Formulas and functions

Formulas and Functions in MediaWiki

1. Introduction

MediaWiki, the software powering Wikipedia and countless other wikis, offers a powerful system for incorporating mathematical formulas and functions into your pages. This capability is invaluable for technical documentation, scientific articles, financial analysis (like Technical Analysis), or any content requiring precise mathematical expression. This article provides a comprehensive guide to using formulas and functions within MediaWiki, aimed at beginners. We'll cover the core concepts, common functions, rendering methods, and best practices. Understanding these tools will significantly enhance the clarity and professionalism of your wiki content.

1. The Foundation: LaTeX and MathML

MediaWiki doesn't inherently understand mathematical notation. Instead, it relies on external markup languages to represent formulas. The two primary languages used are:

**LaTeX:** A widely-used typesetting system, particularly popular in scientific and technical fields. It's a text-based format using commands to define mathematical symbols and structures.
**MathML:** An XML-based language specifically designed for representing mathematical notation. It’s more semantic than LaTeX, meaning it focuses on *what* the math represents rather than *how* it should look.

MediaWiki typically converts LaTeX into MathML for rendering in web browsers. However, the user primarily interacts with LaTeX syntax.

1. Rendering Methods: Inline vs. Display

MediaWiki provides different ways to render formulas, depending on how you want them to appear within the text:

**Inline Formulas:** These are incorporated directly into the text flow. They are enclosed in single dollar signs (`$`). For example: `$E=mc^2$` will render as $E=mc^2$. Inline formulas are best for short expressions within a sentence.
**Display Formulas:** These are displayed on a separate line, centered, and often with more spacing. They are enclosed in double dollar signs (`$$`). For example:

   `$$
   \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}
   $$`

   will render as:

   $$
   \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}
   $$

   Display formulas are suitable for longer equations or those requiring emphasis.

**TeX (HTML) Rendering:** Using single backticks around LaTeX code (`\`LaTeX code\`) renders the LaTeX as HTML. This is helpful for showing code, not executing it. Example: `\`x = y + z\`` renders as `x = y + z`.

1. Basic LaTeX Syntax

Let's explore some fundamental LaTeX commands:

**Greek Letters:** Use backslash followed by the name of the letter.

   *   `\alpha` renders as α
   *   `\beta` renders as β
   *   `\gamma` renders as γ
   *   `\pi` renders as π
   *   `\theta` renders as θ

**Superscripts and Subscripts:** Use `^` for superscripts and `_` for subscripts.

   *   `x^2` renders as x²
   *   `a_i` renders as a_i
   *   `x^{22}` renders as x²² (use curly braces for multiple characters)
   *   `a_{ij}` renders as a_ij

**Fractions:** Use `\frac{numerator}{denominator}`.

   *   `\frac{1}{2}` renders as $\frac{1}{2}$

**Square Roots:** Use `\sqrt{expression}`.

   *   `\sqrt{x}` renders as $\sqrt{x}$
   *   `\sqrt[3]{x}` renders as $\sqrt[3]{x}$ (for cube roots, etc.)

**Summations and Integrals:** Use `\sum` and `\int` respectively.

   *   `\sum_{i=1}^{n} i` renders as $\sum_{i=1}^{n} i$
   *   `\int_a^b f(x) dx` renders as $\int_a^b f(x) dx$

**Matrices:** Use `\begin{matrix} ... \end{matrix}`.

   *   `\begin{matrix} a & b \\ c & d \end{matrix}` renders as $\begin{matrix} a & b \\ c & d \end{matrix}$

**Symbols:**

   *   `\pm` renders as ±
   *   `\times` renders as ×
   *   `\div` renders as ÷
   *   `\leq` renders as ≤
   *   `\geq` renders as ≥
   *   `\neq` renders as ≠
   *   `\infty` renders as ∞
   *   `\rightarrow` renders as →

1. Common Mathematical Functions in LaTeX

LaTeX provides commands for a wide range of mathematical functions:

**Trigonometric Functions:**

   *   `\sin(x)` renders as sin(x)
   *   `\cos(x)` renders as cos(x)
   *   `\tan(x)` renders as tan(x)
   *   `\arcsin(x)` renders as arcsin(x)
   *   `\arccos(x)` renders as arccos(x)
   *   `\arctan(x)` renders as arctan(x)

**Exponential and Logarithmic Functions:**

   *   `e^x` renders as e^x
   *   `\exp(x)` renders as exp(x)
   *   `\ln(x)` renders as ln(x) (natural logarithm)
   *   `\log(x)` renders as log(x) (base 10 logarithm)
   *   `\log_{10}(x)` renders as log₁₀(x) (logarithm with a specified base)

**Other Functions:**

   *   `\lim_{x \to \infty} f(x)` renders as $\lim_{x \to \infty} f(x)$ (limit)
   *   `\frac{d}{dx} f(x)` renders as $\frac{d}{dx} f(x)$ (derivative)
   *   `\int f(x) dx` renders as $\int f(x) dx$ (integral)
   *   `\lfloor x \rfloor` renders as ⌊x⌋ (floor function)
   *   `\lceil x \rceil` renders as ⌈x⌉ (ceiling function)
   *   `|x|` renders as |x| (absolute value)

1. Advanced LaTeX Concepts

**Environments:** LaTeX environments allow you to define specific structures for your formulas, such as equations, arrays, and matrices. We've already seen `matrix`. Other useful environments include:

   *   `equation`: For numbered equations.  `\begin{equation} x + y = z \end{equation}`
   *   `align`: For aligning multiple equations.
   *   `array`: For creating arrays.

