In [1]:
# Trochę magii
from sympy.interactive import printing
printing.init_printing(use_latex=True)

import sympy as sym
from sympy import *

# Definicje zmiennych
x, y, z = symbols("x y z")
k = Symbol("k", integer=True)
f = Function('f')
In [2]:
eq = ((x+y)**2 * (x+1))
eq
Out[2]:
$$\left(x + 1\right) \left(x + y\right)^{2}$$
In [3]:
expand(eq)
Out[3]:
$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$
In [4]:
a = 1/x + (x*y - 1)/x + 1
a
Out[4]:
$$1 + \frac{1}{x} \left(x y - 1\right) + \frac{1}{x}$$
In [5]:
simplify(a)
Out[5]:
$$y + 1$$
In [6]:
eq = Eq(x - 3)
eq
Out[6]:
$$x - 3 = 0$$
In [7]:
solve(eq, x)
Out[7]:
$$\left [ 3\right ]$$
In [8]:
eq = Eq(x**3 + 3*x**2 - 13*x - 15)
eq
Out[8]:
$$x^{3} + 3 x^{2} - 13 x - 15 = 0$$
In [9]:
solve(eq, x)
Out[9]:
$$\left [ -5, \quad -1, \quad 3\right ]$$
In [10]:
eq = Eq(x**3 + 3*x**2 - 13*x + 15)
solve(eq, x)
Out[10]:
$$\left [ -1 - \frac{16}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{5937}}{9} + 15}} - \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{5937}}{9} + 15}, \quad -1 - \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{5937}}{9} + 15} - \frac{16}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{5937}}{9} + 15}}, \quad - \sqrt[3]{\frac{\sqrt{5937}}{9} + 15} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{5937}}{9} + 15}} - 1\right ]$$
In [11]:
eq = Eq(x**3 + 3*x**2 - 13*x + z)
solve(eq, x)
Out[11]:
$$\left [ - \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}} - 1 - \frac{16}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}}}, \quad - \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}} - 1 - \frac{16}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}}}, \quad - \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}} - 1 - \frac{16}{3 \sqrt[3]{\frac{z}{2} + \sqrt{\frac{1}{4} \left(z + 15\right)^{2} - \frac{4096}{27}} + \frac{15}{2}}}\right ]$$
In [12]:
s = Sum(6*k**2 - 1, (k, 1, 10))
s
Out[12]:
$$\sum_{k=1}^{10} \left(6 k^{2} - 1\right)$$
In [13]:
lim = Limit(1/x, x, 0)
lim
Out[13]:
$$\lim_{x \to 0^+} \frac{1}{x}$$
In [14]:
lim.doit()
Out[14]:
$$\infty$$
In [15]:
lim = 2*Limit((1+1/x)**x, x, +oo)
lim
Out[15]:
$$2 \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x}$$
In [16]:
lim.doit()
Out[16]:
$$2 e$$
In [17]:
q = 3*x**2
q
Out[17]:
$$3 x^{2}$$
In [18]:
q.diff(x)
Out[18]:
$$6 x$$
In [19]:
eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)
eqn
Out[19]:
$$9 f{\left (x \right )} + \frac{d^{2}}{d x^{2}} f{\left (x \right )} = 1$$
In [20]:
dsolve(eqn, f(x))
Out[20]:
$$f{\left (x \right )} = C_{1} \sin{\left (3 x \right )} + C_{2} \cos{\left (3 x \right )} + \frac{1}{9}$$