Python | Getting started with SymPy module

Last Updated : 21 Apr, 2022

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.
SymPy only depends on mpmath, a pure Python library for arbitrary floating point arithmetic, making it easy to use.
Installing sympy module:

 pip install sympy

SymPy as a calculator:

SymPy defines following numerical types: Rational and Integer. The Rational class represents a rational number as a pair of two Integers, numerator and denominator, so Rational(1, 2) represents 1/2, Rational(5, 2) 5/2 and so on. The Integer class represents Integer number.
Example #1 :

Python3

# import everything from sympy module
from sympy import *
 
a = Rational(5, 8)
print("value of a is :" + str(a))
 
b = Integer(3.579)
print("value of b is :" + str(b))

Output:

 
value of a is :5/8
value of b is :3

SymPy uses mpmath in the background, which makes it possible to perform computations using arbitrary-precision arithmetic. That way, some special constants, like exp, pi, oo (Infinity), are treated as symbols and can be evaluated with arbitrary precision.
Example #2 :

Python3

# import everything from sympy module
from sympy import *
 
# you can't get any numerical value
p = pi**3
print("value of p is :" + str(p))
 
# evalf method evaluates the expression to a floating-point number
q = pi.evalf()
print("value of q is :" + str(q))
 
# equivalent to e ^ 1 or e ** 1
r = exp(1).evalf()
print("value of r is :" + str(r))
 
s = (pi + exp(1)).evalf()
print("value of s is :" + str(s))
 
rslt = oo + 10000
print("value of rslt is :" + str(rslt))
 
if oo > 9999999 :
    print("True")
else:
    print("False")

Output:

value of p is :pi^3
value of q is :3.14159265358979
value of r is :2.71828182845905
value of s is :5.85987448204884
value of rslt is :oo
True

In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly using Symbol() method.
Example #3 :

Python3

# import everything from sympy module
from sympy import * x = Symbol('x')
y = Symbol('y')
 
z = (x + y) + (x-y)
print("value of z is :" + str(z))

Output:

value of z is :2*x

Calculus:

The real power of a symbolic computation system such as SymPy is the ability to do all sorts of computations symbolically. SymPy can simplify expressions, compute derivatives, integrals, and limits, solve equations, work with matrices, and much, much more, and do it all symbolically. Here is a small sampling of the sort of symbolic power SymPy is capable of, to whet your appetite.
Example #4 : Find derivative, integration, limits, quadratic equation.

Python3

# import everything from sympy module
from sympy import *
 
# make a symbol
x = Symbol('x')
 
# make the derivative of sin(x)*e ^ x
ans1 = diff(sin(x)*exp(x), x)
print("derivative of sin(x)*e ^ x : ", ans1)
 
# Compute (e ^ x * sin(x)+ e ^ x * cos(x))dx
ans2 = integrate(exp(x)*sin(x) + exp(x)*cos(x), x)
print("indefinite integration is : ", ans2)
 
# Compute definite integral of sin(x ^ 2)dx
# in b / w interval of ? and ?? .
ans3 = integrate(sin(x**2), (x, -oo, oo))
print("definite integration is : ", ans3)
 
# Find the limit of sin(x) / x given x tends to 0 
ans4 = limit(sin(x)/x, x, 0)
print("limit is : ", ans4)
 
# Solve quadratic equation like, example : x ^ 2?2 = 0
ans5 = solve(x**2 - 2, x)
print("roots are : ", ans5)

Output :

derivative of sin(x)*e^x :  exp(x)*sin(x) + exp(x)*cos(x)
indefinite integration is :  exp(x)*sin(x)
definite integration is :  sqrt(2)*sqrt(pi)/2
limit is :  1
roots are :  [-sqrt(2), sqrt(2)]

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