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Backport PR #5459: Fix interact animation page jump FF Firefox doesn't render images immediately as the data is available. When animating the way that we animate, this causes the output area to collapse quickly before returning to its original size. When the output area collapses, FireFox scrolls upwards in attempt to compensate for the lost vertical content (so it looks like you are on the same spot in the page, with respect to the contents below the image's prior location). The solution is to resize the image output after the `img onload` event has fired. This PR: - Releases the `clear_output` height lock after the image has been loaded (instead of immediately or using a timeout). - Removes a `setTimeout` call in the `append_output` method. - `clear_output` in zmqshell no longer sends `\r` to the stream outputs. closes #5128
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SymPy: Open Source Symbolic Mathematics

This notebook uses the SymPy package to perform symbolic manipulations, and combined with numpy and matplotlib, also displays numerical visualizations of symbolically constructed expressions.

We first load sympy printing extensions, as well as all of sympy:

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')

Elementary operations

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]:
$$262537412640768743.99999999999925007259719818568888$$

Algebra

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

Calculus

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

Illustrating Taylor series

We will define a function to compute the Taylor series expansions of a symbolically defined expression at various orders and visualize all the approximations together with the original function

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)

With this function defined, we can now use it for any sympy function or expression

In [22]:
plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))
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In [23]:
plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))
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This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by a simple function that has singularities on the real axis:

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))
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