# Quick examples

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

## Elementary operations

 >>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = map(Function, 'fgh')

### Construct a symbolic expression

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

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

### Evaluate a symbolic expression

Calculate the value of ( e^{ix}) for ( x=\pi) :

 >>> x = Symbol('x')
>>> exp(I*x).subs(x,pi).evalf()    #doctest: +SKIP
-1.00000000000000

### Deconstruct an expression

 >>> expr = x + 2*y
>>> expr.__class__
>>> expr.args
(2*y, x)

### 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**3 + 2*x**2*y + x**2 + x*y**2 + 2*x*y + y**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])
{x: -3, y: 1}

or

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

### Calculate a sum

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

 >>> a, b = symbols('a b')
>>> s = Sum(6*n**2 + 2**n, (n, a, b))
>>> s
Sum(2**n + 6*n**2, (n, a, b))
>>> s.doit()
-2**a + 2**(b + 1) - 2*a**3 + 3*a**2 - a + 2*b**3 + 3*b**2 + b

### Calculate a product

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

 >>> product(n*(n+1), (n, 1, b))
RisingFactorial(2, b)*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 + x**2/2 + 5*x**4/24 + O(x**6)

### Calculate a derivative

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

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

### Calculate an integral

Calculate the indefinite integral ( \int x^2 \cos x , dx)

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

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

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

### 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) == C1*cos(3*x) + C2*sin(3*x) + 1/9

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

 >>> f = Function("f")
>>> Eq(f(x).diff(x, x) + 9*f(x), 1)
9*f(x) + Derivative(f(x), x, x) == 1
>>> dsolve(_, f(x))
f(x) == C1*cos(3*x) + C2*sin(3*x) + 1/9