), then \(z\bar{z} = 1\) and \(\frac{1}{z} = \bar{z}\)
If \(z_1z_2 + z_3z_4 = 0\) and \(z_1 + z_3 = 0\), then \(z_2 = z_4\)

Complex Number Formula Handbook | Comprehensive Reference Guide