Implicit Function Theorem Proof

Proof of the Implicit Function Theorem

• Given $F: \mathbb{R}^{n+m} \rightarrow \mathbb{R}^m$ is $C^1$ with $F(a,b) = 0$, $a \in \mathbb{R}^n$, $b \in \mathbb{R}^m$, and $DF(a,b) = [A \quad B]$, where A is an $m \times n$ matrix and B is an invertible $m \times m$ matrix.
• It follows a function $g: U \rightarrow \mathbb{R}^m$ exists, where $U$ is a neighborhood about $a$ in $\mathbb{R}^n$, such that $g(a) = b$ and $F(x, g(x)) = 0$ for all $x \in U$.
• Define $H: \mathbb{R}^{n+m} \rightarrow \mathbb{R}^{n+m}$ by $H(x,y) = (x, F(x,y))$, so $H(a,b) = (a, 0)$.
• The derivative of H at (a,b) is: $$ DH(a,b) = \begin{bmatrix} I_n & 0 \ A & B \end{bmatrix} $$
• $DH(a,b)$ is invertible since B is invertible.
• By the Inverse Function Theorem, there exists $H^{-1}$ defined on a neighborhood $V$ of $(a,0)$ such that $H^{-1}(a,0) = (a,b)$.
• Let $H^{-1}(x,z) = (x, \psi(x,z))$, therefore $H(H^{-1}(x,z)) = (x,z) = H(x, \psi(x,z)) = (x, F(x, \psi(x,z)))$.
• It can be concluded that $F(x, \psi(x,z)) = z$.
• Define: $g(x) = \psi(x,0)$, then $F(x, g(x)) = F(x, \psi(x,0)) = 0$ and $H^{-1}(a,0) = (a,b) = (a, \psi(a,0)) = (a, g(a))$, so $g(a) = b$.

Proof of Rank Theorem

• Given $F: E \subset \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a $C^1$ mapping, and the derivative of F at x, $DF(x)$, has constant rank $r$ in a neighborhood of $a \in E$.
• Need to show that $C^1$ diffeomorphisms $G$ and $H$ exist.
• G maps a neighborhood of $a$ onto a neighborhood of 0 in $\mathbb{R}^n$.
• H maps a neighborhood of $F(a)$ onto a neighborhood of 0 in $\mathbb{R}^m$.
• $H \circ F \circ G^{-1}$ is a linear mapping $A$ which takes $(x_1, \dots, x_n)$ to $(x_1, \dots, x_r, 0, \dots, 0)$.
• Assuming the first $r$ columns of the derivative of F at a, $DF(a)$, are linearly independent then it follows:
  • $x \in \mathbb{R}^n$ as $x = (u,v)$ where $u \in \mathbb{R}^r$ and $v \in \mathbb{R}^{n-r}$.
  • Define $G: E \rightarrow \mathbb{R}^n$ by $G(u,v) = (F(u,v), v)$.
  • The derivative of G at a, $DG(a)$ = $$ \begin{bmatrix} D_uF(a) & D_vF(a) \ 0 & I_{n-r} \end{bmatrix} $$
• Because the first $r$ columns of $DF(a)$ i.e. $D_uF(a)$ are linearly independent, $DG(a)$ is invertible.
• By the Inverse Function Theorem, G is a $C^1$ diffeomorphism of a neighborhood of $a$ onto a neighborhood of $G(a)$.
• Let $G^{-1}(x,y) = (u,v)$, and the derivative of $G^{-1}$ at $G(a)$, $DG^{-1}(G(a))$ = $$ \begin{bmatrix} D_xF(a) & D_yF(a) \ 0 & I_{n-r} \end{bmatrix}^{-1} $$ must exist.
• Consider $F \circ G^{-1}(x,y) = F(u,v)$.
• Because $G(u,v) = (F(u,v), v) = (x,y)$, then $x = F(u,v)$ so $F \circ G^{-1}(x,y) = x$, $F \circ G^{-1}(x,y) = x = (x_1, \dots, x_r, 0, \dots, 0)$.
• Let $F \circ G^{-1} = \phi$.
• The derivative of $\phi$ at $G(a)$, $D\phi(G(a)) = DF(a) \circ DG^{-1}(G(a)) = DF(a) \circ DG(a)^{-1}$.
• Because the derivative of F, $DF$, has rank $r$ near $a$, $D\phi$ has rank $r$ near $G(a)$.
  • If $w \in \mathbb{R}^m$ and $y \in \mathbb{R}^{n-r}$, then $D\phi(w,y) = \begin{bmatrix} I_r & 0 \ 0 & 0 \end{bmatrix}$ so $\phi(w,y) = \phi(w,0)$.
• Define: $H: \mathbb{R}^m \rightarrow \mathbb{R}^m$ by $H(z) = z - \phi(0,z)$.
• $DH(0) = I - D\phi(0)$ is invertible.
• By the Inverse Function Theorem, H is a $C^1$ diffeomorphisim on a neighborhood of 0 in $\mathbb{R}^m$ onto a neighborhood of $H(0) = 0$. Therefore:
  • $H \circ \phi(w,y) = H(\phi(w,0)) = \phi(w,0) - \phi(0) = w$.
  • $H \circ F \circ G^{-1}(x,y) = H \circ \phi(x,y) = x = (x_1, \dots, x_r, 0, \dots, 0)$.

