sympy
Mathematical Programming with Python
https://people.sc.fsu.edu/~jburkardt/classes/...
python_2025/sympy/sympy.html

sympy: symbolic computing, using the sympy() library, doing exact calculations with symbolic variables; differentiation; integration; limits; simplifying products and quotients.

Lecture notes:

• sympy.pdf

• bvp_example.py, manufactures a right hand side for a given boundary value problem.
• divergence_example.py, computes the divergence of a vector field.
• gradient_example.py, computes the gradient vector of a scalar field.
• hessian_example.py, computes the Hessian matrix of a scalar function f(x,y).
• humps_example.py, analyzes a version of the humps function.
• intro_example.py, a quick overview of a variety of sympy() calculations.
• jacobian_example.py, computes the Jacobian matrix of f(x,y,z).
• linalg_example.py, computes inverse, determinant, eigenvalues, singular values, QR and LU factorizations of the second difference matrix.
• newton_example.py, finds first and second derivatives of f(x).
• solve_example.py, finds a root of the humps() function.
• summation_example.py, compare exp(x) to the summation of 5 or 10 terms of MacLaurin series.
• taylor_example.py, computes Taylor series for f(x) around a given value of x, to a given order.

• sympy_derivative.py, symbolic derivative of humps(x), z = x * e^(x*t), sinc(x), sinc(x) derivative limits at 0.
• sympy_evaluate.py, show we can substitute a new expression for the argument of the humps function; we can request a numpy version of the humps function which can then be evaluated at a range of numeric values.
• sympy_integral.py, compute the indefinite integral function (antiderivative) of the humps function, or the definite integral over a given interval. Compute the integral over the plane of f(x,y)=exp(-x^2-y^2).
• sympy_limit.py, limit of humps(x) as x->-oo, sin(x)/x as x->0, log(x)/x as x->+oo, (1+x/n)**n as n->+oo.
• sympy_ode.py, demonstrates the solution of an ordinary differential equation.
• sympy_polynomial.py, works with polynomials in symbolic form.