Lecture_5_Sympy.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zJXndgyAOnwd"
      },
      "source": [
        "# SymPy - Symbolic algebra in Python"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zeHK2c6LOnwh"
      },
      "source": [
        "J.R. Johansson (jrjohansson at gmail.com)\n",
        "\n",
        "The latest version of this [IPython notebook](http://ipython.org/notebook.html) lecture is available at [http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures).\n",
        "\n",
        "The other notebooks in this lecture series are indexed at [http://jrjohansson.github.io](http://jrjohansson.github.io)."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "kEBSnnJIOnwi"
      },
      "outputs": [],
      "source": [
        "%matplotlib inline\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-f2Se1UZOnwj"
      },
      "source": [
        "## Introduction"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LZ3i6KxDOnwk"
      },
      "source": [
        "There are two notable Computer Algebra Systems (CAS) for Python:\n",
        "\n",
        "* [SymPy](http://sympy.org/en/index.html) - A python module that can be used in any Python program, or in an IPython session, that provides powerful CAS features.\n",
        "* [Sage](http://www.sagemath.org/) - Sage is a full-featured and very powerful CAS environment that aims to provide an open source system that competes with Mathematica and Maple. Sage is not a regular Python module, but rather a CAS environment that uses Python as its programming language.\n",
        "\n",
        "Sage is in some aspects more powerful than SymPy, but both offer very comprehensive CAS functionality. The advantage of SymPy is that it is a regular Python module and integrates well with the IPython notebook.\n",
        "\n",
        "In this lecture we will therefore look at how to use SymPy with IPython notebooks. If you are interested in an open source CAS environment I also recommend to read more about Sage.\n",
        "\n",
        "To get started using SymPy in a Python program or notebook, import the module `sympy`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "pqFob53dOnwk"
      },
      "outputs": [],
      "source": [
        "from sympy import *"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v8n2WHGgOnwk"
      },
      "source": [
        "To get nice-looking $\\LaTeX$ formatted output run:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "id": "91QVYr-oOnwk"
      },
      "outputs": [],
      "source": [
        "init_printing()\n",
        "\n",
        "# or with older versions of sympy/ipython, load the IPython extension\n",
        "#%load_ext sympy.interactive.ipythonprinting\n",
        "# or\n",
        "#%load_ext sympyprinting"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fMrj1WldOnwl"
      },
      "source": [
        "## Symbolic variables"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UX2XMukcOnwl"
      },
      "source": [
        "In SymPy we need to create symbols for the variables we want to work with. We can create a new symbol using the `Symbol` class:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "fxJdD8HnOnwl"
      },
      "outputs": [],
      "source": [
        "x = Symbol('x')"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "id": "y25mXk3FOnwl",
        "outputId": "c0424a57-7af1-4279-9985-7e8408f6cda7"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$\\displaystyle \\left(x + \\pi\\right)^{2}$"
            ],
            "text/plain": [
              "       2\n",
              "(x + π) "
            ]
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "(pi + x)**2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "n0w0_HkROnwm"
      },
      "outputs": [],
      "source": [
        "# alternative way of defining symbols\n",
        "a, b, c = symbols(\"a, b, c\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "id": "0UdVKRG2Onwm",
        "outputId": "fa5c3273-a440-4aaf-9bfd-c84e1461c4d6"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "sympy.core.symbol.Symbol"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "type(a)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aaiCzAu3Onwm"
      },
      "source": [
        "We can add assumptions to symbols when we create them:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "id": "kx9mexsnOnwm"
      },
      "outputs": [],
      "source": [
