old wiki Examples

Note: This document was superseded by [http://code.google.com/p/sympy/wiki/Tutorial Tutorial], that contains everything here plus much more.

Introduction

This document contains examples of most of the features of sympy.

You can find all the following examples in the examples directory as regular python scripts. More examples can be found in the tests directory in form of the testing suite.

Note: the code in examples directory is uptodate. We want to generate this page automatically, but until then, if the code on this page doesn't work, try the corresponding example in the svn repository.

Basic manipulation

>>> import sympy
>>> a=sympy.Symbol('a')
>>> b=sympy.Symbol('b')
>>> c=sympy.Symbol('c')
>>> e=( a*b*b+2*b*a*b )**c
>>> print e
b^(2*c)*a^c*3^c

Expansion

>>> import sympy
>>> a=sympy.Symbol('a')
>>> b=sympy.Symbol('b')
>>> e=(a+b)**5
>>> print e
(b+a)^5
>>> print e.expand()
10*a^3*b^2+5*b^4*a+a^5+b^5+5*a^4*b+10*a^2*b^3

Functions

>>> import sympy
>>> a=sympy.Symbol('a')
>>> b=sympy.Symbol('b')
>>> e=sympy.log((a+b)**5)
>>> print e
5*log(b+a)
>>> e=sympy.exp(e)
>>> print e
exp(5*log(b+a))
>>> e=sym.log(sympy.exp((a+b)**5))
>>> print e
(b+a)^5

Differentiation

>>> import sympy
>>> a=sympy.Symbol('a')
>>> b=sympy.Symbol('b')
>>> e=(a+2*b)**5
>>> print e
(2*b+a)^5
>>> print e.diff(a)
5*(2*b+a)^4
>>> print e.diff(b)
10*(2*b+a)^4
>>> print e.diff(b).diffn(a,3)
240*(2*b+a)
>>> print e.expand().diff(b).diffn(a,3)
240*a+480*b

Series expansion

>>> import sympy
>>> x=sympy.Symbol('x')
>>> e=1/sympy.cos(x)
>>> print e.series(x,10)
1+50521/3628800*x^10+61/720*x^6+1/2*x^2+5/24*x^4+277/8064*x^8
>>> e=1/sympy.sin(x)
>>> print e.series(x,4)
x^(-1)+1/36*x^3+1/216*x^5+1/6*x

Substitution

>>> import sympy
>>> x=sympy.Symbol('x')
>>> y=sympy.Symbol('y')
>>> e=1/sympy.cos(x)
>>> print e
cos(x)^(-1)
>>> print e.subs(sympy.cos(x),y)
y^(-1)
>>> print e.subs(sympy.cos(x),y).subs(y,x**2)
x^(-2)
>>> e=1/sympy.log(x)
>>> e=e.subs(x,sympy.Real(2.71828))
>>> print e
log(2.71828)^(-1)
>>> print e.evalf()
1.00000067265

Arbitrary precision integers and rationals

>>> import sympy
>>> e=sympy.Rational(2)**50/sympy.Rational(10)**50
>>> print e
1/88817841970012523233890533447265625

Limits

>>> from sympy import exp,log,Symbol,Rational,sin,limit,limitinf
>>>
>>> x=Symbol("x")
>>> a=Symbol("a")
>>> h=Symbol("h")
>>>
>>> def sqrt(x):
...     return x**Rational(1,2)
...
>>> def sqrt3(x):
...     return x**Rational(1,3)
...
>>> def limitminf(f,x):
...     return limitinf(f.subs(x,-x),x)
...
>>> def show(computed, correct):
...     print "computed:",computed,"correct:",correct
...
>>> show( limitinf(sqrt(x**2-5*x+6)-x,x) , -Rational(5)/2 )
computed: (-5/2) correct: (-5/2)
>>> show( limitinf(x*(sqrt(x**2+1)-x),x) , Rational(1)/2 )
computed: 1/2 correct: 1/2
>>> show( limitinf(x-sqrt3(x**3-1),x) , Rational(0) )
computed: 0 correct: 0
>>> show( limitminf(log(1+exp(x))/x,x) , Rational(0) )
computed: 0 correct: 0
>>> show( limitinf(log(1+exp(x))/x,x) , Rational(1) )
computed: 1 correct: 1
>>> show( limit(sin(3*x)/x,x,0) , Rational(3) )
computed: 3 correct: 3
>>> show( limit(sin(5*x)/sin(2*x),x,0) , Rational(5)/2 )
computed: 5/2 correct: 5/2
>>> show( limitinf(((x-1)/(x+1))**x,x) , exp(-2))
computed: exp((-2)) correct: exp((-2))