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* TOC {:toc}Questions to David Rotermund
pip install sympy| Basic Operations |
| Printing |
| Simplification |
| Calculus |
| Solvers |
| Matrices |
| Basics | Contains a description of operations for the basic modules. Subcategories include: absolute basics, manipulation, assumptions, functions, simplification, calculus, solvers, and some other subcategories. |
| Code Generation | Contains a description of methods for the generation of compilable and executable code. |
| Logic | Contains method details for the logic and sets modules. |
| Matrices | Discusses methods for the matrices, tensor and vector modules. |
| Number Theory | Documents methods for the Number theory module. |
| Physics | Contains documentation for Physics methods. |
| Utilities | Contains docstrings for methods of several utility modules. Subcategories include: Interactive, Parsing, Printing, Testing, Utilities. |
| Topics | Contains method docstrings for several modules. Subcategories include : Plotting, Polynomials, Geometry, Category Theory, Cryptography, Differential, Holonomic, Lie Algebra, and Stats. |
| Assumptions |
| Calculus |
| Combinatorics |
| Functions |
| Integrals |
| Series |
| Simplify |
| Solvers |
| abc |
| Algebras |
| Concrete |
| Core |
| Discrete |
| Numerical Evaluation |
| Numeric Computation |
| Term Rewriting |
import sympy
x, y = sympy.symbols("x y")
expr = sympy.cos(x) + 1
z = expr.subs(x, y**2)
print(z) # -> cos(y**2) + 1import 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)import sympy
x, y = sympy.symbols("x y")
y = sympy.integrate(sympy.cos(x), x)
print(y) # -> sin(x)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)import sympy
x, y, z = sympy.symbols("x y z")
y = sympy.simplify(sympy.sin(x) ** 2 + sympy.cos(x) ** 2)
print(y) # -> 1solveset(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:
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}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)import sympy
x, y = sympy.symbols("x y")
expr = sympy.cos(x) + 1
print(expr) # -> cos(x) + 1
expr = expr.subs(x, 0.333 * sympy.pi)
print(expr) # -> cos(0.333*pi) + 1
print(sympy.N(expr)) # -> 1.50090662536071import sympy
import inspect
import numpy as np
import matplotlib.pyplot as plt
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))
result = sympy.dsolve(diffeq, f(x))
symbols = list(result.rhs.free_symbols)
f = sympy.lambdify(symbols, result.rhs, "numpy")
print("The arguments of the result:")
print(inspect.getfullargspec(f).args)
print("The source code behind f:")
print(inspect.getsource(f))
np_x = np.linspace(0, 2 * np.pi, 100)
plt.plot(f(x=np_x, C1=1.0, C2=1.0))
plt.xlabel("x")
plt.ylabel("f(x)")
plt.show()
C1 = sympy.symbols('C1', positive=True)| real=True | The symbol represents a real number. |
| positive=True | The symbol represents a positive number. |
| negative=True | The symbol represents a negative number. |
| integer=True | The symbol represents an integer. |
| prime=True | The symbol represents a prime number. |
| odd=True | The symbol represents an odd number. |
| even=True | The symbol represents an even number. |
import sympy
x = sympy.symbols("x")
sympy.solve([x**2 - 1, x >= 0.5, x <= 3], x)import sympy
import numpy as np
import matplotlib.pyplot as plt
tau_value: float = 10e-3 # s
a0_value: float = 0.1e3 # V^-1
uc_value: float = -55.0e-3 # V
urest_value: float = -70.0e-3 # V
r_value: float = 10e6 # Ohm
uthr_value: float = -40e-3 # V
u = sympy.symbols("u", cls=sympy.Function, real=True)
urest, uc, uthr = sympy.symbols("urest uc uthr", real=True)
t, tau, a0, r, i = sympy.symbols("t tau a0 r i", real=True, positive=True)
c0, c1, c2 = sympy.symbols("c0 c1 c2", real=True)
diffeq_rhs = (a0 * (u(t) - urest) * (u(t) - uc) + r * i) / tau
diffeq_rhs = sympy.expand(diffeq_rhs)
diffeq_rhs = sympy.collect(diffeq_rhs, u(t))
c0_coeffs = diffeq_rhs.coeff(u(t), 0)
c1_coeffs = diffeq_rhs.coeff(u(t), 1)
c2_coeffs = diffeq_rhs.coeff(u(t), 2)
diffeq_rhs = diffeq_rhs.subs(c0_coeffs, c0)
diffeq_rhs = diffeq_rhs.subs(c1_coeffs, c1)
diffeq_rhs = diffeq_rhs.subs(c2_coeffs, c2)
diffeq = sympy.Eq(u(t).diff(t), diffeq_rhs)
solved_diffeq = sympy.dsolve(diffeq, u(t), ics={u(0): urest}, simplify=False)
solved_diffeq = solved_diffeq.subs(u(t), uthr)
solved_diffeq = sympy.simplify(solved_diffeq)
solved2_diffeq = sympy.solve(solved_diffeq, t)[0]
solved2_diffeq = solved2_diffeq.subs(c0, c0_coeffs)
solved2_diffeq = solved2_diffeq.subs(c1, c1_coeffs)
solved2_diffeq = solved2_diffeq.subs(c2, c2_coeffs)
solved2_diffeq = sympy.simplify(solved2_diffeq)
solved2_diffeq = solved2_diffeq.subs(tau, tau_value)
solved2_diffeq = solved2_diffeq.subs(a0, a0_value)
solved2_diffeq = solved2_diffeq.subs(uc, uc_value)
solved2_diffeq = solved2_diffeq.subs(urest, urest_value)
solved2_diffeq = solved2_diffeq.subs(r, r_value)
solved2_diffeq = solved2_diffeq.subs(uthr, uthr_value)
solved2_diffeq = sympy.simplify(solved2_diffeq)
print("The final function")
print(solved2_diffeq)
print()
thr_func = sympy.lambdify(i, solved2_diffeq, "numpy")
i = 4 * 1e-9 * np.arange(0, 1000, dtype=np.complex128) / 1000
z = thr_func(i)
z = 1.0 / z
z[z < 0] = 0
z = z.astype(dtype=np.float32)
plt.plot(i * 1e9, z)
plt.xlabel("Input [nA]")
plt.ylabel("Rate [Hz]")
plt.show()