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

# # Symbolic mathematics with Sympy
# ## Vince Knight (original repo - [here](https://github.com/drvinceknight/Python-Mathematics-Handbook)) 
# 
# Forked and solved for future reference

# [Sympy](http://www.sympy.org/en/index.html) is described as a:
# 
# > "... Python library for symbolic mathematics."
# 
# This means it can be used to:
# 
# - Manipulate symbolic expressions;
# - Solve symbolic equations;
# - Carry out symbolic Calculus;
# - Plot symbolic function.
# 
# It has other capabilities that we will not go in to in this handbook. But you can read more about it here: http://www.sympy.org/en/index.html
# 
# ## Manipulating symbolic expressions
# 
# Before we can start using the library to manipulate expressions, we need to import it.

# In[1]:


from sympy import *


# The above imports the library and gives us access to it's commands using the shortand `sym` which is conventially used.
# 
# If we wanted to get Python to check that $x - x = 0$ we would get an error if we did not tell Python what $x$ was:

# x - x

# This is where Sympy comes in, we can tell Python to create $x$ as a symbolic variable:

# In[2]:


x = symbols('x')


# Now we can calculate $x - x$:

# In[3]:


x - x


# We can create and manipulate expressions in Sympy. Let us for example verify:
# 
# $$(a + b) ^ 2 = a ^ 2 + 2ab + b ^2$$
# 
# First, we create the symbolic variables $a, b$:

# In[4]:


a, b = symbols('a, b')


# Now let's create our expression:

# In[5]:


expr = (a + b) ** 2 
expr


# **Note** we can get Sympy to use LaTeX so that the output looks nice in a notebook:

# In[6]:


init_printing()


# In[7]:


expr


# Let us expand our expression:

# In[8]:


expr.expand()


# Note that we can also get Sympy to produce the LaTeX code for future use:

# In[9]:


latex(expr.expand())


# ---
# **EXERCISE** Use Sympy to verify the following expressions:
# 
# - $(a - b) ^ 2 = a ^ 2 - 2 a b + b^2$
# - $a ^ 2 - b ^ 2 = (a - b) (a + b)$ (instead of using `expand`, try `factor`)

# In[10]:


expr = (a - b) **2
expr.expand()


# In[11]:


expr = a ** 2 - b **2
expr.factor()


# ## Solving symbolic equations

# We can use Sympy to solve symbolic expression. For example let's find the solution in $x$ of the quadratic equation:
# 
# $$a x ^ 2 + b x + c = 0$$

# In[12]:


# We only really need to define `c` but doing them all again.
a, b, c, x = symbols('a, b, c, x')  


# The Sympy command for solving equations is `solveset`. The first argument is an expression for which the root will be found. The second argument is the value that we are solving for.

# In[13]:


solveset(a * x ** 2 + b * x + c, x)


# ---
# **EXERCISE** Use Sympy to find the solutions to the generic cubic equation:
# 
# $$a x ^ 3 + b x ^ 2 + c  x + d = 0$$
# 
# ---

# In[14]:


a, b, c, d, x = symbols('a, b, c, d, x')


# In[15]:


solveset(a*x**3+b*x**2+c*x+d,x)


# It is possible to pass more arguments to `solveset` for example to constrain the solution space. Let us see what the solution of the following is in $\mathbb{R}$:
# 
# $$x^2=-1$$

# In[16]:


solveset(x ** 2 + 1, x, domain=S.Reals)


# ---
# **EXERCISE** Use Sympy to find the solutions to the following equations:
# 
# - $x ^ 2 == 2$ in $\mathbb{N}$;
# - $x ^ 3 + 2 x = 0$ in $\mathbb{R}$.
# 
# ---

# In[17]:


x = symbols('x')


# In[18]:


solveset(x**2-2, domain= S.Naturals)


# In[19]:


solveset(x**3 + 2 , domain=S.Reals)


# ## Symbolic calculus
# 
# We can use Sympy to compute limits. Let us calculate:
# 
# $$\lim_{x\to 0^+}\frac{1}{x}$$

# In[20]:


limit(1/x, x, 0, dir="+")


# ---
# **EXERCISE** Compute the following limits:
# 
# 1. $\lim_{x\to 0^-}\frac{1}{x}$
# 2.  $\lim_{x\to 0}\frac{1}{x^2}$
# 
# ---

# In[21]:


limit(1/x,x,0,dir="-")


# In[22]:


limit(1/x**2,x,0,dir='+')


# We can use also Sympy to differentiate and integrate. Let us experiment with differentiating the following expression:
# 
# $$x ^ 2 - \cos(x)$$

# In[23]:


diff(x ** 2 - cos(x), x)


# Similarly we can integrate:

# In[24]:


integrate(x ** 2 - cos(x), x)


# In[25]:


integrate(exp(-x**2), x)


# We can also carry out definite integrals:

# In[26]:


integrate(x ** 2 - cos(x), (x, 0, 5))


# ---
# 
# **EXERCISE** Use Sympy to calculate the following:
# 
# 1. $\frac{d\sin(x ^2)}{dx}$
# 2. $\frac{d(x ^2 + xy - \ln(y))}{dy}$
# 3. $\int e^x \cos(x)\;dx$
# 4. $\int_0^5 e^{2x}\;dx$

# In[27]:


x, y = symbols('x,y')


# In[28]:


diff(sin(x**2),x)


# In[29]:


diff(x**2+x*y+ln(y),y)


# In[30]:


diff(x**2+x*y+ln(y),x)


# In[31]:


integrate(exp(x)*cos(x))


# In[32]:


integrate(exp(2*x),(x,0,5))


# ## Plotting with Sympy
# 
# Finally Sympy can be used to plot functions. Note that this makes use of another Python library called [matplotlib](http://matplotlib.org/). Whilst Sympy allows us to not directly need to make use of matplotlib it could be worth learning to use as it's a very powerful and versatile library.
# 
# Before plotting in Jupyter we need to run a command to tell it to display the plots directly in the notebook:

# In[33]:


get_ipython().run_line_magic('matplotlib', 'inline')


# Let us plot $x^2$:

# In[34]:


expr = x ** 2
p = plot(expr);


# We can directly save that plot to a file if we wish to:

# p.save("x_squared.pdf");

# ---
# **EXERCISE** Plot the following functions:
# 
# - $y=x + cos(x)$
# - $y=x ^ 2 - e^x$ (you might find `ylim` helpful as an argument)
# 
# Experiment with saving your plots to a file.
# 
# ---

# In[35]:


expr = cos(x) + x
plot(expr);


# In[36]:


expr = x **2 - exp(x)
plot(expr);


# In[ ]: