symbolic-computation-guide

Symbolic Computation Guide

A skill for using computer algebra systems (CAS) in mathematical research. Covers symbolic differentiation, integration, equation solving, series expansion, linear algebra, and polynomial arithmetic using SymPy, SageMath, and Mathematica, with practical workflows for research mathematics.

SymPy Fundamentals

Symbolic Expressions and Manipulation

from sympy import (
    symbols, expand, factor, simplify, cancel, apart,
    sin, cos, exp, log, sqrt, pi, oo, I,
    Rational, Eq, solve, solveset, S
)

x, y, z, t, n, k = symbols("x y z t n k")
a, b, c = symbols("a b c", real=True)

# Expression manipulation
expr = (x + 1) ** 3
expanded = expand(expr)       # x**3 + 3*x**2 + 3*x + 1
factored = factor(expanded)   # (x + 1)**3

# Trigonometric simplification
from sympy import trigsimp
trig_expr = sin(x)**2 + cos(x)**2
simplified = trigsimp(trig_expr)  # 1

# Partial fraction decomposition
rational = (x**2 + 2*x + 3) / ((x + 1) * (x + 2) * (x + 3))
partial = apart(rational, x)
# 3/(2*(x + 3)) - 2/(x + 2) + 1/(2*(x + 1))

Calculus

from sympy import diff, integrate, limit, series, Sum, Product

# Differentiation
f = x**3 * exp(-x) * sin(x)
f_prime = diff(f, x)
f_double_prime = diff(f, x, 2)

# Integration
# Definite integral
area = integrate(exp(-x**2), (x, -oo, oo))  # sqrt(pi)

# Indefinite integral
antideriv = integrate(x * sin(x), x)  # -x*cos(x) + sin(x)

# Limits
lim_result = limit(sin(x) / x, x, 0)  # 1
lim_inf = limit((1 + 1/n)**n, n, oo)   # E (Euler's number)

# Taylor series
taylor = series(exp(x) * cos(x), x, 0, n=6)
# 1 + x - x**3/3 - x**4/6 + ...

# Summation
harmonic = Sum(1/k, (k, 1, n))
partial_sum = harmonic.doit()  # harmonic(n) -- returns harmonic number

geometric = Sum(x**k, (k, 0, oo))
closed_form = geometric.doit()  # Piecewise(1/(1 - x), Abs(x) < 1)

Equation Solving

# Algebraic equations
solutions = solve(x**3 - 6*x**2 + 11*x - 6, x)  # [1, 2, 3]

# System of equations
system_sol = solve([
    2*x + 3*y - 7,
    x - y + 1
], [x, y])  # {x: 4/5, y: 9/5}

# Differential equations
from sympy import Function, dsolve, Derivative
f = Function("f")

# f''(x) + f(x) = 0  (simple harmonic oscillator)
ode = Eq(f(x).diff(x, 2) + f(x), 0)
general_solution = dsolve(ode, f(x))
# f(x) = C1*sin(x) + C2*cos(x)

# With initial conditions
particular = dsolve(ode, f(x), ics={f(0): 1, f(x).diff(x).subs(x, 0): 0})
# f(x) = cos(x)