# AxiomX 0.0.91596 — Catalan Edition

AxiomX is a **pure‑Python mathematical computation library** built from first principles, with a strong emphasis on *clarity*, *numerical methods*, and *mathematical structure*. It is designed for students, educators, and enthusiasts who want to understand how mathematics is implemented internally — not just call black‑box functions.

This release, **version 0.0.91596**, is titled the **Catalan Edition**, reflecting its focus on constants, convergence‑based methods, and classical analysis techniques.

---

## ✨ Design Philosophy

AxiomX follows a few guiding principles:

- **Pure Python** – no NumPy, SciPy, or external math engines
- **Readable algorithms** – numerical methods are written explicitly
- **Educational value** – suitable for learning numerical analysis
- **Extensible structure** – classes are organized by mathematical domain
- **Deterministic behavior** – avoids hidden optimizations

This makes AxiomX especially useful for:
- Understanding how special functions are computed
- Experimenting with numerical convergence
- Building your own symbolic or numeric math tools

---

## 📦 Package Overview

AxiomX is divided into logical mathematical domains, each implemented as a class:

```
AxiomX
├── Constants        # Mathematical constants
├── Arithmetic       # Basic arithmetic & equation solving
├── Calculus         # Integration & summation
├── Functions        # Roots, Gamma, Zeta, AGM, factorial
├── Exponential      # exp, ln, logarithms
├── Trigonometry     # Circular trigonometric functions
├── Hyperbolic       # Hyperbolic & inverse hyperbolic functions
```

Each class can be used independently or combined for advanced computation.

---

## 🔢 Constants

The `Constants` class exposes commonly used mathematical constants as class attributes.

Included constants:

- π (pi)
- e (Euler’s number)
- τ (tau = 2π)
- Euler–Mascheroni constant (γ)
- Lemniscate constant
- Gelfond constant
- Gelfond–Schneider constant
- Golden ratio (φ)
- Silver ratio
- Ramanujan constant
- Gauss constant
- √2, √3, √5
- Mathematical infinity

### Example

```python
from AxiomX import Constants

print(Constants.pi)
print(Constants.euler_mascheroni)
```

---

## ➗ Arithmetic

The `Arithmetic` module provides essential arithmetic utilities and a **linear equation solver**.

### Features

- Absolute value
- Reciprocal
- Single‑variable linear equation solver

### Linear Equation Solver

Supports equations of the form:

```
ax + b = cx + d
```

Example:

```python
from AxiomX import Arithmetic

solver = Arithmetic()
result = solver.solve_equation("2x - 3 = 7")
print(result)  # 5.0
```

Internally, expressions are parsed term‑by‑term without using `eval`.

---

## 📐 Calculus

The `Calculus` class implements numerical methods commonly taught in introductory analysis courses.

### Numerical Integration

- Uses **Simpson’s Rule**
- Suitable for smooth functions
- Configurable number of subdivisions

```python
from AxiomX import Calculus

calc = Calculus()
area = calc.integrate(lambda x: x**2, 0, 1)
```

### Summation

- Supports finite summations
- Approximates infinite bounds by truncation

```python
calc.summation(1, 1000, lambda n: 1/n**2)
```

> Infinite limits are internally approximated using large finite cutoffs.

---

## 🧮 Functions

The `Functions` class contains advanced numerical algorithms and special functions.

### Implemented Functions

- Square root (Halley’s method)
- Cube root (Newton–Raphson)
- Factorial (integer & real)
- Gamma function (Lanczos approximation)
- Arithmetic–Geometric Mean (AGM)
- Riemann Zeta function (ζ)
- Dirichlet Beta function (β)

### Square Root (Halley’s Method)

```python
from AxiomX import Functions

f = Functions()
print(f.sqrt(2))
```

### Gamma Function

Extends factorials to real numbers:

```python
f.gamma(5.5)
```

---

## 📈 Exponential & Logarithmic Functions

Implemented in the `Exponential` class.

### Available Functions

- exp(x)
- ln(x)
- log₂(x)
- log₁₀(x)
- log₍base₎(x)

### Natural Logarithm

The `ln(x)` implementation:

- Uses argument reduction
- Applies a convergent series
- Achieves high numerical stability

```python
from AxiomX import Exponential

ex = Exponential()
print(ex.ln(2))
```

---

## 📐 Trigonometry

The `Trigonometry` class provides circular trigonometric functions implemented using **Taylor series** and identities.

### Functions

- sin, cos, tan
- cot, sec, cosec
- arcsin, arccos, arctan
- Degree ↔ radian conversion

### Example

```python
from AxiomX import Trigonometry

trig = Trigonometry()
print(trig.sin(Constants.pi / 2))
```

Inverse functions are computed using **Newton’s method**.

---

## 🔁 Hyperbolic Functions

The `Hyperbolic` class implements hyperbolic functions using exponential definitions.

### Functions

- sinh, cosh, tanh
- coth, sech, cosech
- arcsinh, arccosh, arctanh

```python
from AxiomX import Hyperbolic

h = Hyperbolic()
print(h.tanh(1))
```

---

## ⚠️ Numerical Notes & Limitations

- AxiomX prioritizes **clarity over speed**
- Some functions use truncated infinite series
- Floating‑point precision depends on convergence thresholds
- Not intended as a replacement for SciPy in production

---

## 🧠 Intended Audience

AxiomX is ideal for:

- Students learning numerical methods
- Teachers demonstrating algorithmic math
- Math hobbyists and researchers
- Developers building symbolic math engines

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## 📜 License

MIT License — free to use, modify, and distribute.

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## 👤 Author

**Eric**

> “Mathematics should be transparent — not hidden behind black boxes.”