        "x = Symbol('x', real=True)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "3c05oSo4Onwm",
        "outputId": "81aed093-c9d3-44c9-a2cb-25f060a7bc66"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "False"
            ]
          },
          "execution_count": 9,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x.is_imaginary"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "D_n1tH-yOnwm"
      },
      "outputs": [],
      "source": [
        "x = Symbol('x', positive=True)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "id": "ljFfPJnkOnwm",
        "outputId": "f0ec3d3c-cb16-4023-ed47-80fa9d388fa2"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$\\displaystyle \\text{True}$"
            ],
            "text/plain": [
              "True"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x > 0"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "True"
            ]
          },
          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x.is_positive"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {},
      "outputs": [
        {
          "ename": "SyntaxError",
          "evalue": "invalid syntax (1209101337.py, line 1)",
          "output_type": "error",
          "traceback": [
            "\u001b[0;36m  Cell \u001b[0;32mIn[13], line 1\u001b[0;36m\u001b[0m\n\u001b[0;31m    x.\u001b[0m\n\u001b[0m      ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
          ]
        }
      ],
      "source": [
        "# this is where it takes about 5 to 1- seconds every time I put a . across x.\n",
        "x."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KxWwH4tUOnwn"
      },
      "source": [
        "### Complex numbers"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ruQGEVMkOnwn"
      },
      "source": [
        "The imaginary unit is denoted `I` in SymPy."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {
        "id": "GG_-JQDUOnwn",
        "outputId": "9cefe0db-08e2-4df2-93fc-3bece68b11b8"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$\\displaystyle 1 + i$"
            ],
            "text/plain": [
              "1 + ⅈ"
            ]
          },
          "execution_count": 14,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "1+1*I"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {
        "id": "2B8PD7FKOnwn",
        "outputId": "94df0bde-2125-4122-83e1-be6c8ab69112"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAj0lEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhpwFwtBgFIIFJyM1CjtgPIIKOBBwAWJQYgElgHtAvBsAO3sgJ/LwrLIAAAAASUVORK5CYII=",
            "text/latex": [
              "$\\displaystyle -1$"
            ],
            "text/plain": [
              "-1"
            ]
          },
          "execution_count": 15,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "I**2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "S3fqn7XdOnwn",
        "outputId": "0825f0a8-3e41-483e-8ceb-164760cce971"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(i x + 1\\right)^{2}$$"
            ],
            "text/plain": [
              "         2\n",
              "(ⅈ⋅x + 1) "
            ]
          },
          "execution_count": 14,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "(x * I + 1)**2"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EgjSmGrcOnwn"
      },
      "source": [
        "### Rational numbers"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "beOiL_hNOnwn"
      },
      "source": [
        "There are three different numerical types in SymPy: `Real`, `Rational`, `Integer`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "11de6Jk8Onwn"
      },
      "outputs": [],
      "source": [
        "r1 = Rational(4,5)\n",
        "r2 = Rational(5,4)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "4Hz04vrQOnwn",
        "outputId": "528516bc-1fd8-4e9e-b923-2f986d706623"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBUq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAkklEQVQYGWNgYGAQAmIgMAGTrClgqmIKmFoApjgKwNRWBjC1AEzxCICpXQxg6uzdu9+ugnVAtDN8AXOW/L8Bpski/oPAB6K0Tt0gdACoMP//L5DyWVcLQNQFEAGnbrQqgnjeDPUPQDQDiwGYYvrOwMA7gYHrHwMD2wQGpt8MDEwMDMwTGBjYFRjaQMYUrdVmYAAALnIpDsFeUO4AAAAASUVORK5CYII=",
            "text/latex": [
              "$$\\frac{4}{5}$$"
            ],
            "text/plain": [
              "4/5"
            ]
          },
          "execution_count": 16,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "r1"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "QyXDWoRBOnwo",