Introduction to Financial Accounting: Course Overview

• Focus on the basics of financial accounting.
• Understand how to read and interpret financial statements such as the balance sheet, income statement, and statement of cash flows.
• Learn about the accounting equation, different types of assets, liabilities, and equity, and the different accounting methods for recording and reporting financial transactions.

What is Accounting?

• Is a set of rules and methods used to measure and report financial information about a business.
• Accounting is a language, a system of communication, to speak about the financial health of a company.
• Without accounting, it would be very difficult to make informed decisions about a business.

Financial vs. Management Accounting

Who Uses Financial Accounting Information?

• Investors: to decide whether to buy, hold, or sell their investments.
• Creditors: to decide whether to lend money.
• Customers: to assess the company's ability to continue to supply goods and services.
• Employees: to assess the company's ability to pay wages and salaries.
• Regulators: to ensure that companies are complying with accounting standards.

Main Financial Statements

• Balance sheet: reports a company's assets, liabilities, and equity at a specific point in time.
• Income statement: reports a company's financial performance over a period of time.
• Statement of cash flows: reports a company's cash inflows and outflows over a period of time.
• Statement of changes in equity: reports a company's changes in equity over a period of time.

Accounting Equation

• The foundation of the balance sheet.
• The equation states that a company's assets are equal to the sum of its liabilities and equity.
• $Assets = Liabilities + Equity$
• Assets are things a company owns that will provide future economic benefits.
  • Example: cash, accounts receivable, inventory, and equipment.
• Liabilities are amounts a company owes to others.
  • Example: accounts payable, salaries payable, and debt.
• Equity is the owners' stake in the company.
  • Example: common stock and retained earnings.

Balance Sheet Overview

• Reports a company's assets, liabilities, and equity at a specific point in time.
• It is a snapshot of the company's financial position at a particular date.
• The balance sheet is based on the accounting equation: $Assets = Liabilities + Equity$

Balance Sheet: Assets, Liabilities, and Equity

• Assets: things a company owns that will provide future economic benefits.
  • Assets are usually listed in order of liquidity (how quickly they can be converted into cash).
    • Current assets: expected to be converted into cash within one year like Cash, Marketable Securities, Accounts Receivable, Inventory, Prepaid Expenses
    • Non-current assets: expected to be used for more than one year. like Property, Plant, and Equipment (PP&E), Intangible Assets, Long-term Investments
• Liabilities: amounts a company owes to others.
  • Liabilities are usually listed in order of when they are due.
    • Current liabilities: due within one year like Accounts Payable, Salaries Payable, Short-term Debt, Unearned Revenue
    • Non-current liabilities: due in more than one year like Long-term Debt, Deferred Tax Liabilities
• Equity: The owners' stake in the company.
  • Contributed Capital: The amount of money that owners have invested in the company.
  • Retained Earnings: The amount of net income that a company has earned over its lifetime that has not been paid out to owners as dividends.
  • $Equity = Contributed\ Capital + Retained\ Earnings$

Income Statement

• Reports a company's financial performance over a period of time.
• Summary of the company's revenues, expenses, and net income or loss for a specific period.
• also known as the profit and loss (P&L) statement.