**Brackets and Parentheses:** LaTeX automatically adjusts the size of brackets and parentheses based on the content they enclose. Use `\left(` and `\right)` to force specific sizes.
**Spacing:** Use `\,`, `\;`, `\quad`, and `\qquad` to add horizontal space.
**Color:** While generally discouraged for core mathematical notation, you can use the `\color{colorname}` command within a group (using curly braces) to color parts of a formula. Requires the `color` package (see below).

1. Using Packages

MediaWiki supports LaTeX packages that extend its functionality. To use a package, include the following line *before* your formula:

`\usepackage{packagename}`

For example, to use the `amsmath` package (highly recommended for more advanced math), add:

`\usepackage{amsmath}`

Some commonly used packages include:

**amsmath:** Provides enhanced mathematical environments and commands. Essential for complex formulas.
**amssymb:** Provides a wider range of mathematical symbols.
**color:** Allows you to add color to your formulas.
**graphicx:** Enables the inclusion of images within your formulas.

1. Tips for Writing Clear Formulas

**Use Parentheses Wisely:** Parentheses improve readability, especially in complex expressions.
**Break Down Complex Formulas:** Divide long formulas into smaller, more manageable parts. Use `align` or multiple `equation` environments.
**Define Variables:** Clearly define any variables you use in your formulas. Glossary of terms can be helpful.
**Test Your Formulas:** Always preview your formulas to ensure they render correctly.
**Accessibility:** Consider users who may be using screen readers. Provide alternative text descriptions for complex formulas. MathML helps with accessibility.

1. Troubleshooting Common Issues

**Formulas Not Rendering:** Ensure you are using the correct dollar sign delimiters (`$` or `$$`). Check for syntax errors in your LaTeX code.
**Incorrect Symbols:** Double-check the LaTeX command for the symbol you are trying to use. Refer to a LaTeX symbol guide.
**Rendering Problems:** Try a different web browser. Clear your browser's cache. Ensure your MediaWiki installation has the necessary LaTeX and MathML support.
**Package Errors:** If a package isn't loading, verify the `\usepackage{}` command is present and spelled correctly. Some packages may not be enabled on all MediaWiki installations.

1. Formulas in Financial Analysis

Formulas are central to Financial Modeling and Quantitative Analysis. Here are some examples relevant to trading and investment:

**Simple Moving Average (SMA):** `SMA = \frac{\sum_{i=1}^{n} P_i}{n}` (where P_i is the price on day i, and n is the period)
**Exponential Moving Average (EMA):** A more complex formula, often implemented in trading platforms. It’s used in Moving Average Convergence Divergence (MACD).
**Return on Investment (ROI):** `ROI = \frac{Net Profit}{Cost of Investment} \times 100\%`
**Sharpe Ratio:** `Sharpe Ratio = \frac{R_p - R_f}{\sigma_p}` (measures risk-adjusted return)
**Present Value (PV):** `PV = \frac{FV}{(1 + r)^n}` (where FV is future value, r is the discount rate, and n is the number of periods)
**Volatility:** `\sigma = \sqrt{\frac{\sum_{i=1}^{n} (R_i - \mu)^2}{n-1}}` (standard deviation of returns)
**Bollinger Bands:** `Upper Band = MA + 2\sigma`, `Lower Band = MA - 2\sigma` (uses moving average and standard deviation - see Bollinger Bands)
**Fibonacci Retracement Levels:** Calculated based on the Fibonacci sequence. Used in Fibonacci Trading.
**Relative Strength Index (RSI):** A momentum oscillator used to identify overbought or oversold conditions. See Relative Strength Index (RSI).
**Profit Factor:** `Profit Factor = Gross Profit / Gross Loss`. Used in Trading Strategy Evaluation.
**Kelly Criterion:** `f = \frac{bp - q}{b}` (optimal fraction of capital to bet - used in Position Sizing).
**Maximum Drawdown:** Measures the largest peak-to-trough decline during a specific period. Important for Risk Management.
**Black-Scholes Model:** A complex formula for pricing options. See Options Trading.
**Monte Carlo Simulation:** Uses random sampling to model the probability of different outcomes. Used in Algorithmic Trading.
**Value at Risk (VaR):** Estimates the potential loss in value of an asset or portfolio over a defined period. See Portfolio Management.

These are just a few examples. The ability to represent these formulas accurately in your wiki is crucial for clear and informative financial content. Consider exploring resources on Candlestick Patterns and Chart Patterns as well, which often involve geometric representations that benefit from formula integration. Furthermore, understanding Elliott Wave Theory helps in visually representing complex patterns.

1. Resources and Further Learning

**LaTeX Wikibook:** [1]
**Detexify:** [2] (helps you find LaTeX commands by drawing the symbol)
**TeXample:** [3] (online LaTeX editor)
**MediaWiki Help on Formulas:** Help:Math
**Overleaf:** [4] (Collaborative online LaTeX editor)
**Investopedia:** [5] (for financial definitions and examples)

1. Conclusion

Mastering formulas and functions in MediaWiki empowers you to create clear, concise, and professional documentation. While the initial learning curve might seem steep, the benefits are well worth the effort. Practice regularly, utilize the resources provided, and don't hesitate to experiment. Remember to prioritize readability and accessibility to ensure your content is valuable to all users. Understanding these tools is essential for any wiki dealing with technical, scientific, or financial information. Keep exploring Time Series Analysis and Statistical Arbitrage to further apply your knowledge.

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