#!/usr/bin/env python
# coding: utf-8

# # SymPy: Open Source Symbolic Mathematics
# 
# This notebook uses the [SymPy](http://sympy.org) 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')


# <h2>Elementary operations</h2>

# In[2]:


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


# In[3]:


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


# In[4]:


e = x + 2*y


# In[5]:


srepr(e)


# In[6]:


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


# <h2>Algebra<h2>

# In[7]:


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


# In[8]:


expand(eq)


# In[9]:


a = 1/x + (x*sin(x) - 1)/x
a


# In[10]:


simplify(a)


# In[11]:


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


# In[12]:


solve(eq, x)


# In[13]:


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


# <h2>Calculus</h2>

# In[14]:


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


# In[15]:


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


# In[16]:


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


# In[17]:


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


# In[18]:


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


# # 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]:


get_ipython().run_line_magic('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))