Quick examples

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

Elementary operations

Construct a symbolic expression

Construct the formula ( \frac{3 \pi}{2} + \frac{e^{ix}}{x^2 + y}) :

 >>> 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 ( e^{ix}) for ( x=\pi) :

 >>> 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 ( e^{\pi \sqrt{163}}) :

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

Algebra

Expand products and powers

Expand ( (x+y)^2 (x+1)) :

 >>> ((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 ( \frac{1}{x} + \frac{x \sin x - 1}{x}) :

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

Solve a polynomial equation

Find the roots of ( x^3 + 2x^2 + 4x + 8) :

 >>> 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 ( \left(x+5y=2, -3x+6y=15\right)) :

 >>> 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 ( \sum_{n=a}^b 6 n^2 + 2^n) :

 >>> 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 ( \prod_{n=1}^b n (n+1)) :

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

Calculus

Calculate a limit

Evaluate ( \lim_{x\to 0} \frac{\sin x - x}{x^3}) :

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

Calculate a Taylor series

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

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

Calculate a derivative

Differentiate ( \frac{\cos(x^2)^2}{1+x}) :

 >>> 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 ( \int x^2 \cos x , dx)

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

Calculate the definite integral ( \int_0^{\pi/2} x^2 \cos x , dx) :

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

Solve an ordinary differential equation

Solve ( f''(x) + 9 f(x) = 1,!) :

 >>> f = Function('f')
 >>> dsolve(Eq(Derivative(f(x),x,x) + 9*f(x), 1), f(x))
f(x) == 1/9 + 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)
Out[2]:
            2
           d
9⋅f(x) + ─────(f(x)) = 1
         dx dx

In [3]: dsolve(_, f(x))
Out[3]: f(x) = 1/9 + C₁⋅sin(3⋅x) + C₂⋅cos(3⋅x)