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SymPy 1.12 documentation
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SymPy 1.12 documentation


 * Installation
 * Tutorials
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   * Introductory Tutorial
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     * Preliminaries
     * Introduction
     * Gotchas
     * SymPy Features
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       * Basic Operations
       * Printing
       * Simplification
       * Calculus
       * Solvers
       * Matrices
       * Advanced Expression Manipulation
     * What’s Next
 * How-to Guides
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   * Assumptions
   * Symbolic and fuzzy booleans
   * Writing Custom Functions
   * Solve Equations
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     * Solving Guidance
     * Solve an Equation Algebraically
     * Solve a System of Equations Algebraically
     * Solve One or a System of Equations Numerically
     * Solve an Ordinary Differential Equation (ODE) Algebraically
     * Find the Roots of a Polynomial Algebraically or Numerically
     * Solve a Matrix Equation Algebraically
     * Reduce One or a System of Inequalities for a Single Variable
       Algebraically
     * Solve a Diophantine Equation Algebraically
   * Citing SymPy
 * Explanations
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   * Gotchas and Pitfalls
   * Solve Output by Type
   * SymPy Special Topics
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     * Finite Difference Approximations to Derivatives
     * Classification of SymPy objects
   * List of active deprecations
   * Glossary
 * API Reference
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   * Basics
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     * Assumptions
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       * Ask
       * Assume
       * Refine
       * Predicates
     * Calculus
     * Combinatorics
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       * Partitions
       * Permutations
       * Permutation Groups
       * Polyhedron
       * Prufer Sequences
       * Subsets
       * Gray Code
       * Named Groups
       * Galois Groups
       * Nilpotent, Abelian and Cyclic Numbers
       * Utilities
       * Group constructors
       * Test Utilities
       * Tensor Canonicalization
       * Finitely Presented Groups
       * Polycyclic Groups
     * Functions
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       * Elementary
       * Combinatorial
       * Enumeration
       * Special
     * Integrals
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       * Computing Integrals using Meijer G-Functions
       * Integrals
     * Series
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       * Series Expansions
       * Sequences
       * Fourier Series
       * Formal Power Series
       * Limits of Sequences
     * Simplify
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       * Simplify
       * Hypergeometric Expansion
       * Hongguang Fu’s Trigonometric Simplification
     * Solvers
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       * Diophantine
       * Inequality Solvers
       * ODE
       * PDE
       * Solvers
       * Solveset
     * abc
     * Algebras
     * Concrete
     * Core
     * Discrete
     * Numerical Evaluation
     * Numeric Computation
     * Term Rewriting
   * Code Generation
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     * Code Generation
   * Logic
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     * Logic
     * Sets
   * Matrices
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     * Matrices
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       * Matrices (linear algebra)
       * Common Matrices
       * Dense Matrices
       * Sparse Matrices
       * Sparse Tools
       * Immutable Matrices
       * Matrix Expressions
       * Matrix Normal Forms
     * Tensor
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       * N-dim array
       * N-dim array expressions
       * Indexed Objects
       * Methods
       * Tensor
       * Tensor Operators
     * Vector
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       * Introduction
       * Basic Implementation details
       * More about Coordinate Systems
       * Scalar and Vector Field Functionality
       * General examples of usage
       * Applications of Vector Integrals
       * Vector API
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         * Essential Classes in sympy.vector (docstrings)
         * Orienter classes (docstrings)
         * Essential Functions in sympy.vector (docstrings)
   * Number Theory
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     * Number Theory
   * Physics
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     * Hydrogen Wavefunctions
     * Matrices
     * Pauli Algebra
     * Quantum Harmonic Oscillator in 1-D
     * Quantum Harmonic Oscillator in 3-D
     * Second Quantization
     * Wigner Symbols
     * Unit systems
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       * Philosophy behind unit systems
       * More examples
       * Dimensions and dimension systems
       * Unit prefixes
       * Units and unit systems
       * Physical quantities
     * High energy physics
     * The Physics Vector Module
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       * Vector & ReferenceFrame
       * Vector: Kinematics
       * Potential Issues/Advanced Topics/Future Features in Physics/Vector
         Module
       * Scalar and Vector Field Functionality
       * Physics Vector API
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         * Essential Classes
         * Kinematics (Docstrings)
         * Printing (Docstrings)
         * Essential Functions (Docstrings)
         * Docstrings for basic field functions
     * Classical Mechanics
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       * Masses, Inertias, Particles and Rigid Bodies in Physics/Mechanics
       * Kane’s Method in Physics/Mechanics
