Tutorial

Introduction

SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries.

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 SymPy User’s Guide, SymPy Modules Reference. or the sources directly.

First Steps with SymPy

The easiest way to download it is to go to http://code.google.com/p/sympy/ and download the latest tarball from the Featured Downloads:

_images/featured-downloads.png

Unpack it:

$ tar xzf sympy-0.5.12.tar.gz

and try it from a Python interpreter:

$ cd sympy-0.5.12
$ python
Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import Symbol, cos
>>> x = Symbol("x")
>>> (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)

You can use SymPy as shown above and this is indeed the recommended way if you use it in your program. You can also install it using ./setup.py install as any other Python module, or just install a package in your favourite Linux distribution, e.g.:

Installing SymPy in Debian

$ sudo apt-get install python-sympy
Reading package lists... Done
Building dependency tree
Reading state information... Done
The following NEW packages will be installed:
  python-sympy
0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
Need to get 991kB of archives.
After this operation, 5976kB of additional disk space will be used.
Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
Fetched 991kB in 2s (361kB/s)
Selecting previously deselected package python-sympy.
(Reading database ... 232619 files and directories currently installed.)
Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
Setting up python-sympy (0.5.12-1) ...

For other means how to install SymPy, consult the Downloads tab on the SymPy’s webpage.

isympy Console

For experimenting with new features, or when figuring out how to do things, you can use our special wrapper around IPython 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, z and some other things:

$ cd sympy
$ ./bin/isympy
IPython console for SymPy 0.7.2 (Python 2.7.1) (ground types: gmpy)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org

In [1]: (1/cos(x)).series(x, 0, 10)
Out[1]:
     2      4       6        8
    x    5*x    61*x    277*x     / 10\
1 + -- + ---- + ----- + ------ + O\x  /
    2     24     720     8064

Note

Commands entered by you are bold. Thus what we did in 3 lines in a regular Python interpreter can be done in 1 line in isympy.

Using SymPy as a calculator

SymPy has three built-in numeric types: Float, Rational and Integer.

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) represents 5/2, and so on.

>>> from sympy import Rational
>>> 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 int’s and floating point numbers, especially in division, since you may create a Python number, not a SymPy number. A ratio of two Python ints may create a float – the “true division” standard of Python 3 and the default behavior of isympy which imports division from __future__:

>>> 1/2 
0.5

But in earlier Python versions where division has not been imported, a truncated int will result:

>>> 1/2 
0

In both cases, however, you are not dealing with a SymPy Number because Python created its own number. Most of the time you will probably be working with Rational numbers, so make sure to use Rational to get the SymPy result. One might find it convenient to equate R and Rational:

>>> R = Rational
>>> R(1, 2)
1/2
>>> R(1)/2 # R(1) is a SymPy Integer and Integer/int gives a Rational
1/2

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:

>>> from sympy import pi, E
>>> pi**2
pi**2

>>> pi.evalf()
3.14159265358979

>>> (pi + E).evalf()
5.85987448204884

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

The symbol oo is used for a class defining mathematical infinity:

>>> from sympy import oo
>>> oo > 99999
True
>>> oo + 1
oo

Symbols

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

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

On the left is the normal Python variable which has been assigned to the SymPy Symbol class. Predefined symbols (including those for symbols with Greek names) are available for import from abc:

>>> from sympy.abc import x, theta

Symbols can also be created with the symbols or var functions, the latter automatically adding the created symbols to the namespace, and both accepting a range notation:

>>> from sympy import symbols, var
>>> a, b, c = symbols('a,b,c')
>>> d, e, f = symbols('d:f')
>>> var('g:h')
(g, h)
>>> var('g:2')
(g0, g1)

Instances of the Symbol class “play well together” and are the building blocks of expresions:

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

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

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

They can be substituted with other numbers, symbols or expressions using subs(old, new):

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

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

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

For the remainder of the tutorial, we assume that we have run:

>>> from sympy import init_printing
>>> init_printing(use_unicode=False, wrap_line=False, no_global=True)

This will make things look better when printed. See the Printing section below. If you have a unicode font installed, you can pass use_unicode=True for a slightly nicer output.

