References/02 Sympy intro.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 02 Introduction to Sympy Module\n",
    "---\n",
    "\n",
    "## Introduction 👨‍🏫\n",
    "\n",
    "### What is Sympy?\n",
    "[![Sympy](https://img.shields.io/badge/Sympy-3B5526?style=flat&logo=Sympy&logoColor=white)](https://www.sympy.org/en/index.html) is an open-source Python library for symbolic computation. \n",
    "\n",
    "It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live or SymPy Gamma.SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies. This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.\n",
    "\n",
    "SymPy includes features ranging from basic symbolic arithmetic to calculus, algebra, discrete mathematics, and quantum physics. It is capable of formatting the result of the computations as LaTeX code.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np #Importing numpy for arrays\n",
    "import sympy as smp #Importing sympy for symbolic expressions\n",
    "import matplotlib.pyplot as plt #Importing matplotlib for plotting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbols can be defined as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = smp.symbols('x') #Defining a symbol x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can then use various mathematical operators on the defined symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2}$"
      ],
      "text/plain": [
       "x**2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x**2 #Giving power to the symbol x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are many many functions built into sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.sin(x) #Calling sin from sympy module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.cos(x) #Calling cos from sympy module"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can save expressions in new variables\n",
    "Rememeber to use tab key to explore all the options in sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#smp. Hit \"TAB\" for all options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Arithmetic operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "x**2 + 2*x + 1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x**2 + 2*x + 1 #Defining an expression\n",
    "y #CAlling an expression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that it doesn't expand. There are different methods you need to call to do different things with the expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 2 x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "(x**2 + 2*x + 1)**2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = y**2 #Definning a new variable for the expression\n",
    "z #Calling a new variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{4}$"
      ],
      "text/plain": [
       "(x + 1)**4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z.factor() #Calling factor fxn to do the factorization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the notation `smp.solve(f,x)`, sympy finds the value of $x$ that makes $f(x)=0$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.solve(z,x) #Calling solve fxn to find at what value my fxn becomes zero"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, pi]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.solve(smp.sin(x),x) #Calling solve fxn to find at what value my fxn becomes zero"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Good Practice**: \n",
    "\n",
    "If you know that your variable $x$ is real (and in some cases non-negative) it's good to specify this when you define your variables to prevent complicated issues when solving complicated problems (especially with integrals)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-I, I]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = smp.symbols('x') #Defining a variable x\n",
    "smp.solve(x**2+1,x) #Calling solve fxn to do the find out at what valve function becomes zero"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So here you see it has came out as -I and I, imaginary numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = smp.symbols('x',real=True,positive=True) #defining x symbol and saying its real and positive\n",
    "smp.solve(x+4,x) #Calling solve function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here its an empty set indication no posible answer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining many variables at time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + y + z$"
      ],
      "text/plain": [
       "x + y + z"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,z = smp.symbols('x y z') #Defining x y and z\n",
    "F = x+y+z #Defining a fxn\n",
    "F #Calling a fxn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can also solve multivariable expressions, specifying which variable we want to solve for. In each case, $F=0$ is being solved for"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-y - z]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_sols = smp.solve(F,x) #Calling solve to get the values at x = 0\n",
