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))
In [23]:
plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))
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))
