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SymPy
Developer(s)	SymPy Development Team
Initial release	2007; 17 years ago
Stable release	1.12 / 10 May 2023; 19 months ago
Repository	github.com/sympy/sympy ;
Written in	Python
Operating system	Cross-platform
Type	Computer algebra system
License	3-clause BSD
Website	www.sympy.org

SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live^[2] or SymPy Gamma.^[3] SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies.^[4]^[5]^[6] This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.

SymPy includes features ranging from basic symbolic arithmetic to calculus, algebra, discrete mathematics, and quantum physics. It is capable of formatting the result of the computations as LaTeX code.^[4]^[5]

SymPy is free software and is licensed under the 3-clause BSD. The lead developers are Ondřej Čertík and Aaron Meurer. It was started in 2005 by Ondřej Čertík.^[7]

Features

The SymPy library is split into a core with many optional modules.

Currently, the core of SymPy has around 260,000 lines of code^[8] (it also includes a comprehensive set of self-tests: over 100,000 lines in 350 files as of version 0.7.5), and its capabilities include:^[4]^[5]^[9]^[10]^[11]

Core capabilities

Basic arithmetic: *, /, +, -, **
Simplification
Expansion
Functions: trigonometric, hyperbolic, exponential, roots, logarithms, absolute value, spherical harmonics, factorials and gamma functions, zeta functions, polynomials, hypergeometric, special functions, etc.
Substitution
Arbitrary precision integers, rationals and floats
Noncommutative symbols
Pattern matching

Polynomials

Matrices

Basic arithmetic
Eigenvalues and their eigenvectors when the characteristic polynomial is solvable by radicals
Determinants
Inversion
Solving

Geometry

Points, lines, rays, ellipses, circles, polygons, etc.
Intersections
Tangency
Similarity

Plotting

Note, plotting requires the external Matplotlib or Pyglet module.

Coordinate models
Plotting Geometric Entities
2D and 3D
Interactive interface
Colors
Animations

Physics

Statistics

Combinatorics

Permutations
Combinations
Partitions
Subsets
Permutation group: Polyhedral, Rubik, Symmetric, etc.
Prufer sequence and Gray codes

Printing

Pretty-printing: ASCII/Unicode pretty-printing, LaTeX
Code generation: C, Fortran, Python

Related projects

SageMath: an open source alternative to Mathematica, Maple, MATLAB, and Magma (SymPy is included in Sage)
SymEngine: a rewriting of SymPy's core in C++, in order to increase its performance. Work is currently in progress^{[as of?]} to make SymEngine the underlying engine of Sage too.^[14]
mpmath: a Python library for arbitrary-precision floating-point arithmetic^[15]
SympyCore: another Python computer algebra system^[16]
SfePy: Software for solving systems of coupled partial differential equations (PDEs) by the finite element method in 1D, 2D and 3D.^[17]
GAlgebra: Geometric algebra module (previously sympy.galgebra).^[18]
Quameon: Quantum Monte Carlo in Python.^[19]
Lcapy: Experimental Python package for teaching linear circuit analysis.^[20]
LaTeX Expression project: Easy LaTeX typesetting of algebraic expressions in symbolic form with automatic substitution and result computation.^[21]
Symbolic statistical modeling: Adding statistical operations to complex physical models.^[22]
Diofant: a fork of SymPy, started by Sergey B Kirpichev^[23]

Dependencies

Since version 1.0, SymPy has the mpmath package as a dependency.

There are several optional dependencies that can enhance its capabilities:

gmpy: If gmpy is installed, SymPy's polynomial module will automatically use it for faster ground types. This can provide a several times boost in performance of certain operations.
matplotlib: If matplotlib is installed, SymPy can use it for plotting.
Pyglet: Alternative plotting package.

Usage examples

Pretty-printing

Sympy allows outputs to be formatted into a more appealing format through the pprint function. Alternatively, the init_printing() method will enable pretty-printing, so pprint need not be called. Pretty-printing will use unicode symbols when available in the current environment, otherwise it will fall back to ASCII characters.

>>> from sympy import pprint, init_printing, Symbol, sin, cos, exp, sqrt, series, Integral, Function
>>>
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> f = Function("f")
>>> # pprint will default to unicode if available
>>> pprint(x ** exp(x))
 ⎛ x⎞
 ⎝ℯ ⎠
x   
>>> # An output without unicode
>>> pprint(Integral(f(x), x), use_unicode=False)
  /       
 |        
 | f(x) dx
 |        
/        
>>> # Compare with same expression but this time unicode is enabled
>>> pprint(Integral(f(x), x), use_unicode=True)
⌠        
⎮ f(x) dx
⌡     
>>> # Alternatively, you can call init_printing() once and pretty-print without the pprint function.
>>> init_printing()
>>> sqrt(sqrt(exp(x)))
   ____
4 ╱  x 
╲╱  ℯ  
>>> (1/cos(x)).series(x, 0, 10)
     2      4       6        8         
    x    5⋅x    61⋅x    277⋅x     ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x  ⎠
    2     24     720     8064