Algebra

For partial fraction decomposition, use apart(expr, x):

>>> from sympy import apart
>>> from sympy.abc import x, y, z

>>> 1/( (x+2)*(x+1) )
       1
---------------
(x + 1)*(x + 2)

>>> apart(1/( (x+2)*(x+1) ), x)
    1       1
- ----- + -----
  x + 2   x + 1

>>> (x+1)/(x-1)
x + 1
-----
x - 1

>>> apart((x+1)/(x-1), x)
      2
1 + -----
    x - 1

To combine things back together, use together(expr, x):

>>> from sympy import together
>>> together(1/x + 1/y + 1/z)
x*y + x*z + y*z
---------------
     x*y*z

>>> together(apart((x+1)/(x-1), x), x)
x + 1
-----
x - 1

>>> together(apart(1/( (x+2)*(x+1) ), x), x)
       1
---------------
(x + 1)*(x + 2)

Calculus

Limits

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 limit, Symbol, sin, oo
>>> 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

for some non-trivial examples on limits, you can read the test file test_demidovich.py

Differentiation

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

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

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

You can check, that it is correct by:

>>> from sympy import limit
>>> from sympy.abc import delta
>>> limit((tan(x + delta) - tan(x))/delta, delta, 0)
   2
tan (x) + 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)

Series expansion

Use .series(var, point, order):

>>> from sympy import Symbol, cos
>>> x = Symbol('x')
>>> cos(x).series(x, 0, 10)
     2    4     6      8
    x    x     x      x      / 10\
1 - -- + -- - --- + ----- + O\x  /
    2    24   720   40320
>>> (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

Another simple example:

>>> from sympy import Integral, pprint

>>> y = Symbol("y")
>>> e = 1/(x + y)
>>> s = e.series(x, 0, 5)

>>> print(s)
1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)
>>> pprint(s)
          2    3    4
1   x    x    x    x     / 5\
- - -- + -- - -- + -- + O\x /
y    2    3    4    5
    y    y    y    y

Summation

Compute the summation of f with respect to the given summation variable over the given limits.

summation(f, (i, a, b)) computes the sum of f with respect to i from a to b, i.e.,

                            b
                          ____
                          \   `
summation(f, (i, a, b)) =  )    f
                          /___,
                          i = a

If it cannot compute the sum, it prints the corresponding summation formula. Repeated sums can be computed by introducing additional limits:

>>> from sympy import summation, oo, symbols, log
>>> i, n, m = symbols('i n m', integer=True)

>>> summation(2*i - 1, (i, 1, n))
 2
n
>>> summation(1/2**i, (i, 0, oo))
2
>>> summation(1/log(n)**n, (n, 2, oo))
  oo
 ___
 \  `
  \     -n
  /   log (n)
 /__,
n = 2
>>> summation(i, (i, 0, n), (n, 0, m))
      3    2
m    m    m
-- + -- + -
6    2    3
>>> from sympy.abc import x
>>> from sympy import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
 x
e

Integration

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 integrate, erf, exp, sin, log, oo, pi, sinh, symbols
>>> x, y = symbols('x,y')

You can integrate elementary functions:

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

Also special functions are handled easily:

>>> integrate(exp(-x**2)*erf(x), x)
  ____    2
\/ pi *erf (x)
--------------
      4

It is possible to compute definite integrals:

>>> 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

Complex numbers

Besides the imaginary unit, I, which is imaginary, symbols can be created with attributes (e.g. real, positive, complex, etc...) and this will affect how they behave:

>>> from sympy import Symbol, exp, I
>>> x = Symbol("x") # a plain x with no attributes
>>> exp(I*x).expand()
 I*x
e
>>> exp(I*x).expand(complex=True)
   -im(x)               -im(x)
I*e      *sin(re(x)) + e      *cos(re(x))
>>> x = Symbol("x", real=True)
>>> exp(I*x).expand(complex=True)
I*sin(x) + cos(x)

