Understanding Complex Numbers

Complex Numbers

• A complex number is in the form $a + bi$.
• $a$ and $b$ are real numbers.
• $i$ is the imaginary unit, where $i = \sqrt{-1}$.

Components

• Real Part: $a$
• Imaginary Part: $b$

Notation

• $z = a + bi$
• Re$(z) = a$, which represents the Real part of $z$.
• Im$(z) = b$, which represents the Imaginary part of $z$.

Operations with Complex Numbers

Addition

• $z_1 + z_2 = (a + c) + (b + d)i$ where $z_1 = a + bi$ and $z_2 = c + di$

Subtraction

• $z_1 - z_2 = (a - c) + (b - d)i$ where $z_1 = a + bi$ and $z_2 = c + di$

Multiplication

• $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$ where $z_1 = a + bi$ and $z_2 = c + di$
• Achieved using distributive property and $i^2 = -1$.

Division

• $\frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$ where $z_1 = a + bi$ and $z_2 = c + di$
• Achieved by multiplying numerator and denominator by the conjugate of the denominator.

Complex Conjugate

• Denoted as $\bar{z} = a - bi$ for a complex number $z = a + bi$.

Properties of Conjugates

• $\overline{(z_1 + z_2)} = \bar{z_1} + \bar{z_2}$
• $\overline{(z_1 - z_2)} = \bar{z_1} - \bar{z_2}$
• $\overline{(z_1 \cdot z_2)} = \bar{z_1} \cdot \bar{z_2}$
• $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$
• $z \cdot \bar{z} = a^2 + b^2$
• $\overline{\bar{z}} = z$

Modulus (Absolute Value)

• Denoted as $|z| = \sqrt{a^2 + b^2}$ for a complex number $z = a + bi$.

Properties of Modulus

• $|z| \geq 0$ for all $z \in \mathbb{C}$
• $|z| = 0$ if and only if $z = 0$
• $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
• $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$
• $|z_1 + z_2| \leq |z_1| + |z_2|$ (Triangle Inequality)
• $|z| = |\bar{z}|$
• $z \cdot \bar{z} = |z|^2$

Geometric Representation

• Complex numbers can be represented as points on complex plane or Argand diagram.

Coordinates

• Real part $a$ is on the horizontal axis (real axis).
• Imaginary part $b$ is on the vertical axis (imaginary axis).

Polar Form

• $z = r(\cos\theta + i\sin\theta)$
• $r = |z| = \sqrt{a^2 + b^2}$ is the modulus of $z$.
• $\theta = \arg(z)$ is the argument of $z$.

Euler's Formula

• $e^{i\theta} = \cos\theta + i\sin\theta$
• Using Euler's formula, $z = re^{i\theta}$.

De Moivre's Theorem

• $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$ where $z = r(\cos\theta + i\sin\theta)$
• In exponential form, $(re^{i\theta})^n = r^ne^{in\theta}$.

Roots of Complex Numbers

• $w_k = \sqrt[n]{r}\left(\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right)$
• Where $k = 0, 1, 2, \dots, n-1$.
• In exponential form: $w_k = \sqrt[n]{r}e^{i\left(\frac{\theta + 2\pi k}{n}\right)}$

Algorithmes de tri

Complexité

Complexité temporelle

• Measures the time needed for execution.
• Expressed as O, Θ, Ω.

Types de complexité

• Meilleur cas (Best case): Optimal performance.
• Cas moyen (Average case): Expected average performance.
• Pire cas (Worst case): Most unfavorable performance.

Complexité Spatiale

• Measures memory amount used by algorithm.
• Includes memory for variables, data structures, and execution stack.

Tris Classiques

Tri à bulles (Bubble Sort)

• Compares/exchanges adjacent elements to move largest element to end of array.

Algorithm

pour i de 0 à n-2 faire:
    pour j de 0 à n-2-i faire:
        si tab[j] > tab[j+1] alors:
            échanger(tab[j], tab[j+1])

Complexités

• Temps:
  • Meilleur cas: $O(n)$
  • Moyen cas: $O(n^2)$
  • Pire cas: $O(n^2)$
• Espace: $O(1)$

Tri par sélection (Selection Sort)

• Selects smallest element and places it in correct position.

Algorithm

pour i de 0 à n-1 faire:
    min_index = i
    pour j de i+1 à n faire:
        si tab[j] < tab[min_index] alors:
            min_index = j
    échanger(tab[i], tab[min_index])

Complexités

• Temps:
  • Meilleur cas: $O(n^2)$
  • Moyen cas: $O(n^2)$
  • Pire cas: $O(n^2)$
• Espace: $O(1)$

Tri par insertion (Insertion Sort)

• Inserts each element in correct position in already sorted part of array.

Algorithm

pour i de 1 à n faire:
    clé = tab[i]
    j = i - 1
    tant que j >= 0 et tab[j] > clé faire:
        tab[j+1] = tab[j]
        j = j - 1
    tab[j+1] = clé

Complexités

• Temps:
  • Meilleur cas: $O(n)$
  • Moyen cas: $O(n^2)$
  • Pire cas: $O(n^2)$
• Espace: $O(1)$

Tri rapide (Quicksort)

• Chooses pivot, partitions array around pivot, and recursively sorts two sub-arrays.