    "x_sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-x - z]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_sols = smp.solve(F,y) #Calling solve to get the values at y = 0\n",
    "y_sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-x - y]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_sols = smp.solve(F,z) #Calling solve to get the values at z = 0\n",
    "z_sols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, it may be the case that one has a sympy expression but wants to convert it into a numerical function so they can plot on a computer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - y - z$"
      ],
      "text/plain": [
       "-y - z"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x_sols[0] #Solve analytically\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_f = smp.lambdify([z,y],expr) #This module provides convenient functions to transform SymPy expressions to lambda functions which can be used to calculate numerical values very fast.\n",
    "expr_f(2,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_num = np.linspace(0,1,100) #Defining an array\n",
    "y_num = 2 #Defining a constant\n",
    "plt.plot(x_num, expr_f(x_num,y_num)) #Ploting x and y\n",
    "plt.grid() #Calling grid\n",
    "plt.show() #Calling plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For multivariable expressions, can also substitute values in"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + y + z$"
      ],
      "text/plain": [
       "x + y + z"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F #Calling function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + 8$"
      ],
      "text/plain": [
       "x + 8"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.subs([(y,5),(z,3)]) #Substituting y = 5 and z = 3 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = smp.symbols('x') #Defining x symbol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\lim_{x \\to \\pi} \\tan(x/2+ \\tan(x))$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -\\infty$"
      ],
      "text/plain": [
       "-oo"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit(smp.tan(x/2+smp.tan(x)),x,smp.pi) #Calling limit fxn and defining the fxn for a range of x to pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\frac{d}{dx} \\left( \\frac{1+\\cos x}{1 - \\sin x} \\right)^2 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{2 \\left(\\cos{\\left(x \\right)} + 1\\right) \\sin{\\left(x \\right)}}{\\left(1 - \\sin{\\left(x \\right)}\\right)^{2}} + \\frac{2 \\left(\\cos{\\left(x \\right)} + 1\\right)^{2} \\cos{\\left(x \\right)}}{\\left(1 - \\sin{\\left(x \\right)}\\right)^{3}}$"
      ],
      "text/plain": [
       "-2*(cos(x) + 1)*sin(x)/(1 - sin(x))**2 + 2*(cos(x) + 1)**2*cos(x)/(1 - sin(x))**3"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.diff(((1+smp.cos(x))/(1-smp.sin(x)))**2,x) #Calling diff fxn for analytical differentiation "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{d}{dx} f(x+g(x)) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\int \\sin(x) \\cos (x)  dx $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sin^{2}{\\left(x \\right)}}{2}$"
      ],
      "text/plain": [
       "sin(x)**2/2"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate(smp.sin(x)*smp.cos(x),x) #Callng integrate fxn for indefinite range"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definite Integrals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int_{0}^{\\ln(4)}\\frac{e^x dt}{\\sqrt{e^{2x}+9}} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\operatorname{asinh}{\\left(\\frac{1}{3} \\right)} + \\operatorname{asinh}{\\left(\\frac{4}{3} \\right)}$"
      ],
      "text/plain": [
       "-asinh(1/3) + asinh(4/3)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate(smp.exp(x)/smp.sqrt(smp.exp(2*x)+9),(x,0,smp.log(4))) #calling integrate to integrate a fxn for a definte range "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int_1^t x^{3} e^x dx $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(t^{3} - 3 t^{2} + 6 t - 6\\right) e^{t} + 2 e$"
      ],
      "text/plain": [
       "(t**3 - 3*t**2 + 6*t - 6)*exp(t) + 2*E"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t= smp.symbols('t') #Defining a new variable t\n",
    "smp.integrate(x**3*smp.exp(x), (x, 1, t)) #Calling integrate "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multivariable Calculus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y,z,p,q,r,a,b,c = smp.symbols('x y z p q r a b c') #Defining multi-variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vectors and Geometry"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "U = smp.Matrix([p,q,r]) #Defining Matrix U\n",
    "V = smp.Matrix([a,b,c]) #Defining Matrix V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Addition and multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a + 2 p\\\\b + 2 q\\\\c + 2 r\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[a + 2*p],\n",
       "[b + 2*q],\n",
       "[c + 2*r]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*U + V #Adding matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a p + b q + c r$"
      ],
      "text/plain": [