Functions

trigonometric:

>>> from sympy import asin, asinh, cos, sin, sinh, symbols, I
>>> x, y = symbols('x,y')

>>> sin(x+y).expand(trig=True)
sin(x)*cos(y) + sin(y)*cos(x)

>>> cos(x+y).expand(trig=True)
-sin(x)*sin(y) + cos(x)*cos(y)

>>> sin(I*x)
I*sinh(x)

>>> sinh(I*x)
I*sin(x)

>>> asinh(I)
I*pi
----
 2

>>> asinh(I*x)
I*asin(x)

>>> sin(x).series(x, 0, 10)
     3     5     7       9
    x     x     x       x       / 10\
x - -- + --- - ---- + ------ + O\x  /
    6    120   5040   362880

>>> sinh(x).series(x, 0, 10)
     3     5     7       9
    x     x     x       x       / 10\
x + -- + --- + ---- + ------ + O\x  /
    6    120   5040   362880

>>> asin(x).series(x, 0, 10)
     3      5      7       9
    x    3*x    5*x    35*x     / 10\
x + -- + ---- + ---- + ----- + O\x  /
    6     40    112     1152

>>> asinh(x).series(x, 0, 10)
     3      5      7       9
    x    3*x    5*x    35*x     / 10\
x - -- + ---- - ---- + ----- + O\x  /
    6     40    112     1152

spherical harmonics:

>>> from sympy import Ylm
>>> from sympy.abc import theta, phi

>>> Ylm(1, 0, theta, phi)
  ___
\/ 3 *cos(theta)
----------------
        ____
    2*\/ pi

>>> Ylm(1, 1, theta, phi)
   ___  I*phi
-\/ 6 *e     *sin(theta)
------------------------
            ____
        4*\/ pi

>>> Ylm(2, 1, theta, phi)
   ____  I*phi
-\/ 30 *e     *sin(theta)*cos(theta)
------------------------------------
                  ____
              4*\/ pi

factorials and gamma function:

>>> from sympy import factorial, gamma, Symbol
>>> x = Symbol("x")
>>> n = Symbol("n", integer=True)

>>> factorial(x)
x!

>>> factorial(n)
n!

>>> gamma(x + 1).series(x, 0, 3) # i.e. factorial(x)
                      /          2     2\
                    2 |EulerGamma    pi |    / 3\
1 - EulerGamma*x + x *|----------- + ---| + O\x /
                      \     2         12/

zeta function:

>>> from sympy import zeta
>>> zeta(4, x)
zeta(4, x)

>>> zeta(4, 1)
  4
pi
---
 90

>>> zeta(4, 2)
       4
     pi
-1 + ---
      90

>>> zeta(4, 3)
         4
  17   pi
- -- + ---
  16    90

polynomials:

>>> from sympy import assoc_legendre, chebyshevt, legendre, hermite
>>> chebyshevt(2, x)
   2
2*x  - 1

>>> chebyshevt(4, x)
   4      2
8*x  - 8*x  + 1

>>> legendre(2, x)
   2
3*x    1
---- - -
 2     2

>>> legendre(8, x)
      8         6         4        2
6435*x    3003*x    3465*x    315*x     35
------- - ------- + ------- - ------ + ---
  128        32        64       32     128

>>> assoc_legendre(2, 1, x)
        __________
       /    2
-3*x*\/  - x  + 1

>>> assoc_legendre(2, 2, x)
     2
- 3*x  + 3

>>> hermite(3, x)
   3
8*x  - 12*x

Differential Equations

In isympy:

>>> from sympy import Function, Symbol, dsolve
>>> f = Function('f')
>>> x = Symbol('x')
>>> f(x).diff(x, x) + f(x)
        2
       d
f(x) + ---(f(x))
         2
       dx

>>> dsolve(f(x).diff(x, x) + f(x), f(x))
f(x) = C1*sin(x) + C2*cos(x)

Algebraic equations

In isympy:

>>> from sympy import solve, symbols
>>> x, y = symbols('x,y')
>>> solve(x**4 - 1, x)
[-1, 1, -I, I]

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

Linear Algebra

Matrices

Matrices are created as instances from the Matrix class:

>>> from sympy import Matrix, Symbol
>>> Matrix([[1,0], [0,1]])
[1  0]
[    ]
[0  1]

They can also contain symbols:

>>> 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 about Matrices, see the Linear Algebra tutorial.

