In [1]:

    

from IPython.display import display

from sympy.interactive import printing
printing.init_printing(use_latex='mathjax')

from __future__ import division
import sympy as sym
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')


    

In [2]:

    

Rational(3,2)*pi + exp(I*x) / (x**2 + y)


    

    
Out[2]:





$$\frac{3 \pi}{2} + \frac{e^{i x}}{x^{2} + y}$$

In [3]:

    

exp(I*x).subs(x,pi).evalf()


    

    
Out[3]:





$$-1.0$$

In [4]:

    

e = x + 2*y


    

In [5]:

    

srepr(e)


    

    
Out[5]:





"Add(Symbol('x'), Mul(Integer(2), Symbol('y')))"

In [6]:

    

exp(pi * sqrt(163)).evalf(50)


    

    
Out[6]:





$$262537412640768743.99999999999925007259719818568888$$

In [7]:

    

eq = ((x+y)**2 * (x+1))
eq


    

    
Out[7]:





$$\left(x + 1\right) \left(x + y\right)^{2}$$

In [8]:

    

expand(eq)


    

    
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
a


    

    
Out[9]:





$$\frac{1}{x} \left(x \sin{\left (x \right )} - 1\right) + \frac{1}{x}$$

In [10]:

    

simplify(a)


    

    
Out[10]:





$$\sin{\left (x \right )}$$

In [11]:

    

eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)
eq


    

    
Out[11]:





$$x^{3} + 2 x^{2} + 4 x + 8 = 0$$

In [12]:

    

solve(eq, x)


    

    
Out[12]:





$$\begin{bmatrix}-2, & - 2 i, & 2 i\end{bmatrix}$$

In [13]:

    

a, b = symbols('a b')
Sum(6*n**2 + 2**n, (n, a, b))


    

    
Out[13]:





$$\sum_{n=a}^{b} \left(2^{n} + 6 n^{2}\right)$$

In [14]:

    

limit((sin(x)-x)/x**3, x, 0)


    

    
Out[14]:





$$- \frac{1}{6}$$

In [15]:

    

(1/cos(x)).series(x, 0, 6)


    

    
Out[15]:





$$1 + \frac{x^{2}}{2} + \frac{5 x^{4}}{24} + \mathcal{O}\left(x^{6}\right)$$

In [16]:

    

diff(cos(x**2)**2 / (1+x), x)


    

    
Out[16]:





$$- \frac{4 x \cos{\left (x^{2} \right )}}{x + 1} \sin{\left (x^{2} \right )} - \frac{\cos^{2}{\left (x^{2} \right )}}{\left(x + 1\right)^{2}}$$

In [17]:

    

integrate(x**2 * cos(x), (x, 0, pi/2))


    

    
Out[17]:





$$-2 + \frac{\pi^{2}}{4}$$

In [18]:

    

eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)
display(eqn)
dsolve(eqn, f(x))


    

    






$$9 f{\left (x \right )} + \frac{d^{2}}{d x^{2}}  f{\left (x \right )} = 1$$






    
Out[18]:





$$f{\left (x \right )} = C_{1} \sin{\left (3 x \right )} + C_{2} \cos{\left (3 x \right )} + \frac{1}{9}$$

In [19]:

    

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt


    

In [20]:

    

# You can change the default figure size to be a bit larger if you want,
# uncomment the next line for that:
#plt.rc('figure', figsize=(10, 6))


    

In [21]:

    

def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):
    """Plot the Taylor series approximations to a function at various orders.

    Parameters
    ----------
    func : a sympy function
    x0 : float
      Origin of the Taylor series expansion.  If not given, x0=xrange[0].
    orders : list
      List of integers with the orders of Taylor series to show.  Default is (2, 4).
    xrange : 2-tuple or array.
      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),
      or the actual array of values to use.
    yrange : 2-tuple
      (ymin, ymax) tuple indicating the y range for the plot.  If not given,
      the full range of values will be automatically used. 
    npts : int
      Number of points to sample the x range with.  Default is 200.
    """
    if not callable(func):
        raise ValueError('func must be callable')
    if isinstance(xrange, (list, tuple)):
        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)
    else:
        x = xrange
    if x0 is None: x0 = x[0]
    xs = sym.Symbol('x')
    # Make a numpy-callable form of the original function for plotting
    fx = func(xs)
    f = sym.lambdify(xs, fx, modules=['numpy'])
    # We could use latex(fx) instead of str(), but matploblib gets confused
    # with some of the (valid) latex constructs sympy emits.  So we play it safe.
    plt.plot(x, f(x), label=str(fx), lw=2)
    # Build the Taylor approximations, plotting as we go
    apps = {}
    for order in orders:
        app = fx.series(xs, x0, n=order).removeO()
        apps[order] = app
        # Must be careful here: if the approximation is a constant, we can't
        # blindly use lambdify as it won't do the right thing.  In that case, 
        # evaluate the number as a float and fill the y array with that value.
        if isinstance(app, sym.numbers.Number):
            y = np.zeros_like(x)
            y.fill(app.evalf())
        else:
            fa = sym.lambdify(xs, app, modules=['numpy'])
            y = fa(x)
        tex = sym.latex(app).replace('$', '')
        plt.plot(x, y, label=r'$n=%s:\, %s$' % (order, tex) )
        
    # Plot refinements
    if yrange is not None:
        plt.ylim(*yrange)
    plt.grid()
    plt.legend(loc='best').get_frame().set_alpha(0.8)


    

In [22]:

    

plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))


    

In [23]:

    

plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))


    

In [24]:

    

# For an expression made from elementary functions, we must first make it into
# a callable function, the simplest way is to use the Python lambda construct.
plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))