Expansion

>>> from sympy import init_printing, Symbol, expand
>>> init_printing()
>>>
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> e = (a + b) ** 3
>>> e
(a + b)³
>>> e.expand()
a³ + 3⋅a²⋅b + 3⋅a⋅b²  + b³

Arbitrary-precision example

>>> from sympy import Rational, pprint
>>> e = 2**50 / Rational(10) ** 50
>>> pprint(e)
1/88817841970012523233890533447265625

Differentiation

>>> from sympy import init_printing, symbols, ln, diff
>>> init_printing()
>>> x, y = symbols("x y")
>>> f = x**2 / y + 2 * x - ln(y)
>>> diff(f, x)
 2⋅x    
 ─── + 2
  y 
>>> diff(f, y)
    2    
   x    1
 - ── - ─
    2   y
   y
>>> diff(diff(f, x), y)
 -2⋅x
 ────
   2 
  y

Plotting

>>> from sympy import symbols, cos
>>> from sympy.plotting import plot3d
>>> x, y = symbols("x y")
>>> plot3d(cos(x * 3) * cos(y * 5) - y, (x, -1, 1), (y, -1, 1))
<sympy.plotting.plot.Plot object at 0x3b6d0d0>

Limits

>>> from sympy import init_printing, Symbol, limit, sqrt, oo
>>> init_printing()
>>> 
>>> x = Symbol("x")
>>> limit(sqrt(x**2 - 5 * x + 6) - x, x, oo)
-5/2
>>> limit(x * (sqrt(x**2 + 1) - x), x, oo)
1/2
>>> limit(1 / x**2, x, 0)
∞
>>> limit(((x - 1) / (x + 1)) ** x, x, oo)
 -2
ℯ

Differential equations

>>> from sympy import init_printing, Symbol, Function, Eq, dsolve, sin, diff
>>> init_printing()
>>>
>>> x = Symbol("x")
>>> f = Function("f")
>>>
>>> eq = Eq(f(x).diff(x), f(x))
>>> eq
d              
──(f(x)) = f(x)
dx         
>>>    
>>> dsolve(eq, f(x))
           x
f(x) = C₁⋅ℯ

>>>
>>> eq = Eq(x**2 * f(x).diff(x), -3 * x * f(x) + sin(x) / x)
>>> eq
 2 d                      sin(x)
x ⋅──(f(x)) = -3⋅x⋅f(x) + ──────
   dx                       x   
>>>
>>> dsolve(eq, f(x))
       C₁ - cos(x)
f(x) = ───────────   
            x³

Integration

>>> from sympy import init_printing, integrate, Symbol, exp, cos, erf
>>> init_printing()
>>> x = Symbol("x")
>>> # Polynomial Function
>>> f = x**2 + x + 1
>>> f
 2        
x  + x + 1
>>> integrate(f, x)
 3    2    
x    x     
── + ── + x
3    2     
>>> # Rational Function
>>> f = x / (x**2 + 2 * x + 1)
>>> f
     x      
────────────
 2          
x  + 2⋅x + 1

>>> integrate(f, x)
               1  
log(x + 1) + ─────
             x + 1
>>> # Exponential-polynomial functions
>>> f = x**2 * exp(x) * cos(x)
>>> f
 2  x       
x ⋅ℯ ⋅cos(x)
>>> integrate(f, x)
 2  x           2  x                         x           x       
x ⋅ℯ ⋅sin(x)   x ⋅ℯ ⋅cos(x)      x          ℯ ⋅sin(x)   ℯ ⋅cos(x)
──────────── + ──────────── - x⋅ℯ ⋅sin(x) + ───────── - ─────────
     2              2                           2           2    
>>> # A non-elementary integral
>>> f = exp(-(x**2)) * erf(x)
>>> f
   2       
 -x        
ℯ   ⋅erf(x)
>>> integrate(f, x)

  ___    2   
╲╱ π ⋅erf (x)
─────────────
      4

Series

>>> from sympy import Symbol, cos, sin, pprint
>>> x = Symbol("x")
>>> e = 1 / cos(x)
>>> pprint(e)
  1   
──────
cos(x)
>>> pprint(e.series(x, 0, 10))
     2      4       6        8         
    x    5⋅x    61⋅x    277⋅x     ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x  ⎠
    2     24     720     8064          
>>> e = 1/sin(x)
>>> pprint(e)
  1   
──────
sin(x)
>>> pprint(e.series(x, 0, 4))
           3        
1   x   7⋅x     ⎛ 4⎞
─ + ─ + ──── + O⎝x ⎠
x   6   360

Logical reasoning

Example 1

>>> from sympy import *
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> facts = Q.positive(x), Q.positive(y)
>>> with assuming(*facts):
...     print(ask(Q.positive(2 * x + y)))
True

Example 2

>>> from sympy import *
>>> x = Symbol("x")
>>> # Assumption about x
>>> fact = [Q.prime(x)]
>>> with assuming(*fact):
...     print(ask(Q.rational(1 / x)))
True