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Wrap os.path functions in method calls...
Wrap os.path functions in method calls Some functions from os.path are now references to C functions (e.g. isdir on Windows). This breaks the path module, because compiled functions do not get bound to an object instance. All os.path functions have been wrapped in method calls, out of general caution. Closes gh-737
Brian E. Granger - - Load All Authors

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SymPy: Open Source Symbolic Mathematics

In [1]:
%load_ext sympyprinting
In [2]:
from __future__ import division
from sympy import *
x, y, z = symbols("x y z")
k, m, n = symbols("k m n", integer=True)
f, g, h = map(Function, 'fgh')

Elementary operations

In [3]:
Rational(3,2)*pi + exp(I*x) / (x**2 + y)
Out[3]:
$$\frac{3}{2} \pi + \frac{e^{\mathbf{\imath} x}}{x^{2} + y}$$
In [4]:
exp(I*x).subs(x,pi).evalf()
Out[4]:
$$-1.0$$
In [5]:
e = x + 2*y
In [6]:
srepr(e)
Out[6]:
Add(Symbol('x'), Mul(Integer(2), Symbol('y')))
In [7]:
exp(pi * sqrt(163)).evalf(50)
Out[7]:
$$262537412640768743.99999999999925007259719818568888$$

Algebra

In [8]:
((x+y)**2 * (x+1)).expand()
Out[8]:
$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$
In [9]:
a = 1/x + (x*sin(x) - 1)/x
display(a)
simplify(a)
$$\frac{x \operatorname{sin}\left(x\right) -1}{x} + \frac{1}{x}$$
Out[9]:
$$\operatorname{sin}\left(x\right)$$
In [10]:
solve(Eq(x**3 + 2*x**2 + 4*x + 8, 0), x)
Out[10]:
[-2⋅ⅈ, 2⋅ⅈ, -2]
In [11]:
a, b = symbols('a b')
Sum(6*n**2 + 2**n, (n, a, b))
Out[11]:
$$\sum_{n=a}^{b} \left(2^{n} + 6 n^{2}\right)$$

Calculus

In [12]:
limit((sin(x)-x)/x**3, x, 0)
Out[12]:
$$- \frac{1}{6}$$
In [13]:
(1/cos(x)).series(x, 0, 6)
Out[13]:
$$1 + \frac{1}{2} x^{2} + \frac{5}{24} x^{4} + \operatorname{\mathcal{O}}\left(x^{6}\right)$$
In [14]:
diff(cos(x**2)**2 / (1+x), x)
Out[14]:
$$- 4 \frac{x \operatorname{sin}\left(x^{2}\right) \operatorname{cos}\left(x^{2}\right)}{x + 1} - \frac{\operatorname{cos}^{2}\left(x^{2}\right)}{\left(x + 1\right)^{2}}$$
In [15]:
integrate(x**2 * cos(x), (x, 0, pi/2))
Out[15]:
$$-2 + \frac{1}{4} \pi^{2}$$
In [16]:
eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)
display(eqn)
dsolve(eqn, f(x))
$$9 \operatorname{f}\left(x\right) + \frac{\partial^{2}}{\partial^{2} x} \operatorname{f}\left(x\right) = 1$$
Out[16]:
$$\operatorname{f}\left(x\right) = C_{1} \operatorname{cos}\left(3 x\right) + C_{2} \operatorname{sin}\left(3 x\right) + \frac{1}{9}$$