Algorithm

fonction quicksort(tab, gauche, droite):
    si gauche < droite alors:
        pivot_index = partitionner(tab, gauche, droite)
        quicksort(tab, gauche, pivot_index - 1)
        quicksort(tab, pivot_index + 1, droite)

fonction partitionner(tab, gauche, droite):
    pivot = tab[droite]
    i = gauche - 1
    pour j de gauche à droite - 1 faire:
        si tab[j] # Chemical Engineering Thermodynamics

## 3 Volumetric Properties of Pure Fluids

### 3.1 PVT Behavior of Pure Substances

#### 3.1.1 The Virial Equation
- $$\frac{PV}{RT} = 1 + \frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3} +...$$
- $$Z = 1 + \frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3} +...$$
 - $V$ is the molar volume
 - $B, C, D,...$ are virial coefficients, function of temperature and unique to chemical species
- $$\frac{PV}{RT} = 1 + B'P + C'P^2 + D'P^3 +...$$
- $$Z = 1 + B'P + C'P^2 + D'P^3 +...$$
- $B' = \frac{B}{RT}$
- $C' = \frac{C-B^2}{(RT)^2}$
- Virial coefficients relate to interactions between molecules.
- $B$ relates to interactions between pairs of molecules
- $C$ relates to interactions between triplets of molecules
- Virial equations are used as empirical equations.
- Truncated for engineering applications:
- $Z = 1 + \frac{B}{V}$
- $Z = 1 + B'P$

### 3.1.2 The Ideal Gas Equation
- $$PV = RT$$

### 3.1.3 Cubic Equations of State
- $$P = \frac{RT}{V-b} - \frac{a(T)}{\left( V+c \right) \left( V+d \right)}$$
 - $a(T)$ is temperature-dependent
 - $b, c, d$ are temperature-independent

#### van der Waals Equation of State
- $$P = \frac{RT}{V-b} - \frac{a}{V^2}$$
 - $a = \frac{27}{64} \frac{R^2 T_c^2}{P_c}$
 - $b = \frac{RT_c}{8P_c}$

#### Soave-Redlich-Kwong (SRK) Equation of State
- $$P = \frac{RT}{V-b} - \frac{\alpha(T)}{V(V+b)}$$
 - $a = \Omega_a \frac{R^2 T_c^2}{P_c}$
 - $b = \Omega_b \frac{RT_c}{P_c}$
 - $\Omega_a = 0.42748$
 - $\Omega_b = 0.08664$
 - $\alpha = \left[1+m(1-T_r^{0.5})\right]^2$
 - $T_r = \frac{T}{T_c}$
 - $m = 0.480 + 1.574\omega - 0.176\omega^2$

#### Peng-Robinson Equation of State
- $$P = \frac{RT}{V-b} - \frac{a(T)}{V(V+b) + b(V-b)}$$
 - $a = \Omega_a \frac{R^2 T_c^2}{P_c}$
 - $b = \Omega_b \frac{RT_c}{P_c}$
 - $\Omega_a = 0.45724$
 - $\Omega_b = 0.07780$
 - $\alpha = \left[1+m(1-T_r^{0.5})\right]^2$
 - $T_r = \frac{T}{T_c}$
 - $m = 0.37464 + 1.54226\omega - 0.26992\omega^2$

### 3.2 Generalized Correlations for Gases

#### 3.2.1 Corresponding States
- $$Z = Z(T_r, P_r)$$
 - $T_r = \frac{T}{T_c}$ is the reduced temperature
 - $P_r = \frac{P}{P_c}$ is the reduced pressure

#### 3.2.2 Acentric Factor
- $$\omega = -log_{10}(P_r^{sat})|_{T_r=0.7} - 1$$
 - $P_r^{sat}$ is the reduced saturation pressure

#### 3.2.3 Correlations for the Compressibility Factor
- $$Z = Z^0 + \omega Z^1$$
 - $Z^0$ and $Z^1$ are functions of $T_r$ and $P_r$.

#### 3.2.4 Correlations for the Virial Coefficients
- $$B = \frac{RT_c}{P_c}(B^0 + \omega B^1)$$
 - $B^0 = 0.083 - \frac{0.422}{T_r^{1.6}}$
 - $B^1 = 0.139 - \frac{0.172}{T_r^{4.2}}$

## Física

Here's a summary of the physics concepts described in the text:

### Vectores (Vectors)

#### Suma de vectores (Vector Addition)

##### Método Analítico (Analytical Method)
- Represents vectors using components.
- Vector components listed.
  - $\vec{A} = A_x\hat{i} + A_y\hat{j}$
  - $A_x = A\cos\theta$
  - $A_y = A\sin\theta$

- Resultant vector, found through formula.
  - $\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$
  - $R_x = A_x + B_x$
  - $R_y = A_y + B_y$
  - $R = \sqrt{R_x^2 + R_y^2}$
  - $\theta = \arctan(\frac{R_y}{R_x})$

##### Ejemplo (Example)

- $\vec{A} = 2\hat{i} + 2\hat{j}$
- $\vec{B} = -3\hat{i} + 4\hat{j}$
- $\vec{R} = \vec{A} + \vec{B} = (2-3)\hat{i} + (2+4)\hat{j} = -\hat{i} + 6\hat{j}$
- $R_x = -1$
- $R_y = 6$
- $R = \sqrt{(-1)^2 + 6^2} = \sqrt{37} = 6.08$
- $\theta = \arctan(\frac{6}{-1}) = -80.5^{\circ}$

#### Producto escalar (punto) (Dot Product)
- $\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$

#### Producto vectorial (cruz) (Cross Product)
- $\vec{A} \times \vec{B} = AB\sin\theta \hat{n}$
- $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$