pytutorial/sympy/intro/README.md

Symbolic Computation

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Top

Questions to David Rotermund

pip install sympy

Basic Operations
Printing
Simplification
Calculus
Solvers

Some examples

Substitution

import sympy

x, y = sympy.symbols("x y")

expr = sympy.cos(x) + 1
z = expr.subs(x, y**2)
print(z) # -> cos(y**2) + 1

Derivatives

import sympy

x, y = sympy.symbols("x y")

y = sympy.diff(sympy.sin(x) * sympy.exp(x), x)
print(y) # -> exp(x)*sin(x) + exp(x)*cos(x)

Integrals

import sympy

x, y = sympy.symbols("x y")

y = sympy.integrate(sympy.cos(x), x)
print(y)  # -> sin(x)

(Taylor) Series Expansion

import sympy

x, y, z = sympy.symbols("x y z")
y = sympy.cos(x)
z = y.series(x, 0, 8) # around x = 0 , up order 7

print(z)  # -> 1 - x**2/2 + x**4/24 - x**6/720 + O(x**8)

simplify

import sympy

x, y, z = sympy.symbols("x y z")
y = sympy.simplify(sympy.sin(x) ** 2 + sympy.cos(x) ** 2)

print(y)  # -> 1

Solving Equations Algebraically

solveset(equation, variable=None, domain=S.Complexes)

Recall from the gotchas section of this tutorial that symbolic equations in SymPy are not represented by = or ==, but by Eq.

import sympy

x, y, z = sympy.symbols("x y z")

z = sympy.Eq(x, y)

Output:

x=y

import sympy

x, y, z = sympy.symbols("x y z")

y = sympy.Eq(x**2 - x, 0)
z = sympy.solveset(y, x)
print(z) # -> {0, 1}

Solving Differential Equations

import sympy

# Undefined functions
f = sympy.symbols("f", cls=sympy.Function)

x = sympy.symbols("x")


diffeq = sympy.Eq(f(x).diff(x, x) - 2 * f(x).diff(x) + f(x), sympy.sin(x))

print(diffeq)  # -> Eq(f(x) - 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)), sin(x))

result = sympy.dsolve(diffeq, f(x))
print(result)  # -> Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)