        "outputId": "4cbbb51e-c747-46a9-f156-4ce262c96a1d"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\frac{41}{20}$$"
            ],
            "text/plain": [
              "41\n",
              "──\n",
              "20"
            ]
          },
          "execution_count": 17,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "r1+r2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "j1W_tbRBOnwo",
        "outputId": "26bba194-e6f0-474a-d741-fbef53dcebe4"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\frac{16}{25}$$"
            ],
            "text/plain": [
              "16\n",
              "──\n",
              "25"
            ]
          },
          "execution_count": 18,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "r1/r2"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2MxBjZM0Onwo"
      },
      "source": [
        "## Numerical evaluation"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "C7mIwOHqOnwo"
      },
      "source": [
        "SymPy uses a library for arbitrary precision as numerical backend, and has predefined SymPy expressions for a number of mathematical constants, such as: `pi`, `e`, `oo` for infinity.\n",
        "\n",
        "To evaluate an expression numerically we can use the `evalf` function (or `N`). It takes an argument `n` which specifies the number of significant digits."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "qMPP5X4BOnwo",
        "outputId": "1cd85361-7880-4e5f-8b19-81220bb2c569"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$3.1415926535897932384626433832795028841971693993751$$"
            ],
            "text/plain": [
              "3.1415926535897932384626433832795028841971693993751"
            ]
          },
          "execution_count": 19,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "pi.evalf(n=50)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "imq8VW_3Onwo"
      },
      "outputs": [],
      "source": [
        "y = (x + pi)**2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "AMVHTImqOnwo",
        "outputId": "0227eb69-0be0-40fa-da71-8b0681b5ab4f"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(x + 3.1416\\right)^{2}$$"
            ],
            "text/plain": [
              "            2\n",
              "(x + 3.1416) "
            ]
          },
          "execution_count": 21,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "N(y, 5) # same as evalf"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ARTO7UjgOnwo"
      },
      "source": [
        "When we numerically evaluate algebraic expressions we often want to substitute a symbol with a numerical value. In SymPy we do that using the `subs` function:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "TQXE6XM8Onwp",
        "outputId": "57c4e614-20ae-4292-f339-5396f6cd5234"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(1.5 + \\pi\\right)^{2}$$"
            ],
            "text/plain": [
              "         2\n",
              "(1.5 + π) "
            ]
          },
          "execution_count": 22,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "y.subs(x, 1.5)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "GCyItb82Onwp",
        "outputId": "3d4e1007-0520-4b62-cead-1e13d4266713"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$21.5443823618587$$"
            ],
            "text/plain": [
              "21.5443823618587"
            ]
          },
          "execution_count": 23,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "N(y.subs(x, 1.5))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l_PZMlRUOnwp"
      },
      "source": [
        "The `subs` function can of course also be used to substitute Symbols and expressions:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "0Kk7yplMOnws",
        "outputId": "bad63637-6b44-487a-c558-628c0f32ab3a"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(a + 2 \\pi\\right)^{2}$$"
            ],
            "text/plain": [
              "         2\n",
              "(a + 2⋅π) "
            ]
          },
          "execution_count": 24,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "y.subs(x, a+pi)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Tk6VBGGSOnws"
      },
      "source": [
        "We can also combine numerical evaluation of expressions with NumPy arrays:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "8kexkb4HOnwt"
      },
      "outputs": [],
      "source": [
        "import numpy"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "bMahtfELOnwt"