Key Components of an Income Statement

• Revenues: The amount of money that a company earns from its business activities.
• Expenses: The costs that a company incurs to generate revenue.
• Net Income: The difference between revenues and expenses.
  • If revenues are greater than expenses, the company has net income.
  • If expenses are greater than revenues, the company has a net loss.
  • $Net\ Income = Revenues - Expenses$

Statement of Cash Flows

• Reports a company's cash inflows and outflows over a period of time.
• Summary of the company's cash receipts and cash payments for a specific period.
• Divided into three sections:
  • Cash flows from operating activities
  • Cash flows from investing activities
  • Cash flows from financing activities

Cash Flows from Operating Activities

• Cash flows from operating activities are the cash flows that result from the normal day-to-day operations of a business.
• Examples of cash inflows from operating activities include:
  • Cash receipts from sales of goods or services
  • Cash receipts from interest and dividends
• Examples of cash outflows from operating activities include:
  • Cash payments for purchases of inventory
  • Cash payments for salaries and wages
  • Cash payments for rent, utilities, and other operating expenses
  • Cash payments for interest
  • Cash payments for income taxes

Cash Flows from Investing Activities

• Investing activities are the cash flows that result from the purchase and sale of long-term assets.
• Examples of cash inflows from investing activities include:
  • Cash receipts from the sale of property, plant, and equipment (PP&E)
  • Cash receipts from the sale of investments
• Examples of cash outflows from investing activities include:
  • Cash payments for the purchase of property, plant, and equipment (PP&E)
  • Cash payments for the purchase of investments

Cash Flows from Financing Activities

• Financing activities are the cash flows that result from debt and equity financing.
• Examples of cash inflows from financing activities include:
  • Cash receipts from the issuance of debt
  • Cash receipts from the issuance of stock
• Examples of cash outflows from financing activities include:
  • Cash payments for the repayment of debt
  • Cash payments for the purchase of treasury stock
  • Cash payments for dividends

Financial Ratios

• Calculations based on financial statement data.
• Provide insights into a company's performance and financial health.
• Help to standardize and compare financial information across different companies and time periods.
• Can be used by investors, creditors, and managers to make informed decisions.

Categories of Financial Ratios

• Liquidity Ratios: Measure a company's ability to meet its short-term obligations.
• Solvency Ratios: Measure a company's ability to meet its long-term obligations.
• Profitability Ratios: Measure a company's ability to generate profits.
• Efficiency Ratios: Measure how efficiently a company is using its assets.
• Valuation Ratios: Measure the value of a company's stock.

Liquidity Ratios

• Current Ratio:
  • Formula: $\frac{Current\ Assets}{Current\ Liabilities}$
  • Interpretation: Measures a company's ability to pay its current liabilities with its current assets. A ratio of 2 or higher is generally considered good.
• Quick Ratio:
  • Formula: $\frac{Current\ Assets - Inventory}{Current\ Liabilities}$
  • Interpretation: Measures a company's ability to pay its current liabilities with its most liquid assets. A ratio of 1 or higher is generally considered good.

Solvency Ratios

• Debt-to-Equity Ratio:
  • Formula: $\frac{Total\ Debt}{Total\ Equity}$
  • Interpretation: Measures the amount of debt a company uses to finance its assets relative to the amount of equity. A lower ratio is generally considered better.

Profitability Ratios

• Gross Profit Margin:
  • Formula: $\frac{Gross\ Profit}{Net\ Sales}$
  • Interpretation: Measures the percentage of revenue that remains after deducting the cost of goods sold. A higher ratio is generally considered better.
• Net Profit Margin:
  • Formula: $\frac{Net\ Income}{Net\ Sales}$
  • Interpretation: Measures the percentage of revenue that remains after deducting all expenses. A higher ratio is generally considered better.
• Return on Equity (ROE):
  • Formula: $\frac{Net\ Income}{Average\ Equity}$
  • Interpretation: Measures how much profit a company generates with the money shareholders have invested. A higher ratio is generally considered better.

Course Intro: Program Manipulation

• Course is about programs that manipulate programs.
  • Examples: Compilers, interpreters, static analyzers, program transformers, macro systems.
• Focus on compilation to turn high-level source code into low-level machine code.

Compilers

• Read in source code, output machine code.
• Components:
  • Lexing: Break source file into tokens
  • Parsing: Turn tokens into an Abstract Syntax Tree (AST)
  • Semantic Analysis: Validate the AST, resolve names, check types, etc.
  • Optimization: Transform the AST to make the program faster/smaller
  • Code Generation: Translate AST into machine code

Why This Course?