Pattern matching

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 Symbol, Wild
>>> x = Symbol('x')
>>> p = Wild('p')
>>> (5*x**2).match(p*x**2)
{p: 5}

>>> q = Wild('q')
>>> (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 use the exclude parameter of the Wild class to ensure that certain things do not show up in the result:

>>> 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}

Printing

There are many ways to print expressions.

Standard

This is what str(expression) returns and it looks like this:

>>> from sympy import Integral
>>> from sympy.abc import x
>>> print x**2
x**2
>>> print 1/x
1/x
>>> print Integral(x**2, x)
Integral(x**2, x)

Pretty printing

Nice ascii-art printing is produced by the pprint function:

>>> from sympy import Integral, pprint
>>> from sympy.abc import x
>>> pprint(x**2)
 2
x
>>> pprint(1/x)
1
-
x
>>> pprint(Integral(x**2, x))
  /
 |
 |  2
 | x  dx
 |
/

If you have a unicode font installed, the pprint function will use it by default. You can override this using the use_unicode option.:

>>> pprint(Integral(x**2, x), use_unicode=True)

⎮  2
⎮ x  dx

See also the wiki Pretty Printing for more examples of a nice unicode printing.

Tip: To make pretty printing the default in the Python interpreter, use:

$ python
Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
[GCC 4.3.1] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import init_printing, var, Integral
>>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
>>> var("x")
x
>>> x**3/3
 3
x
--
3
>>> Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
  /
 |
 |  2
 | x  dx
 |
/

Python printing

>>> from sympy.printing.python import python
>>> from sympy import Integral
>>> from sympy.abc import x
>>> print python(x**2)
x = Symbol('x')
e = x**2
>>> print python(1/x)
x = Symbol('x')
e = 1/x
>>> print python(Integral(x**2, x))
x = Symbol('x')
e = Integral(x**2, x)

LaTeX printing

>>> from sympy import Integral, latex
>>> from sympy.abc import x
>>> latex(x**2)
x^{2}
>>> latex(x**2, mode='inline')
$x^{2}$
>>> latex(x**2, mode='equation')
\begin{equation}x^{2}\end{equation}
>>> latex(x**2, mode='equation*')
\begin{equation*}x^{2}\end{equation*}
>>> latex(1/x)
\frac{1}{x}
>>> latex(Integral(x**2, x))
\int x^{2}\, dx

MathML

>>> from sympy.printing.mathml import mathml
>>> from sympy import Integral, latex
>>> from sympy.abc import x
>>> print mathml(x**2)
<apply><power/><ci>x</ci><cn>2</cn></apply>
>>> print mathml(1/x)
<apply><power/><ci>x</ci><cn>-1</cn></apply>

Pyglet

>>> from sympy import Integral, preview
>>> from sympy.abc import x
>>> preview(Integral(x**2, x)) 

If pyglet is installed, a pyglet window will open containing the LaTeX rendered expression:

_images/pngview1.png

Notes

isympy calls pprint automatically, so that’s why you see pretty printing by default.

Note that there is also a printing module available, sympy.printing. Other printing methods available through 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.

Further documentation

Now it’s time to learn more about SymPy. Go through the SymPy User’s Guide and SymPy Modules Reference.

Be sure to also browse our public wiki.sympy.org, that contains a lot of useful examples, tutorials, cookbooks that we and our users contributed, and feel free to edit it.

Translations

This tutorial is also available in other languages: