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:

- 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')
```

- 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)
```

- Equation Solving: SymPy can solve equations symbolically and find solutions for variables.

```
equation = sp.Eq(x**2 - 4, 0)
solutions = sp.solve(equation, x)
```

- Calculus: It supports differentiation and integration of expressions.

```
derivative = sp.diff(x**3, x)
integral = sp.integrate(x**2, x)
```

- 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
```

- Trigonometry and Special Functions: It includes trigonometric functions, logarithms, exponentials, and various special functions.

```
sin_expr = sp.sin(x)
log_expr = sp.log(x)
```

- Calculations with Constants: You can work with mathematical constants like pi and Euler’s number (e).

```
pi_value = sp.pi
e_value = sp.E
```

- 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.