Arithmetical Operations in Python
Common Arithmetic Operators
a, b = 7, 2
print("a + b = ", a + b)
print("a - b = ", a - b)
print("a * b = ", a * b)
print("a / b = ", a / b)
print("a // b = ", a // b)
print("a ** b = ", a ** b)
print("a % b = ", a % b)
a + b = 9
a - b = 5
a * b = 14
a / b = 3.5
a // b = 3
a ** b = 49
a % b = 1
Integration
integrate.quad
About scipy.integrate.quad
from scipy import integrate
def f(x, a, b):
return a * x + b
v, err = integrate.quad(f, 1, 2, args = (-1, 1))
print(v)
-0.5
For $\int_{-1}^1\frac{1}{\sqrt{\vert x\vert}}dx$, because $x\ne 0$, we need to add points=[0], otherwise we will get an report like:
inf
/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:4: RuntimeWarning: divide by zero encountered in double_scalars after removing the cwd from sys.path.
from scipy import integrate
def f(x):
return 1 / np.sqrt(abs(x))
v, err = integrate.quad(f, -1, 1, points=[0])
print(v)
3.9999999999999813
About sympy
SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python.
from sympy import *
x, y, z = symbols('x, y, z')
f = (2/3)*x**2 + (1/3)*x**2 + x + x + 1
g = (y+1)**3 + (z+2)**2 + 6*y + 2*(z-4)
Simplification and Expansion
print(f.simplify())
print(g.expand())
1.0*x**2 + 2*x + 1
y**3 + 3*y**2 + 9*y + z**2 + 6*z - 3
Solve equations
f1 = 2*x - y + z - 10
f2 = 3*x + 2*y - z - 16
f3 = x + 6*y - z - 28
print(solve([f1, f2, f3]))
{x: 46/11, y: 56/11, z: 74/11}
Limitation
limit(function, independent variable, value)
f = (x+1)**2 + y**(1/2) + 3
print(limit(f, x, z+1))
print(limit(f, y, z-1))
print(limit(f, x, 1))
Classical limit example: $\lim_{x\to 0}\frac{\sin x}{x}$
f = sin(x)/x
print(limit(f, x, 0))
Negative direction approximation: dir='-'
print(limit(f, x, 0, dir='-'))
Value can also be equal to $\infty$ using oo; while for $-\infty$ can be expressed by -oo.
Differential
f = 3 * x**2 + x
g = cos(x*y)
print(diff(f, x))
print(diff(f, x, 2))
print(diff(g, x, 2, y, 4))
6*x + 1
6
x**2*(-x**2*y**2*cos(x*y) - 8*x*y*sin(x*y) + 12*cos(x*y))
Integration
\[\int_{-\infty}^0e^x\]f = exp(x)
integrate(f, (x, -oo, 0))
f = 3*x**2 + 1
integrate(f, x)
f = x + 2*y
integrate(f, (x,0,1), (y,-3,4))