SymPy in Python

Symbolic Mathematics with SymPy

Symbolic Mathematics with SymPy allows us to work with algebraic expressions, solve equations symbolically, compute derivatives and integrals, and perform various other symbolic math operations in Python. SymPy is a powerful library for handling symbolic mathematics and provides functionalities that are similar to what you might find in a Computer Algebra System (CAS) like Mathematica or Maple.

1. Introduction to SymPy

SymPy is a Python library for symbolic computation. It allows you to define mathematical symbols, perform algebraic manipulations, and find analytical solutions to mathematical problems.

   # Install SymPy if needed
   # !pip install sympy

   from sympy import symbols, Symbol

2. Defining Symbols and Expressions

Symbols are the building blocks of symbolic math in SymPy. You define variables and use them to create symbolic expressions.

   from sympy import symbols

   x, y, z = symbols('x y z')
   expr = x**2 + 2*x*y + y**2  # Define a symbolic expression

• Single Symbol: x = Symbol('x')
• Multiple Symbols: x, y, z = symbols('x y z')
• Subscripts: Use symbols('x1:4') to create symbols x1, x2, x3.

3. Simplifying Expressions

SymPy can simplify expressions using algebraic rules to create a more compact form.

   from sympy import simplify, expand, factor

   expr = x**2 + 2*x + 1
   simplified_expr = simplify(expr)         # x**2 + 2*x + 1 -> (x + 1)**2
   expanded_expr = expand((x + y)**3)       # Expands to x**3 + 3*x**2*y + 3*x*y**2 + y**3
   factored_expr = factor(x**2 - y**2)      # Factorization -> (x + y)(x - y)

• simplify(): General simplification.
• expand(): Expands expressions.
• factor(): Factors expressions.

4. Solving Equations

SymPy can solve both algebraic and transcendental equations symbolically.

   from sympy import Eq, solve

   equation = Eq(x**2 - 4, 0)   # Defines equation x**2 - 4 = 0
   solutions = solve(equation)  # Solves for x, returns [2, -2]

• Eq(): Defines an equation.
• solve(): Solves equations for variables.

Systems of Equations: SymPy can also solve multiple equations.

   eq1 = Eq(x + y, 10)
   eq2 = Eq(x - y, 4)
   solutions = solve((eq1, eq2), (x, y))  # Solves for x and y

5. Calculus with SymPy

SymPy provides tools for symbolic differentiation and integration.

#### a. Differentiation

   from sympy import diff

   expr = x**3 + 3*x**2 + 5
   derivative = diff(expr, x)  # Differentiate with respect to x

• Higher-order Derivatives: Use diff(expr, x, n) to differentiate n times.

  second_derivative = diff(expr, x, 2)  # Second derivative
  

#### b. Integration

   from sympy import integrate

   integral = integrate(expr, x)          # Indefinite integral
   definite_integral = integrate(expr, (x, 0, 2))  # Definite integral from 0 to 2

• Indefinite Integrals: integrate(expr, x)
• Definite Integrals: integrate(expr, (x, a, b))

6. Limits and Series Expansions

#### a. Limits

Compute limits of expressions as a variable approaches a particular value.

   from sympy import limit

   limit_expr = limit((x**2 - 1) / (x - 1), x, 1)  # Calculates limit as x -> 1

#### b. Series Expansion

Find the Taylor or Maclaurin series expansion of an expression.

   series_expr = expr.series(x, 0, 5)  # Expands expr around x=0 up to x^4

• series(): Expands around a point to a specified order.

7. Linear Algebra Operations

SymPy supports symbolic linear algebra, allowing you to work with matrices and solve systems of linear equations.

   from sympy import Matrix

   A = Matrix([[2, 1], [1, 3]])
   B = Matrix([5, 10])

   # Basic matrix operations
   C = A + B                       # Matrix addition
   product = A * B                 # Matrix multiplication
   determinant = A.det()           # Determinant of matrix A
   inverse = A.inv()               # Inverse of matrix A
   solution = A.solve(B)           # Solves Ax = B for x

8. Working with Polynomials

Polynomials can be defined and manipulated using SymPy. This includes finding roots, expanding, factoring, and simplifying.

   from sympy import Poly

   poly = Poly(x**3 + 3*x**2 + 3*x + 1, x)
   roots = poly.roots()            # Finds the roots of the polynomial

9. Discrete Mathematics Functions

SymPy supports combinatorial functions and sequences.

• Factorials and Combinations:

  from sympy import factorial, binomial
  
  factorial_5 = factorial(5)      # 5!
  combinations = binomial(5, 3)   # "5 choose 3"
  

• Summation and Products:

  from sympy import summation, Product
  
  summation_expr = summation(x**2, (x, 1, 5))  # Sum of x^2 from x=1 to 5
  product_expr = Product(x, (x, 1, 5)).doit()  # Product of x from 1 to 5
  

10. Plotting with SymPy

SymPy integrates with Matplotlib to enable symbolic plotting.

   from sympy.plotting import plot

   plot(x**2, (x, -10, 10))  # Plots x^2 from -10 to 10

Summary Table of Key Functions in SymPy