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SymPy 1.15.0.dev3298+gc508f90ea0.uninstalled documentation
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SymPy 1.15.0.dev3298+gc508f90ea0.uninstalled documentation
  • Installation
  • Tutorials
    • Introductory Tutorial
      • Preliminaries
      • Introduction
      • Gotchas
      • SymPy Features
        • Basic Operations
        • Printing
        • Simplification
        • Calculus
        • Solvers
        • Matrices
        • Advanced Expression Manipulation
      • What’s Next
    • Physics Tutorials
      • Biomechanics Tutorials
        • Biomechanical Model Example
      • Mechanics Tutorials
        • Duffing Oscillator with a Pendulum
        • A rolling disc
          • A rolling disc, with Kane’s method
          • A rolling disc, with Kane’s method and constraint forces
          • A rolling disc using Lagrange’s Method
        • Multi Degree of Freedom Holonomic System
        • Nonminimal Coordinates Pendulum
        • A four bar linkage
        • Linearized Carvallo-Whipple Bicycle Model
        • Atwood Machine Example
        • Muscle Wrapping over a Spherical Bone in 3D Space
      • Continuum Mechanics Tutorials
        • Solving Beam Bending Problems using Singularity Functions
      • Control Tutorials
        • Control Package Examples
        • Electrical Problems using StateSpace
        • Mechanics Problems using StateSpace
  • How-to Guides
    • Assumptions
    • Symbolic and fuzzy booleans
    • Writing Custom Functions
    • Physics
    • Solve Equations
      • 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
      • Solve Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) Algebraically with the Laplace Transform
      • 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
    • SymPy Logo
    • Citing SymPy
  • Explanations
    • Best Practices
    • Gotchas and Pitfalls
    • Solve Output by Type
    • Physics
      • Vector
        • Vector & ReferenceFrame
        • Vector: Kinematics
        • Scalar and Vector Field Functionality
        • Potential Issues/Advanced Topics/Future Features in Physics/Vector Module
      • Classical Mechanics
        • 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
        • References for Physics/Mechanics
        • Autolev Parser
        • SymPy Mechanics for Autolev Users
        • Potential Issues/Advanced Topics/Future Features in Physics/Mechanics
      • Biomechanics
        • Introduction to Biomechanical Modeling
    • Matrix derivatives
    • SymPy Special Topics
      • Finite Difference Approximations to Derivatives
      • Classification of SymPy objects
    • Implementation Details
      • Laplace Transform: Design Decisions and Implementation Details
    • List of active deprecations
    • Glossary
  • API Reference
    • Basics
      • Assumptions
        • Ask
        • Assume
        • Refine
        • Predicates
      • Calculus
      • Combinatorics
        • Partitions
        • Permutations
        • Permutation Groups
        • Polyhedron
        • Prufer Sequences
        • Subsets
        • Gray Code
        • Named Groups
        • Galois Groups
        • Number of groups
        • Utilities
        • Group constructors
        • Test Utilities
        • Tensor Canonicalization
        • Finitely Presented Groups
        • Polycyclic Groups
        • Character Tables
      • Functions
        • Elementary
        • Combinatorial
        • Enumeration
        • Special
      • Integrals
        • Computing Integrals using Meijer G-Functions
        • Integrals
      • Series
        • Approximants
        • Formal Power Series
        • Fourier Series
        • Limits of Sequences
        • Sequences
        • Series Expansions
      • Simplify
        • Simplify
        • Hypergeometric Expansion
        • Hongguang Fu’s Trigonometric Simplification
      • Solvers
        • Diophantine
        • Inequality Solvers
        • ODE
        • PDE
        • Solvers
        • Solveset
      • abc
      • Algebras
      • Concrete
      • Core
      • Discrete
      • Numerical Evaluation
      • Numeric Computation
      • Term Rewriting
    • Code Generation
      • Code Generation
    • Logic
      • Logic
      • Sets
    • Matrices
      • Matrices
        • Matrices (linear algebra)
        • Matrix Kind
        • Dense Matrices
        • Sparse Matrices
        • Sparse Tools
        • Immutable Matrices
        • Matrix Expressions
        • Matrix Normal Forms