• Become a better programmer:
  • Understand what your code is "doing."
  • How high-level languages work.
  • How to design a language
• Jobs: compiler engineer or related fields.

Activities

• Implementation a compiler for a simple language, a subset of Java--µJava.
• Compile and run simple µJava programs using your compiler
• Learn each stage of a compiler and modern compiler tools:
• ocamllex/menhir for lexing/parsing
• LLVM for code generation

LLVM

• Is a Low Level Virtual Machine
• is a compiler infrastructure project.
• Provides reusable libraries for all stages of compilation.
• Supports many languages and architectures.

Why LLVM?

• It's widely used
• It's well-documented
• It's modular
• It has a C++ API (and OCaml bindings)

Népérien Logarithm Function

• The Népérien Logarithm Function is denoted as $ln$
• Defined on $]0, +\infty[$
• Is the primitive of the function $x \longmapsto \frac{1}{x}$ that cancels to 1.

Népérien Function Properties

• $ln(1) = 0$
• $ln(e) = 1$
• For all real $x > 0$, $e^{ln(x)} = x$
• For all real $x$, $ln(e^x) = x$

Algebraic Properties

For all real numbers $a > 0$ and $b > 0$:

• $ln(ab) = ln(a) + ln(b)$
• $ln(\frac{a}{b}) = ln(a) - ln(b)$
• For all real numbers $r$, $ln(a^r) = r \cdot ln(a)$
• $ln(\frac{1}{b}) = -ln(b)$

Limits

• $\lim_{x \to +\infty} ln(x) = +\infty$
• $\lim_{x \to 0} ln(x) = -\infty$
• $\lim_{x \to +\infty} \frac{ln(x)}{x} = 0$
• $\lim_{x \to 0} x \cdot ln(x) = 0$
• $\lim_{h \to 0} \frac{ln(1+h)}{h} = 1$

Derivative

• The $ln$ function is differentiable on $]0, +\infty[$ and its derivative is defined by: $(ln(x))' = \frac{1}{x}$
• If $u$ is a differentiable and strictly positive function on an interval $I$, then the function $ln(u)$ is differentiable on $I$ and: $(ln(u))' = \frac{u'}{u}$

Variations

The $ln$ function is strictly increasing on $]0, +\infty[$.

Variation Table

Representative Curve

The curve starts from negative infinity as x approaches 0, crosses the x-axis at x=1, and increases slowly towards positive infinity as x increases.

Equations with $ln$

For all real numbers $a > 0$ and $b > 0$:

• $ln(a) = ln(b) \Leftrightarrow a = b$
• $ln(a) < ln(b) \Leftrightarrow a < b$
• $ln(a) > ln(b) \Leftrightarrow a > b$

Intro to Code Theory

Coding theory is the study of the properties of codes and their suitability for specific applications.

• Used for data compression, cryptography, error detection and correction, and network coding.
• Studied in information theory, computer science, math, and electronics.
• The goal is to find codes that transmit data reliably and quickly.

History of Coding Theory

• Started in 1948 with Claude Shannon's "A Mathematical Theory of Communication".
• Paper laid the theoretical foundation and showed that codes could reliably transmit info through noisy channels.
• One of the first codes was the Hamming code (1950), which can detect and correct single errors (used in computer memory).
• Reed-Solomon codes (1960s) are very efficient for correcting burst errors and are used in CD, DVD, and QR codes.

Types of Methods

There are many types of methodologies. Some of the most common methods are:

• Block codes
• Convolutional codes
• Turbo codes
• LDPC codes (Low-Density Parity-Check)

Applications

Has a wide range of applications in various areas, including:

• Data storage
• Communication
• Cryptography

Basic Concepts in Code Theory

• Alphabet
• A finite set of symbols. For example, the binary alphabet is Σ = {0, 1}.
• String
• A finite sequence of symbols from an alphabet. For example, 01101 is a string over the binary alphabet.
• Code
• A set of strings, where each string represents a message. For example, C = {00, 01, 10, 11}.
• Encoding
• The process of converting a message into a code string.
• Decoding
• Converting a code string into a message.
• Hamming Distance
• The number of positions in which the strings differ. For example, the Hamming distance between 10110 and 10011 is 2.
• Hamming Weight
• The number of non-zero symbols in the string. For example, the Hamming weight of 10110 is 3.

