Understanding Complex Numbers

Complex Numbers: Definition

• A complex number has the form $a + bi$.
• $a$ is the real part, denoted as $Re(z)$.
• $b$ is the imaginary part, denoted as $Im(z)$.
• $i$ is the imaginary unit, where $i^2 = -1$.
• $\mathbb{C}$ represents the set of complex numbers.

Geometric Representation

• Represented on a 2D plane called the complex plane or Argand diagram.
• The x-axis represents the real part.
• The y-axis represents the imaginary part.

Forms of Complex Numbers

• These are the several mathematical constructs representing a complex number.

Binomial Form

• $z = a + bi$, where $a, b \in \mathbb{R}$.

Conjugate

• The conjugate of $z = a + bi$ is $\overline{z} = a - bi$.
• Geometrically, it is the reflection of $z$ across the real axis.

Opposite

• The opposite of $z = a + bi$ is $-z = -a - bi$.

Polar Form (or Modulus-Argument Form)

• $z = r(\cos\theta + i\sin\theta)$.
• $r = |z| = \sqrt{a^2 + b^2}$ is the modulus (or absolute value). It represents the distance from the origin.
• $\theta = \arg(z)$ is the argument, representing the angle between the positive real axis and the line connecting the origin to to z.

Euler's Form

• $z = re^{i\theta}$.
• $r = |z|$ is the modulus.
• $\theta = \arg(z)$ is the argument.
• $e^{i\theta} = \cos\theta + i\sin\theta$ (Euler's formula).

Operations with Complex Numbers

• Let $z_1 = a + bi$ and $z_2 = c + di$.

Addition

• $z_1 + z_2 = (a + c) + (b + d)i$

Subtraction

• $z_1 - z_2 = (a - c) + (b - d)i$

Multiplication

• $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$

Division

• $\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Modulus Properties

• $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
• $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$
• $|z^n| = |z|^n$

Argument Properties

• $\arg(z_1 \cdot z_2) = \arg(z_1) + \arg(z_2)$
• $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$
• $\arg(z^n) = n \cdot \arg(z)$

Powers of $i$

• $i^0 = 1$
• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$
• $i^{4k} = 1$
• $i^{4k+1} = i$
• $i^{4k+2} = -1$
• $i^{4k+3} = -i$

De Moivre's Theorem

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

nth Roots of a Complex Number

• If $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, the $n$th roots of $z$ are given by:
• $w_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$ or
• $w_k = \sqrt[n]{r} e^{i\left(\frac{\theta + 2\pi k}{n}\right)}$, where $k = 0, 1, 2,..., n-1$.
• The nth roots of a complex number are equally spaced around a circle in the complex plane with radius $\sqrt[n]{r}$.

Fourier Transform Properties

• Describe how different operations on a function in either the time or frequency domain affect its Fourier transform in the other domain.

Linearity

• The Fourier transform is a linear operation: $a \cdot f(t) + b \cdot g(t) \iff a \cdot F(f) + b \cdot G(f)$

Time Shifting

• Shifting in the time domain corresponds to multiplying the Fourier transform by a complex exponential: $f(t - t_0) \iff e^{-j2\pi ft_0}F(f)$.

Frequency Shifting

• Shifting in the frequency domain involves multiplying the time-domain function by a complex exponential: $e^{j2\pi f_0t}f(t) \iff F(f - f_0)$.

Scaling

• Scaling in the time domain scales the frequency domain and multiplies by a factor: $f(at) \iff \frac{1}{|a|}F(\frac{f}{a})$.

Time Reversal

• Flipping a function in the time domain corresponds to flipping it in the frequency domain: $f(-t) \iff F(-f)$.

Multiplication

• Multiplication in the time domain corresponds to convolution in the frequency domain: $f(t) \cdot g(t) \iff F(f) * G(f)$.

Convolution

• Convolution in the time domain equates to multiplication in the frequency domain: $f(t) * g(t) \iff F(f) \cdot G(f)$.

Differentiation in Time

• Differentiation in the time domain corresponds to multiplication by $j2\pi f$ in the frequency domain: $\frac{d}{dt}f(t) \iff j2\pi fF(f)$.

Integration in Time

• Integration in the time domain equates to division by $j2\pi f$ in the frequency domain plus an impulse at the origin: $\int_{-\infty}^{t}f(\tau)d\tau \iff \frac{1}{j2\pi f}F(f) + \frac{1}{2}F(0)\delta(f)$.

Differentiation in Frequency

• Differentiation in the frequency domain corresponds to multiplication by $j2\pi t$ in the time domain: $-j2\pi tf(t) \iff \frac{d}{df}F(f)$.

Parseval's Theorem

• $\int_{-\infty}^{\infty} |f(t)|^2 dt = \int_{-\infty}^{\infty} |F(f)|^2 df$