      • Tensor
        • N-dim array
        • N-dim array expressions
        • Indexed Objects
        • Methods
        • Tensor
        • Tensor Operators
      • Vector
        • Introduction
        • Basic Implementation details
        • More about Coordinate Systems
        • Scalar and Vector Field Functionality
        • General examples of usage
        • Applications of Vector Integrals
        • Vector API
          • Essential Classes in sympy.vector (docstrings)
          • Orienter classes (docstrings)
          • Essential Functions in sympy.vector (docstrings)
    • Number Theory
      • Number Theory
    • Physics
      • Physics Vector API
        • Essential Classes
        • Kinematics (Docstrings)
        • Printing (Docstrings)
        • Essential Functions (Docstrings)
        • Docstrings for basic field functions
      • Mechanics API Reference
        • Bodies, Inertias, Loads & Other Functions (Docstrings)
        • Kane’s Method & Lagrange’s Method (Docstrings)
        • Joints Framework (Docstrings)
        • System (Docstrings)
        • Linearization (Docstrings)
        • Expression Manipulation (Docstrings)
        • Printing (Docstrings)
        • Pathway (Docstrings)
        • Actuator (Docstrings)
        • Wrapping Geometry (Docstrings)
        • Deprecated Classes (Docstrings)
        • Method Abstract Base Class (Docstrings)
      • Biomechanics API Reference
        • Musculotendon (Docstrings)
        • Activation (Docstrings)
        • Curve (Docstrings)
      • Control
        • Control
        • Control API
        • Control System Plots
        • Routh-Hurwitz Table
      • Optics
        • Gaussian Optics
        • Medium
        • Polarization
        • Utilities
        • Waves
      • Unit Systems
        • Philosophy behind unit systems
        • More examples
        • Dimensions and dimension systems
        • Unit prefixes
        • Units and unit systems
        • Physical quantities
      • Continuum Mechanics
        • Beam (Docstrings)
        • Column (Docstrings)
        • Truss (Docstrings)
        • Cable (Docstrings)
        • Arch (Docstrings)
      • High Energy Physics
      • Quantum Mechanics
        • Anticommutator
        • Clebsch-Gordan Coefficients
        • Commutator
        • Constants
        • Dagger
        • Inner Product
        • Tensor Product
        • Boson
        • Cartesian Operators and States
        • Fermion
        • Hilbert Space
        • Operator
        • Operator/State Helper Functions
        • Pauli
        • Qapply
        • Represent
        • Spin
        • State
        • Circuit Plot
        • Gates
        • Grover’s Algorithm
        • QFT
        • Qubit
        • Shor’s Algorithm
        • Particle in a Box
      • Hydrogen Wavefunctions
      • Matrices
      • Pauli Algebra
      • Quantum Harmonic Oscillator in 1-D
      • Second Quantization
      • Quantum Harmonic Oscillator in 3-D
      • Wigner Symbols
    • Utilities
      • Testing
        • pytest
        • Randomised Testing
        • Run Tests
      • Utilities
        • Autowrap Module
        • Codegen
        • Decorator
        • Enumerative
        • Exceptions and Warnings
        • Iterables
        • Lambdify
        • Memoization
        • Miscellaneous
        • Source Code Inspection
        • Timing Utilities
      • Interactive
      • Parsing
      • Printing
    • Topics
      • Geometry
        • Entities
        • Utils
        • Points
        • Lines
        • Curves
        • Ellipses
        • Polygons
        • Plane
      • Holonomic
        • 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
        • 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
        • Power Series Ring
        • Literature
        • Poly solvers
        • Introducing the domainmatrix of the poly module
        • Number Fields
      • Category Theory
      • Cryptography
      • Differential Geometry
      • Plotting
      • Stats
  • Contributing
    • Introduction to Contributing
    • Guide for New Contributors
      • Setup Development Environment
      • Development Workflow Process
      • Writing Tests
      • Building the Documentation
    • Writing a pull request description
    • Dependencies
    • Debugging
    • Docstrings Style Guide
    • Documentation Style Guide
    • Deprecation Policy
    • AI Generated Code and Communication Policy
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Last updated on Nov 06, 2026