# 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¶

Unpack it:

:math: tar xzf sympy-0.5.12.tar.gz


and try it from a Python intepreter:

 cd sympy-0.5.12
:math: 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
>>> 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
Building dependency tree
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:

:math: cd sympy
 ./bin/isympy


IPython console for SymPy 0.7.0 (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]: input: Out[1]:

2 4 6 8

x 5*x 61*x 277*x

1 + ── + ──── + ───── + ────── + O(x**10)
2 24 720 8064

Note

Commands entered by you are bold. Thus what we did in 3 lines in a regular Python interpeter 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) 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 int’s 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

>>> 1/2
0.5


True division is going to be standard in python3k and isympy does that too.

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 precission:

>>> pi**2
pi**2

>>> pi.evalf()
3.14159265358979

>>> (pi+exp(1)).evalf()
5.85987448204884


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


### Symbols¶

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(old, new):

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

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


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

>>> import sys
>>> oldhook = sys.displayhook
>>> sys.displayhook = pprint


So that things pretty print. See the Printing section below. If you have a unicode font installed, your output may look a litte different (it will look slightly nicer).

## Algebra¶

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

>>> 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):

>>> 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 *
>>> 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 *
>>> 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:

>>> limit((tan(x+y)-tan(x))/y, y, 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 *
>>> x = Symbol('x')
>>> cos(x).series(x, 0, 10)
2    4     6      8
x    x     x      x
1 - -- + -- - --- + ----- + O(x**10)
2    24   720   40320
>>> (1/cos(x)).series(x, 0, 10)
2      4       6        8
x    5*x    61*x    277*x
1 + -- + ---- + ----- + ------ + O(x**10)
2     24     720     8064


Another simple example:

from sympy import Integral, Symbol, pprint

x = Symbol("x")
y = Symbol("y")

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

print(s)
pprint(s)


That should print the following after the execution:

1/y + x**2*y**(-3) + x**4*y**(-5) - x*y**(-2) - x**3*y**(-4) + O(x**5)
2    4         3
1   x    x    x    x
─ + ── + ── - ── - ── + O(x**5)
y    3    5    2    4
y    y    y    y

### Integration¶

SymPy has support for indefinite and definite integration of transcendental elementary and special functions via 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)
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 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


### Complex numbers¶

>>> from sympy import Symbol, exp, I
>>> x = Symbol("x")
>>> 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:

>>> 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
x - -- + --- - ---- + ------ + O(x**10)
6    120   5040   362880

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

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

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


spherical harmonics:

>>> 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:

>>> x = Symbol("x")

>>> y = Symbol("y", integer=True)

>>> factorial(x)
x!

>>> factorial(y)
y!

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


zeta function:

>>> 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:

>>> 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:

>>> f(x).diff(x, x) + f(x)
2
d
f(x) + -----(f(x))
dx dx

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


### Algebraic equations¶

In isympy:

>>> 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
>>> 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]


## 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 *
>>> 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:

>>> 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}


## Printing¶

There are many ways how expressions can be printed.

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

This is a nice ascii-art printing produced by a pprint function:

>>> sys.displayhook = oldhook
>>> 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, it should use unicode pretty printing by default. You can override this using the 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 the pretty printing default in the python interpreter, use:

:math: python
Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
[GCC 4.3.1] on linux2
>>> from sympy import *
>>> import sys
>>> sys.displayhook = pprint
>>> 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')
$$x^{2}$$
>>> 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))


And a pyglet window with the LaTeX rendered expression will popup:

### 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 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.

## 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 we encourage you to edit it.