This tutorial gives an overview and introduction to SymPy. Read this to have an idea what SymPy can do for you (and how) and if you want to know more, read the API documentation (that is generated from the sources) or the sources directly.

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To import SymPy from the standard Python shell, just type

>>> from sympy import *

and most common functions will be imported. Other modules of interest include sympy.limits, sympy.solvers, etc.

There is also a little app called isympy (located in bin/isympy if you are running from the source directory) which is just a standard Python shell that has already imported the relevant SymPy modules and defined the symbols x, y and z.

Sympy has two built-in numeric types: Real and Rational.

The Rational class represents a rational number as a pair of two integers: the numerator and the denominator, so Rational(1,2) represents 1/2, Rational(5,2) 5/2 and so on.

>>> from sympy import *
>>> a = Rational(1,2)

>>> a
1/2

>>> a*2
1

>>> Rational(2)**50/Rational(10)**50
1/88817841970012523233890533447265625

Proceed with caution while working with Python's int since they truncate integer division, and that's why:

>>> 1/2
0

>>> 1.0/2
0.5

You can however do:

>>> from __future__ import division # doctest: +SKIP

>>> 1/2 # doctest: +SKIP
0.5

It's going to be standard in Python, hopefully soon...

We also have some special constants, like E and pi, that are treated as symbols (1+pi won't evaluate to something numeric, rather it will remain as 1+pi), and have arbitrary precision:

>>> pi**2
pi**2

>>> pi.evalf()                  # doctest: +SKIP
3.141592653589793238462643383

>>> (pi+exp(1)).evalf()         # doctest: +SKIP
5.859874482049203234066343309

As you see, evalf evaluates the expression to a floating-point number.

There is also a class representing mathematical infinity, called oo:

>>> oo > 99999
True
>>> oo + 1
oo

In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly:

>>> from sympy import *
>>> x = Symbol('x')
>>> y = Symbol('y')

Then you can play with them:

>>> x+y+x-y
2*x

>>> (x+y)**2
(x + y)**2

>>> ((x+y)**2).expand()
 x**2 + 2*x*y + y**2

And substitute them for other symbols or numbers using subs(var, substitution):

>>> ((x+y)**2).subs(x, 1)
(y + 1)**2

>>> ((x+y)**2).subs(x, y)
4*y**2

Limits are easy to use in SymPy. They follow the syntax limit(function, variable, point), so to compute the limit of f(x) as x -> 0, you would issue limit(f, x, 0).

>>> from sympy import *
>>> x=Symbol("x")
>>> limit(sin(x)/x, x, 0)
1

You can also calculate the limit at infinity:

>>> limit(x, x, oo)
oo

>>> limit(1/x, x, oo)
0

>>> limit(x**x, x, 0)
1

>>> limit((5**x + 3**x)**(1/x), x, oo)  # doctest: +SKIP
5

For some non-trivial examples on limits, you can read the test file [[test_demidovich.py|http://hg.sympy.org/sympy/file/tip/sympy/series/tests/test_demidovich.py]].

You can differentiate any SymPy expression using diff(func, var). Examples:

>>> from sympy import *
>>> x = Symbol('x')
>>> diff(sin(x), x)
cos(x)
>>> diff(sin(2*x), x)
2*cos(2*x)

>>> diff(tan(x), x)
tan(x)**2 + 1

You can check, that it is correct by:

>>> limit((tan(x+y)-tan(x))/y, y, 0)
tan(x)**2 + 1

Higher derivatives can be calculated using the diff(func, var, n) method:

>>> diff(sin(2*x), x, 1)
2*cos(2*x)

>>> diff(sin(2*x), x, 2)
-4*sin(2*x)

>>> diff(sin(2*x), x, 3)
-8*cos(2*x)

Use .series(var, point, order):

>>> from sympy import *
>>> x = Symbol('x')
>>> cos(x).series(x, 0, 10)
1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**10)
>>> (1/cos(x)).series(x, 0, 10)
1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
>>> sin(x).series(x, 2, 8)
sin(2) + (x - 2)*cos(2) - (x - 2)**2*sin(2)/2 - (x - 2)**3*cos(2)/6 + (x - 2)**4*sin(2)/24 + (x - 2)**5*cos(2)/120 - (x - 2)**6*sin(2)/720 - (x - 2)**7*cos(2)/5040 + O((x - 2)**8, (x, 2))

SymPy has support for indefinite and definite integration of transcendental elementary and special functions via integrate() facility, which uses powerful extended Risch-Norman algorithm and some heuristics and pattern matching.

