Quick examples

From SymPy

This page gives quick examples of common symbolic calculations in SymPy. Print it and keep it under your pillow!

Contents 1 Elementary operations 1.1 Construct a symbolic expression 1.2 Evaluate a symbolic expression 1.3 Deconstruct an expression 1.4 Calculate a numerical value 2 Algebra 2.1 Expand products and powers 2.2 Simplify a formula 2.3 Solve a polynomial equation 2.4 Solve an equation system 2.5 Calculate a sum 2.6 Calculate a product 3 Calculus 3.1 Calculate a limit 3.2 Calculate a Taylor series 3.3 Calculate a derivative 3.4 Calculate an integral 3.5 Solve an ordinary differential equation

Elementary operations

Construct a symbolic expression

Construct the formula <math>\frac{3 \pi}{2} + \frac{e^{ix}}{x^2 + y}</math>:

>>> var('x y')
>>> Rational(3,2)*pi + exp(I*x) / (x**2 + y)
(3/2)*pi + 1/(y + x**2)*exp(I*x)

Evaluate a symbolic expression

Calculate the value of <math>e^{ix}</math> for <math>x=\pi</math>:

>>> x = Symbol('x')
>>> exp(I*x).subs(x,pi).evalf()
-1

Deconstruct an expression

>>> expr = x + 2*y
>>> expr.__class__
<class 'sympy.core.add.Add'>
>>> expr.args
(x, 2*y)

Calculate a numerical value

Calculate 50 digits of <math>e^{\pi \sqrt{163}}</math>:

>>> exp(pi * sqrt(163)).evalf(50)
262537412640768743.99999999999925007259719818568888

Algebra

Expand products and powers

Expand <math>(x+y)^2 (x+1)</math>:

>>> ((x+y)**2 * (x+1)).expand()
x**2 + x**3 + y**2 + x*y**2 + 2*x*y + 2*y*x**2

Simplify a formula

Simplify <math>\frac{1}{x} + \frac{x \sin x - 1}{x}</math>:

>>> a = 1/x + (x*sin(x) - 1)/x
>>> simplify(a)
sin(x)

Solve a polynomial equation

Find the roots of <math>x^3 + 2x^2 + 4x + 8</math>:

>>> solve(Eq(x**3 + 2*x**2 + 4*x + 8, 0), x)
[-2*I, 2*I, -2]

or more easily:

>>> solve(x**3 + 2*x**2 + 4*x + 8, x)
[-2*I, 2*I, -2]

For details, see: Finding roots of polynomials.

Solve an equation system

Solve the equation system <math>\left(x+5y=2, -3x+6y=15\right)</math>:

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

or

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

Calculate a sum

Evaluate <math>\sum_{n=a}^b 6 n^2 + 2^n</math>:

>>> sum(6*n**2 + 2**n, (n, a, b))
b + 2**(1 + b) - a - 2**a - 2*a**3 + 2*b**3 + 3*a**2 + 3*b**2

Calculate a product

Evaluate <math>\prod_{n=1}^b n (n+1)</math>:

>>> product(n*(n+1), (n, 1, b))
RisingFactorial(2, b)*gamma(1 + b)

Calculus

Calculate a limit

Evaluate <math>\lim_{x\to 0} \frac{\sin x - x}{x^3}</math>:

>>> limit((sin(x)-x)/x**3, x, 0)
-1/6

Calculate a Taylor series

Find the Maclaurin series of <math>\frac{1}{\cos x}</math> up to the <math>O(x^6)</math> term:

>>> (1/cos(x)).series(x, 0, 6)
1 + (1/2)*x**2 + (5/24)*x**4 + O(x**6)

Calculate a derivative

Differentiate <math>\frac{\cos(x^2)^2}{1+x}</math>:

>>> diff(cos(x**2)**2 / (1+x), x)
-(1 + x)**(-2)*cos(x**2)**2 - 4*x/(1 + x)*cos(x**2)*sin(x**2)

Calculate an integral

Calculate the indefinite integral <math>\int x^2 \cos x \, dx</math>

>>> integrate(x**2 * cos(x), x)
-2*sin(x) + x**2*sin(x) + 2*x*cos(x)

Calculate the definite integral <math>\int_0^{\pi/2} x^2 \cos x \, dx</math>:

>>> integrate(x**2 * cos(x), (x, 0, pi/2))
(-2) + (1/4)*pi**2

Solve an ordinary differential equation

Solve <math>f(x) + 9 f(x) + 1 = 1\,\!</math>:

>>> f = Function('f')
>>> dsolve(Eq(Derivative(f(x),x,x) + 9*f(x) + 1, 1), f(x))
C1*sin(3*x) + C2*cos(3*x)

You can also use .diff(), like here (an example in isympy)

In [1]: f = Function("f")

In [2]: Eq(f(x).diff(x, x) + 9*f(x) + 1, 1)
Out[2]: 
                2           
               d            
1 + 9*f(x) + ─────(f(x)) = 1
             dx dx          

In [3]: dsolve(_, f(x))
Out[3]: C₁*sin(3*x) + C₂*cos(3*x)