SymPy

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 or SymPy Gamma. SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies. 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.
SymPy is free software and is licensed under New BSD License. The lead developers are Ondřej Čertík and Aaron Meurer. It was started in 2005 by Ondřej Čertík.
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 (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:
Plotting.
Note, plotting requires the external matplotlib or Pyglet module.
Dependencies.
Since version 1.0, SymPy has the mpmath package as a dependency.
There are several optional dependencies that can enhance its capabilities:
Usage examples.
Pretty-printing.
Sympy allows outputs to be formatted into a more appealing format through the codice_1 function. Alternatively, the codice_2 method will enable pretty-printing, so codice_1 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