SymPy is a Python library for symbolic mathematics. It provides a wide range of mathematical operations and capabilities for symbolic computation, making it a powerful tool for tasks like algebraic manipulation, calculus, equation solving, and more.

Here are some of the key features and functionalities of SymPy:

1. Symbolic Variables: SymPy allows you to define symbolic variables and symbols, which can represent mathematical entities like numbers, variables, and expressions.
``````import sympy as sp
x, y = sp.symbols('x y')``````
1. Algebraic Manipulation: You can perform various algebraic operations, such as simplification, expansion, and factorization of expressions.
``````expr = (x + y)**2
simplified_expr = sp.simplify(expr)``````
1. Equation Solving: SymPy can solve equations symbolically and find solutions for variables.
``````equation = sp.Eq(x**2 - 4, 0)
solutions = sp.solve(equation, x)``````
1. Calculus: It supports differentiation and integration of expressions.
``````derivative = sp.diff(x**3, x)
integral = sp.integrate(x**2, x)``````
1. Linear Algebra: SymPy can perform operations on matrices and vectors.
``````A = sp.Matrix([[1, 2], [3, 4]])
B = sp.Matrix([x, y])
product = A * B``````
1. Trigonometry and Special Functions: It includes trigonometric functions, logarithms, exponentials, and various special functions.
``````sin_expr = sp.sin(x)
log_expr = sp.log(x)``````
1. Calculations with Constants: You can work with mathematical constants like pi and Euler’s number (e).
``````pi_value = sp.pi
e_value = sp.E``````
1. Equation Solving for Symbols: SymPy can solve equations symbolically for specific symbols.
``solution_for_x = sp.solve(equation, x)``

SymPy is a powerful tool for symbolic mathematics and is widely used in mathematics, engineering, and scientific computing. It is an open-source library and can be easily integrated into Python projects for various mathematical tasks.