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# Mathematical Expressions
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Mathematical typesetting with equations, matrices, aligned environments, and vector notation. PyLaTeX provides comprehensive support for LaTeX's mathematical capabilities, enabling the creation of complex equations, matrices, and mathematical symbols with proper formatting.
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## Capabilities
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### Math Environment
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The Math class provides the foundation for mathematical expressions, supporting both inline and display math modes.
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```python { .api }
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class Math(Container):
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    def __init__(self, *, inline=False, data=None, escape=None):
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        """
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        Create a math environment for equations.
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        Parameters:
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        - inline: bool, if True creates inline math ($...$), if False creates display math (\\[...\\])
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        - data: list or str, initial mathematical content
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        - escape: bool, whether to escape special characters (usually False for math)
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        Requires:
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        - amsmath package (automatically added)
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        """
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```
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Usage example:
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```python
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from pylatex import Document, Section
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from pylatex.math import Math
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doc = Document()
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with doc.create(Section('Mathematical Expressions')):
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    # Display math (block)
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    with doc.create(Math()) as math:
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        math.append(r'E = mc^2')
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    # Inline math
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    doc.append('The famous equation ')
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    doc.append(Math(data=r'E = mc^2', inline=True))
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    doc.append(' was derived by Einstein.')
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    # Complex expressions
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    with doc.create(Math()) as math:
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        math.append(r'\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}')
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```
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### Aligned Equations
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The Alignat environment creates aligned multi-line equations with proper spacing and alignment.
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```python { .api }
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class Alignat(Environment):
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    def __init__(self, aligns=2, numbering=True, escape=None):
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        """
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        Create an aligned equation environment.
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        Parameters:
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        - aligns: int, number of alignment points (default 2)
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        - numbering: bool, whether to number equations
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        - escape: bool, whether to escape special characters
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        Requires:
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        - amsmath package (automatically added)
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        """
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```
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Usage example:
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```python
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Alignat
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doc = Document()
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with doc.create(Section('System of Equations')):
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    # Basic aligned equations
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    with doc.create(Alignat(numbering=True)) as agn:
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        agn.append(r'2x + 3y &= 7 \\')
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        agn.append(r'x - y &= 1')
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    # Multiple alignment points
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    with doc.create(Alignat(aligns=3)) as agn:
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        agn.append(r'f(x) &= (x+a)(x+b) &= x^2 + (a+b)x + ab \\')
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        agn.append(r'g(x) &= (x-a)(x-b) &= x^2 - (a+b)x + ab')
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    # Unnumbered equations
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    with doc.create(Alignat(numbering=False)) as agn:
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        agn.append(r'\nabla \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\')
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        agn.append(r'\nabla \times \vec{B} &= \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}')
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```
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### Matrix Environments
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The Matrix class creates various types of mathematical matrices with automatic formatting.
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```python { .api }
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class Matrix(Environment):
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    def __init__(self, matrix, *, mtype="p", alignment=None):
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        """
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        Create a matrix environment.
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        Parameters:
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        - matrix: numpy.ndarray, matrix data as numpy array
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        - mtype: str, matrix type:
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          - 'p': parentheses ()
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          - 'b': brackets []
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          - 'B': braces {}
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          - 'v': vertical bars ||
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          - 'V': double vertical bars ||||
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        - alignment: str, column alignment ('c', 'l', 'r')
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        Requires:
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        - amsmath package (automatically added)
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        """
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```
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Usage example:
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```python
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import numpy as np
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Matrix, Math
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doc = Document()
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with doc.create(Section('Matrices')):
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    # Basic matrix with parentheses (requires numpy)
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    matrix_data = np.array([[1, 2, 3], 
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                           [4, 5, 6], 
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                           [7, 8, 9]])
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    with doc.create(Math()) as math:
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        math.append('A = ')
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        math.append(Matrix(matrix_data, mtype='p'))
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    # Matrix with brackets
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    identity = [['1', '0', '0'],
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                ['0', '1', '0'], 
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                ['0', '0', '1']]
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    with doc.create(Math()) as math:
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        math.append('I = ')
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        math.append(Matrix(identity, mtype='b'))
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    # Determinant with vertical bars
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    det_matrix = [['a', 'b'], 
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                  ['c', 'd']]
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    with doc.create(Math()) as math:
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        math.append(r'\det(M) = ')
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        math.append(Matrix(det_matrix, mtype='v'))
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        math.append(' = ad - bc')
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    # Matrix with mathematical expressions
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    symbolic_matrix = [[NoEscape(r'\alpha'), NoEscape(r'\beta')], 
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                       [NoEscape(r'\gamma'), NoEscape(r'\delta')]]
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    with doc.create(Math()) as math:
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        math.append('Greek = ')
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        math.append(Matrix(symbolic_matrix, mtype='B'))
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```
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### Vector Notation
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The VectorName class creates properly formatted vector names using LaTeX's mathbf command.
