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Use environment variable to identify conda / mamba (#14515)...
Use environment variable to identify conda / mamba (#14515) Conda and mamba both set an environment variable which refers to the base environment's executable path, use that in preference to less reliable methods, but fall back on the other approaches if unable to locate the executable this way. Additionally, change the search to look for the bare command name rather than the command within the top level of the active environment, I'm dubious this approach works with any current conda / mamba version which usually place their executable links in a `condabin` directory or elsewhere not at the same level as the Python executable. I believe this will also address https://github.com/ipython/ipython/issues/14350, which I'm also seeing in a Windows context where the regex fails to parse and causes a traceback.
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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]:
$$\left [ -2, \quad - 2 i, \quad 2 i\right ]$$
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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