>>> from sympy import *
>>> x, y = symbols('x y')

You can integrate elementary functions:

>>> integrate(6*x**5, x)
x**6
>>> integrate(sin(x), x)
-cos(x)
>>> integrate(log(x), x)
x*log(x) - x
>>> integrate(2*x + sinh(x), x)     # doctest: +PRETTY
     2          
    x  + cosh(x)

Also special functions are handled easily:

>>> integrate(exp(-x**2)*erf(x), x)     #doctest: +RELEASE_ONLY
pi**(1/2)*erf(x)**2/4
>>> integrate(exp(-x**2)*erf(x), x)     #doctest: +FUTURE_ONLY
sqrt(pi)*erf(x)**2/4

It is possible to compute definite integral:

>>> integrate(x**3, (x, -1, 1))
0
>>> integrate(sin(x), (x, 0, pi/2))
1
>>> integrate(cos(x), (x, -pi/2, pi/2))
2

Also improper integrals are supported as well:

>>> integrate(exp(-x), (x, 0, oo))
1
>>> integrate(log(x), (x, 0, 1))
-1

>>> from sympy import Symbol, exp, I
>>> x = Symbol("x")
>>> exp(I*x).expand()
exp(I*x)
>>> exp(I*x).expand(complex=True)
I*exp(-im(x))*sin(re(x)) + exp(-im(x))*cos(re(x))
>>> x = Symbol("x", real=True)
>>> exp(I*x).expand(complex=True)
I*sin(x) + cos(x)

In isympy:

In [9]: Eq(f(x).diff(x, x) + f(x), 0)
Out[9]:
   2
  d
─────(f(x)) + f(x) = 0
dx dx

In [10]: dsolve(Eq(f(x).diff(x, x) + f(x), 0), f(x))
Out[10]: C1*sin(x) + C2*cos(x)

In isympy:

In [19]: solve(Eq(x**4, 1), x)
Out[19]: [1, -1, -I, I]

In [20]: solve([Eq(x + 5*y, 2), Eq(-3*x + 6*y, 15)], [x, y])
Out[20]: {y: 1, x: -3}

Matrices are created as instances from the [[Matrix|http://sympy.googlecode.com/svn/api/sympy.modules.matrices.Matrix-class.html]] class. This class is located in sympy.matrices, but at usual, isympy imports this for you:

>>> from sympy import *
>>> from sympy.matrices import Matrix
>>> Matrix([[1,0], [0,1]])
[1, 0]
[0, 1]

You can also put symbols in it:

>>> x = Symbol('x')
>>> y = Symbol('y')
>>> A = Matrix([[1,x], [y,1]])
>>> A
[1, x]
[y, 1]

>>> A**2
[x*y + 1,     2*x]
[    2*y, x*y + 1]

For more information and examples with matrices, see the Linear Algebra tutorial.

Use the .match() method, along with the Wild class, to perform pattern matching on expressions. The method will return a dictionary with the required substitutions, as follows:

>>> from sympy import *
>>> x = Symbol('x')
>>> p = Wild('p', exclude=[x])
>>> q = Wild('q', exclude=[x])
>>> (5*x**2 + 3*x).match(p*x**2 + q*x)
{p_: 5, q_: 3}

>>> (x**2).match(p*x**q)
{p_: 1, q_: 2}

If the match is unsuccessful, it returns None:

>>> print (x+1).match(p**x)
None

One can also make use of the WildFunction class to perform more specific matches with functions and their arguments:

>>> f = WildFunction('f', nofargs=1)
>>> (5*cos(x)).match(p*f)
{p_: 5, f_: cos(x)}
>>> (cos(3*x)).match(f(p*x))   # XFail #doctest: +FUTURE_ONLY
{p_: 3, f_: cos}
>>> g = WildFunction('g', nofargs=2)
>>> (5*cos(x)).match(p*g)
{p_: 5, g_: cos(x)}

One can also use the exclude parameter of the Wild class to ensure that certain things do not show up in the result:

>>> x = Symbol('x')
>>> p = Wild('p', exclude=[1,x])
>>> print (x+1).match(x+p) # 1 is excluded
None
>>> print (x+1).match(p+1) # x is excluded
None
>>> print (x+1).match(x+2+p) # -1 is not excluded
{p_: -1}

SymPy comes pre-packed with several ways of printing expressions. The most basic way to print an expression is simply through the use of str(expression) or repr(expression).

Also a printing module available, sympy.printing. Level 2 printing is made possible through the pretty printing component of the printing module. Other printing methods available trough this module are:

• pretty(expr), pretty_print(expr), pprint(expr)
  • Return or print, respectively, a pretty representation of expr. This is the same as the second level of representation described above.
• latex(expr), print_latex(expr)
  • Return or print, respectively, a LaTeX representation of expr.
• mathml(expr), print_mathml(expr)
  • Return or print, respectively, a MathML representation of expr.
• print_gtk(expr)
  • Print expr to Gtkmathview, a GTK widget that displays MathML code. The Gtkmathview program is required.
• print_pygame(expr)
  • Print expr to a PyGame window. The PyGame library is required, along with the Python pexpect library, the LaTeX program, and the dvipng program.

The following list provides links to documentation for some of the various modules SymPy offers

• Geometry Module
• Linear Algebra Module
• Numerics Module
• Plotting Module
• Concrete Mathematics Module
• Polynomials Module
• Statistics Module
• Pretty Printing