Block Codes

• Divide a message into fixed-size blocks and encode each block separately.
• Parameters:
  • n: The length of the encoded string.
  • k: The length of the original message.
  • d: The minimum Hamming distance between two distinct code strings.
• A code is denoted as a code (n, k, d). An example is the Hamming code (7, 4, 3)
  • Encodes 4-bit messages into 7-bit code strings.
  • The minimum Hamming distance between two distinct code strings is 3, detects up to 2 errors and corrects up to 1 error.

Fourier Transform Basics

• Any periodic function $f(t)$ with period $T$ can be expressed as a Fourier series:

  $f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)$

  where $\omega_0 = \frac{2\pi}{T}$, $a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \cos(n\omega_0 t) dt$, and $b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \sin(n\omega_0 t) dt$.

• Using Euler's formula, the Fourier series can be rewritten in complex form:

  $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$

  where $c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-jn\omega_0 t} dt$.

Properties of Non-Periodic Signal as Periodic Signal

• A non-periodic signal can be considered as a periodic signal with period approaching infinity.
• The Fourier transform represents non-periodic signals using complex exponentials.

Defining Fourier Transform

• Start with the complex Fourier series:

  $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$ and $c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-jn\omega_0 t} dt$

• Let $\Delta \omega = \omega_0 = \frac{2\pi}{T}$. Then $c_n = \frac{\Delta \omega}{2\pi} \int_{-T/2}^{T/2} f(t) e^{-jn\Delta \omega t} dt$.

• Define $F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$. Then $c_n = \frac{\Delta \omega}{2\pi} F(jn\Delta \omega)$.

• Substituting $c_n$ into the Fourier series:

  $f(t) = \sum_{n=-\infty}^{\infty} \frac{\Delta \omega}{2\pi} F(jn\Delta \omega) e^{jn\Delta \omega t}$

• As $T \rightarrow \infty$, $\Delta \omega \rightarrow d\omega$, and the summation becomes an integral:

  $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) e^{j\omega t} d\omega$

• This is the inverse Fourier transform.

• The Fourier transform is:

  $F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$

Conditions for Existence

• A sufficient (but not necessary) condition for the existence of the Fourier transform is that $f(t)$ is absolutely integrable:

  $\int_{-\infty}^{\infty} |f(t)| dt < \infty$

• This ensures that the integral defining $F(j\omega)$ converges.

Fourier Transform Properties

• Linearity: $a f(t) + b g(t) \overset{\mathcal{F}}{\longleftrightarrow} a F(j\omega) + b G(j\omega)$

• Time Shifting: $f(t - t_0) \overset{\mathcal{F}}{\longleftrightarrow} e^{-j\omega t_0} F(j\omega)$

• Frequency Shifting: $e^{j\omega_0 t} f(t) \overset{\mathcal{F}}{\longleftrightarrow} F(j(\omega - \omega_0))$

• Time Scaling: $f(at) \overset{\mathcal{F}}{\longleftrightarrow} \frac{1}{|a|} F(j\frac{\omega}{a})$

• Differentiation in Time: $\frac{df(t)}{dt} \overset{\mathcal{F}}{\longleftrightarrow} j\omega F(j\omega)$

• Integration in Time: $\int_{-\infty}^{t} f(\tau) d\tau \overset{\mathcal{F}}{\longleftrightarrow} \frac{1}{j\omega} F(j\omega) + \pi F(0)\delta(\omega)$

• Convolution in Time: $(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau \overset{\mathcal{F}}{\longleftrightarrow} F(j\omega) G(j\omega)$

• Multiplication in Time: $f(t) g(t) \overset{\mathcal{F}}{\longleftrightarrow} \frac{1}{2\pi} F * G = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\theta) G(j(\omega - \theta)) d\theta$

Parseval's Theorem

• Relates the energy of a signal in the time domain to the energy of its Fourier transform in the frequency domain.

  $\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(j\omega)|^2 d\omega$

• Useful for calculating the energy of a signal from its Fourier transform, and vice versa.

Linear Algebra and Analytical Geometry I

• Introduction
  • A linear equation system consists of equations of the form: $a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1$ $a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2$ ... $a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m$ where $$ are the unknowns, $$ are the coefficients, and $$ are the constant terms.