       "a*p + b*q + c*r"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(V) #Dot product of U and V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cross product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- b r + c q\\\\a r - c p\\\\- a q + b p\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-b*r + c*q],\n",
       "[ a*r - c*p],\n",
       "[-a*q + b*p]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.cross(V) #Cross product of U and V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{\\left|{p}\\right|^{2} + \\left|{q}\\right|^{2} + \\left|{r}\\right|^{2}}$"
      ],
      "text/plain": [
       "sqrt(Abs(p)**2 + Abs(q)**2 + Abs(r)**2)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.norm() #Returns the norm of the matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 t\\\\\\sin{\\left(t \\right)}\\\\t^{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   3*t],\n",
       "[sin(t)],\n",
       "[  t**2]])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = smp.Matrix([3*t, smp.sin(t), t**2]) #Defining a matrix\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3\\\\\\cos{\\left(t \\right)}\\\\2 t\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[     3],\n",
       "[cos(t)],\n",
       "[   2*t]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.diff(r,t) #Calling dif to get first order differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector Integrals\n",
    "\n",
    "(Note: Analytical results wont all the +C(constant term in the end))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{t} \\cos{\\left(t \\right)}\\\\t^{4}\\\\\\frac{1}{t^{2} + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[exp(t)*cos(t)],\n",
       "[         t**4],\n",
       "[ 1/(t**2 + 1)]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = smp.Matrix([smp.exp(t)*smp.cos(t), t**4, 1/(1+t**2)]) #Defining a new matrix\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{e^{t} \\sin{\\left(t \\right)}}{2} + \\frac{e^{t} \\cos{\\left(t \\right)}}{2}\\\\\\frac{t^{5}}{5}\\\\\\operatorname{atan}{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[exp(t)*sin(t)/2 + exp(t)*cos(t)/2],\n",
       "[                           t**5/5],\n",
       "[                          atan(t)]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.Integral(r,t).doit() #Calling integral for integration of the elements in matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Laplace Transforms and Sympy\n",
    "\n",
    "Laplace transforms convert a function $f(t)$ in the time domain into function in the Laplace domain $F(s)$.\n",
    "\n",
    "$F(s) = \\mathcal{L}\\left(f(t)\\right) = \\int_0^\\infty f(t)e^{-s\\,t}dt$\n",
    "\n",
    "As an example of the Laplace transform, consider a constant $c$. The function $f(t) = c$ and the following expression is integrated. \n",
    "\n",
    "$\\mathcal{L}(c)=\\int_0^\\infty c \\, e^{-s\\,t} dt = -\\frac{c}{s}e^{-s\\,t} \\biggr\\rvert_0^\\infty = 0 - \\left(-\\frac{c}{s} \\right) = \\frac{c}{s}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5}{s^{2}}$"
      ],
      "text/plain": [
       "5/s**2"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import t,s #Calling sympy.abc to import multivariables\n",
    "U = smp.laplace_transform(5*t,t,s) #Calling laplace transform\n",
    "U[0] #Calling zeroth element"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 5 t \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "5*t*Heaviside(t)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import inverse_laplace_transform #Importing inverse laplace from sympy module\n",
    "\n",
    "inverse_laplace_transform(U[0],s,t) #calling inverse for the defined fxn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5 s + 5}{\\left(s + 9\\right)^{3}}$"
      ],
      "text/plain": [
       "(5*s + 5)/(s + 9)**3"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = 5*(s+1)/(s+9)**3 #Defining a fxn\n",
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5}{\\left(s + 9\\right)^{2}} - \\frac{40}{\\left(s + 9\\right)^{3}}$"
      ],
      "text/plain": [
       "5/(s + 9)**2 - 40/(s + 9)**3"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.apart(F) #Partial fraciton decompostion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle s^{2} \\left(s + 1\\right) \\left(s + 3\\right)$"
      ],
      "text/plain": [
       "s**2*(s + 1)*(s + 3)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1 = (s+1)*(s+3)*(s**2) #Defining a transfer fxn\n",
    "A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle s^{4} + 4 s^{3} + 3 s^{2}$"
      ],
      "text/plain": [
       "s**4 + 4*s**3 + 3*s**2"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D2 = smp.expand(A1) #expand polynomial\n",
    "D2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-1: 1, -3: 1, 0: 2}"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.roots(D2) #Finding roots"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.8.5 ('base')",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  },
  "vscode": {
   "interpreter": {
    "hash": "284910bcbfdb55250f81d592016322a3782b5b0f8147b0f25add58b1c562c53f"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}