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```python { .api }
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class VectorName(Command):
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    def __init__(self, name):
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        """
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        Create a bold vector name.
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        Parameters:
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        - name: str, vector name (will be rendered in bold math font)
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        """
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```
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Usage example:
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```python
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from pylatex import Document, Section
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from pylatex.math import Math, VectorName
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doc = Document()
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with doc.create(Section('Vector Operations')):
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    # Vector definitions
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    with doc.create(Math()) as math:
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        math.append(VectorName('v'))
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        math.append(' = ')
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        math.append(VectorName('a'))
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        math.append(' + ')
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        math.append(VectorName('b'))
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    # Vector components
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    with doc.create(Math()) as math:
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        math.append(VectorName('r'))
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        math.append(r' = \begin{pmatrix} x \\ y \\ z \end{pmatrix}')
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    # Cross product
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    with doc.create(Math()) as math:
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        math.append(VectorName('c'))
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        math.append(' = ')
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        math.append(VectorName('a'))
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        math.append(r' \times ')
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        math.append(VectorName('b'))
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```
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## Advanced Mathematical Constructs
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### Complex Equations
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```python
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Math, Alignat
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doc = Document()
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with doc.create(Section('Advanced Mathematics')):
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    # Integral equations
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    with doc.create(Math()) as math:
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        math.append(NoEscape(r'''
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            \oint_{\partial S} \vec{F} \cdot d\vec{l} = 
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            \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS
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        '''))
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    # Series expansions
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    with doc.create(Math()) as math:
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        math.append(NoEscape(r'''
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            e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 
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            1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
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        '''))
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    # Limit definitions
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    with doc.create(Math()) as math:
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        math.append(NoEscape(r'''
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            \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x)
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        '''))
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```
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### Matrix Operations
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```python
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import numpy as np
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Math, Matrix
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doc = Document()
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with doc.create(Section('Matrix Operations')):
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    # Matrix multiplication (symbolic)
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    A = np.array([['a_{11}', 'a_{12}'], 
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                  ['a_{21}', 'a_{22}']])
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    B = np.array([['b_{11}', 'b_{12}'], 
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                  ['b_{21}', 'b_{22}']])
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    with doc.create(Math()) as math:
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        math.append('AB = ')
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        math.append(Matrix(A, mtype='b'))
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        math.append(Matrix(B, mtype='b'))
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        math.append(' = ')
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        result = [['a_{11}b_{11} + a_{12}b_{21}', 'a_{11}b_{12} + a_{12}b_{22}'],
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                  ['a_{21}b_{11} + a_{22}b_{21}', 'a_{21}b_{12} + a_{22}b_{22}']]
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        math.append(Matrix(result, mtype='b'))
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    # Eigenvalue equation
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    with doc.create(Math()) as math:
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        math.append('A')
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        math.append(VectorName('v'))
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        math.append(r' = \lambda ')
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        math.append(VectorName('v'))
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```
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### Systems of Equations
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```python
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Alignat
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doc = Document()
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with doc.create(Section('Linear Systems')):
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    # Augmented matrix representation
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    with doc.create(Alignat(aligns=4, numbering=False)) as agn:
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        agn.append(r'2x &+ 3y &- z &= 7 \\')
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        agn.append(r'x &- y &+ 2z &= 3 \\') 
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        agn.append(r'3x &+ 2y &+ z &= 8')
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    # Solution steps
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    with doc.create(Alignat(numbering=False)) as agn:
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        agn.append(r'x &= \frac{D_x}{D} \\')
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        agn.append(r'y &= \frac{D_y}{D} \\')
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        agn.append(r'z &= \frac{D_z}{D}')
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```
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## Mathematical Symbols and Notation
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### Common Mathematical Constructs
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```python
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Math
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doc = Document()
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# Greek letters and symbols
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\alpha, \beta, \gamma, \delta, \epsilon'))
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# Calculus notation
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\frac{d}{dx}f(x), \frac{\partial}{\partial x}f(x,y)'))
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# Set theory
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'A \cup B, A \cap B, A \subset B, x \in A'))
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# Logic symbols
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\forall x \in \mathbb{R}, \exists y \in \mathbb{N}'))
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# Operators
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\sum_{i=1}^{n} x_i, \prod_{i=1}^{n} x_i, \int_a^b f(x)dx'))
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```
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### Custom Mathematical Functions
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```python
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from pylatex import Document, Command, NoEscape
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from pylatex.math import Math
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doc = Document()
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# Define custom operators
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doc.preamble.append(Command('DeclareMathOperator', arguments=[NoEscape(r'\Tr'), 'Tr']))
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doc.preamble.append(Command('DeclareMathOperator', arguments=[NoEscape(r'\rank'), 'rank']))
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\Tr(A) = \sum_{i} A_{ii}'))
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'\rank(A) \leq \min(m, n)'))
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```
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## Equation Numbering and References
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### Numbered Equations
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```python
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from pylatex import Document, Section, NoEscape
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from pylatex.math import Alignat
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from pylatex import Label, Ref
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doc = Document()
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with doc.create(Section('Referenced Equations')):
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    # Numbered alignat environment
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    with doc.create(Alignat()) as agn:
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        agn.append(NoEscape(r'E &= mc^2'))
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        agn.append(Label('einstein'))
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        agn.append(NoEscape(r'\\'))
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        agn.append(NoEscape(r'F &= ma'))
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        agn.append(Label('newton'))
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    doc.append(NoEscape('Einstein\'s equation '))
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    doc.append(Ref('einstein'))
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    doc.append(NoEscape(' and Newton\'s law '))
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    doc.append(Ref('newton'))
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    doc.append(NoEscape(' are fundamental to physics.'))
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```
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## Package Dependencies
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Mathematical environments automatically include necessary packages:
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- **Math, Alignat, Matrix**: `amsmath` package (automatically added)
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- **Additional symbols**: May require `amssymb`, `amsfonts`
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- **Bold math**: `bm` package for bold mathematical symbols
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- **Custom operators**: Use `\DeclareMathOperator` in preamble
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```python
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from pylatex import Document, Package
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doc = Document()
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doc.packages.append(Package('amsmath'))
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doc.packages.append(Package('amssymb'))
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doc.packages.append(Package('amsfonts'))
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doc.packages.append(Package('bm'))
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```
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## Mathematical Best Practices
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### Equation Formatting
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```python
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from pylatex import Document, NoEscape
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from pylatex.math import Math, Alignat
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# Use proper spacing
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'a \cdot b \quad \text{not} \quad a.b'))
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# Use text in equations
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with doc.create(Math()) as math:
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    math.append(NoEscape(r'P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}'))
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# Break long equations
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with doc.create(Alignat(numbering=False)) as agn:
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    agn.append(NoEscape(r'f(x) &= a_0 + a_1x + a_2x^2 + a_3x^3 \\'))
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    agn.append(NoEscape(r'&\quad + a_4x^4 + a_5x^5 + \cdots'))
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```
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Mathematical expressions in PyLaTeX maintain LaTeX's high-quality typesetting while providing programmatic control over equation generation and formatting.