Resolution of Linear Systems

• Gauss Elimination Method
• Involves transforming the system of equations into a simpler, equivalent system by performing elementary row operations on the augmented matrix.
• Elementary Row Operations:
  • Interchange two rows.
  • Multiply a row by a nonzero scalar.
  • Add a multiple of one row to another row.

Geometric Interpretations

• For two variables, each linear equation represents a line in the plane.
• The solution of the system is the intersection of these lines. Three outcomes are possible include:
  • If the lines are parallel and distinct, there is no solution.
  • If the lines are coincident, there are infinitely many solutions.
  • If the lines intersect at a single point, there is a unique solution.

Applications

Linear equation systems are utilized in engineering, economics, and computer science.

Matrices

• A matrix is a rectangular array of numbers. A matrix with $$ rows and $$ columns is called an $$ matrix.

Operations on Matrices

• Additions, Subtractions, Multiplication

Matrix Transposition

The transpose of a matrix $$ is obtained by interchanging the rows and columns of $$

Specials Matricides

• Identity, and inverse

Determinants

• The determinant of a square matrix is a scalar value that provides information about the properties of the matrix. For matrix determinant: $det(A) = ad - bc$, where $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$. Property: $det(A^T) = det(A)$ $det(AB) = det(A)det(B)$

Vector Spaces

• A vector space is a set equipped with two operations (addition and scalar multiplication) that satisfy certain properties.
• A vector subspace is a subset of a vector space that is itself a vector space.

Basis and Dimensions

• A basis is a set of linearly independent vectors that span the vector space.
• The dimension is the number of vectors in a basis.
• Preserves operation with addition and scalar multiplication.
• Eigen operators linear and apply with matrices to find new values.

Complex Numbers: Definition and Representation

• The definition of a complex number, its real and imaginary parts, and the imaginary unit $i$.
• It can be written as $a + bi$ Real Part: $a$, denoted as $\operatorname{Re}(z)$ Imaginary Part: $b$, denoted as $\operatorname{Im}(z)$

Complex Plane

A complex number $z = a + bi$ can be represented as a point $(a, b)$ in the complex plane.

• Real Axis: The horizontal axis represents the real part ($a$).
• Imaginary Axis: The vertical axis represents the imaginary part ($b$).

Modulus and Argument

Modulus: $|z| = \sqrt{a^2 + b^2}$

Argument: $\theta = \arctan\left(\frac{b}{a}\right)$

Polar Form

• Can be expressed in polar form as:

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

Where:

• $r = |z|$ is the modulus of $z$
• $\theta = \arg(z)$ is the argument of $z$

Using Euler's formula, $e^{i\theta} = \cos \theta + i \sin \theta$, the polar form can be written more compactly as:

$z = re^{i\theta}$

Complex Numbers: Operations

Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ then Addition: $z_1 + z_2 = (a + c) + (b + d)i$ Subtractions: $z_1 - z_2 = (a - c) + (b - d)i$

Multiplication: $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$ In polar form: $z_1 = r_1e^{i\theta_1}$ and $z_2 = r_2e^{i\theta_2}$

$z_1 \cdot z_2 = r_1r_2e^{i(\theta_1 + \theta_2)}$ Division: $\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ In polar form:

$\frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$

Complex Conjugate

The complex conjugate of a complex number $z = a + bi$, denoted as $\overline{z}$, is obtained by changing the sign of the imaginary part:

$\overline{z} = a - bi$

In polar form:

If $z = re^{i\theta}$, then $\overline{z} = re^{-i\theta}$

• Properties:*
• $z \cdot \overline{z} = |z|^2 = a^2 + b^2$
• $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
• $\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$
• $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$

De Moivre's Theorem

For any complex number $z = r(\cos \theta + i \sin \theta)$ and any integer $n$:

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

Or, using Euler's formula:

$(re^{i\theta})^n = r^ne^{in\theta}$

Roots of Complex Numbers

To find the $n$th roots of a complex number $z = re^{i\theta}$, we use the formula:

$w_k = \sqrt[n]{r}e^{i\left(\frac{\theta + 2\pi k}{n}\right)}$

Where $k = 0, 1, 2, \dots, n-1$

This formula gives $n$ distinct roots, equally spaced around a circle of radius $\sqrt[n]{r}$ in the complex plane.

Calculus 1: The Derivative

The formal definition of derivatives: $f'(x)