       * Lagrange’s Method in Physics/Mechanics
       * Joints Framework in Physics/Mechanics
       * Symbolic Systems in Physics/Mechanics
       * Linearization in Physics/Mechanics
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         * Nonminimal Coordinates Pendulum
       * Examples for Physics/Mechanics
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         * A rolling disc
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           * A rolling disc, with Kane’s method
           * A rolling disc, with Kane’s method and constraint forces
           * A rolling disc using Lagrange’s Method
         * A bicycle
         * Nonminimal Coordinates Pendulum
         * Multi Degree of Freedom Holonomic System
         * A four bar linkage
       * Potential Issues/Advanced Topics/Future Features in Physics/Mechanics
       * References for Physics/Mechanics
       * Autolev Parser
       * SymPy Mechanics for Autolev Users
       * Mechanics API Reference
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         * Masses, Inertias & Particles, RigidBodys (Docstrings)
         * Body (Docstrings)
         * Kane’s Method & Lagrange’s Method (Docstrings)
         * Joints Framework (Docstrings)
         * SymbolicSystem (Docstrings)
         * Linearization (Docstrings)
         * Expression Manipulation (Docstrings)
         * Printing (Docstrings)
     * Quantum Mechanics
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       * Anticommutator
       * Clebsch-Gordan Coefficients
       * Commutator
       * Constants
       * Dagger
       * Inner Product
       * Tensor Product
       * Cartesian Operators and States
       * Hilbert Space
       * Operator
       * Operator/State Helper Functions
       * Qapply
       * Represent
       * Spin
       * State
       * Circuit Plot
       * Gates
       * Grover’s Algorithm
       * QFT
       * Qubit
       * Shor’s Algorithm
       * Particle in a Box
     * Optics Module
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       * Gaussian Optics
       * Medium
       * Polarization
       * Utilities
       * Waves
     * Control Module
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       * Control
       * Control API
       * Control System Plots
     * Continuum Mechanics
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       * Beam (Docstrings)
       * Solving Beam Bending Problems using Singularity Functions
       * Truss (Docstrings)
   * Utilities
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     * Testing
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       * pytest
       * Randomised Testing
       * Run Tests
     * Utilities
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       * Autowrap Module
       * Codegen
       * Decorator
       * Enumerative
       * Exceptions and Warnings
       * Iterables
       * Lambdify
       * Memoization
       * Miscellaneous
       * PKGDATA
       * Source Code Inspection
       * Timing Utilities
     * Interactive
     * Parsing
     * Printing
   * Topics
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     * Geometry
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       * Entities
       * Utils
       * Points
       * Lines
       * Curves
       * Ellipses
       * Polygons
       * Plane
     * Holonomic
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       * About Holonomic Functions
       * Representation of holonomic functions in SymPy
       * Operations on holonomic functions
       * Converting other representations to holonomic
       * Uses and Current limitations
       * Internal API
     * Lie Algebra
     * Polynomial Manipulation
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       * Basic functionality of the module
       * Examples from Wester’s Article
       * Polynomials Manipulation Module Reference
       * AGCA - Algebraic Geometry and Commutative Algebra Module
       * Introducing the Domains of the poly module
       * Reference docs for the Poly Domains
       * Internals of the Polynomial Manipulation Module
       * Series Manipulation using Polynomials
       * Literature
       * Poly solvers
       * Introducing the domainmatrix of the poly module
       * Number Fields
     * Category Theory
     * Cryptography
     * Differential Geometry
     * Plotting
     * Stats
 * Contributing
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   * Development Environment Setup
   * Dependencies
   * Build the Documentation
   * Debugging
   * SymPy Docstrings Style Guide
   * Docstring Style Guide
   * Making a Contribution
   * Deprecation Policy

DOCUMENTATION VERSION

 * SymPy 1.12 (latest version)
 * SymPy 1.13-dev (dev version)

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WELCOME TO SYMPY’S DOCUMENTATION!#

A PDF version of these docs is also available.

SymPy is a Python library for symbolic mathematics. If you are new to SymPy,
start with the introductory tutorial.

This is the central page for all of SymPy’s documentation.

--------------------------------------------------------------------------------


INSTALLATION#

Instructions on how to install SymPy.


TUTORIALS#

Tutorials are the best place to start for anyone new to SymPy or one of SymPy’s
features.


HOW-TO GUIDES#

How-to guides are step-by-step instructions on how to do specific tasks.


EXPLANATIONS#

Explanations provide in-depth discussions about select SymPy features. These
topic guides talk about things like the motivation behind design decisions,
technical implementation details, and opinionated recommendations.


API REFERENCE#

The API reference is a detailed description of the SymPy API. Every SymPy
function and class is enumerated here with detailed information on what each
function does.


CONTRIBUTING#

The contributing guide goes over the details necessary to contribute to SymPy.
See also the full Development Workflow guide on the SymPy wiki.


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Installation
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Last updated on May 09, 2023