      },
      "outputs": [],
      "source": [
        "x_vec = numpy.arange(0, 10, 0.1)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "MVmdp4eoOnwt"
      },
      "outputs": [],
      "source": [
        "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "mF27nGYsOnwt",
        "outputId": "9e2b35a3-e08d-460d-e946-bc879b264eae"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x1093af4d0>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.plot(x_vec, y_vec);"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VCOB7CXpOnwt"
      },
      "source": [
        "However, this kind of numerical evaluation can be very slow, and there is a much more efficient way to do it: Use the function `lambdify` to \"compile\" a SymPy expression into a function that is much more efficient to evaluate numerically:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "CUS9X62SOnwt"
      },
      "outputs": [],
      "source": [
        "f = lambdify([x], (x + pi)**2, 'numpy')  # the first argument is a list of variables that\n",
        "                                         # f will be a function of: in this case only x -> f(x)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "xcfKZt1KOnwt"
      },
      "outputs": [],
      "source": [
        "y_vec = f(x_vec)  # now we can directly pass a numpy array and f(x) is efficiently evaluated"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "drL0xvSfOnwt"
      },
      "source": [
        "The speedup when using \"lambdified\" functions instead of direct numerical evaluation can be significant, often several orders of magnitude. Even in this simple example we get a significant speed up:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "9lphhXV3Onwt",
        "outputId": "33c53d04-7d9c-48f4-8983-408474d959a4"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "10 loops, best of 3: 28.2 ms per loop\n"
          ]
        }
      ],
      "source": [
        "%%timeit\n",
        "\n",
        "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "9-Y_WJsdOnwt",
        "outputId": "ade86c9d-37e5-4945-d2c3-8aeddcd98dc2"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "The slowest run took 8.86 times longer than the fastest. This could mean that an intermediate result is being cached \n",
            "100000 loops, best of 3: 2.93 µs per loop\n"
          ]
        }
      ],
      "source": [
        "%%timeit\n",
        "\n",
        "y_vec = f(x_vec)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2f-JRUM_Onwu"
      },
      "source": [
        "## Algebraic manipulations"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZZOBEVboOnwu"
      },
      "source": [
        "One of the main uses of an CAS is to perform algebraic manipulations of expressions. For example, we might want to expand a product, factor an expression, or simplify an expression. The functions for doing these basic operations in SymPy are demonstrated in this section."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MToA4XrHOnwu"
      },
      "source": [
        "### Expand and factor"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "R71kEik_Onwu"
      },
      "source": [
        "The first steps in an algebraic manipulation"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "KDn3QXwBOnwu",
        "outputId": "1c057bd6-c4fa-4e5d-ba32-b0366ec662b7"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAUBAMAAADGn0QzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQqzLsm4+cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACtklEQVQ4EY2Vv2tTURTHv+/l10vyUmOlDrqkRZ2jf4B94sNBUIKlkxReQVERMQ6lg5XGRReHgi6KQybFyYCKQ4U+QSwihdZF3Lo4G6SoQzCec38k75QX6h1uzj3nez/f3PN+AU4NI8akyY+PqMOt6opTG6VQiAOjqqhogFNPU0zMnAS2dCUV4ay+amnEqbT98J4CfqBKWfNPhc7fxFobH3QuFTEB76dCuI3kTq+pVxtzPQreqsXRpAD39aoSo7iCspJLhFUst/BLIfLiH1oHFNjhk6LNpzkUI1T+wl/hmkRYhxdtf0chDor90iHbpqITCYU5Q7bHDjjLNYmwDuAugRALJFm/c+i2bo90yDWoWKDJX9x6rls23I88HXOWHSRiqPjYAQhxgvZ33Cv5gLWQDm5AqUwN+IxjzUsU0zBnoOglAY5wSiIGisOXY8ANcBUox+WeVyXpbgcvolS+BbzDWnyXYhpDhxu0uscpiRgqyk8IGIFmH9k6K8theOZRGG5yrK70WJeiYhuIcZqTeB2G18LwnIpVC39wmEAIBR7HIASVgX1NnmnILg0cgN+6njgDdx/WIYGwZ9gPTLfYgY5Iwm2eaUiHUkQp7hLG6L7Rw3bJa4AYzzgpEdahH7MDIegyefEyCrECSAc3oCRf6QeFLpYopmEdvgO3hlc6gbCKh8DNKtwAb+imaE1jnHfvPkOuQ6lyA6U/ma5To5iGcXCur85FwEVOSYRVTMGj1hKCzrmxuH6hxtKhQ+7bzhd6XLYpRef0vy4tTHGZhnHI9vv9CDjPKYmwitLsTFsh5CM/6BLvNA/rvIrtZLuk1k7EPxJhHZSAEfK15Vd1Qc9pb773SUHqm08oGHE8uUXE5u2dEa5CYd7eeyBSPx+KY9BOXVCTC4PeA+HUknuS8aRZmPssWdLxf31F/wED/Kt4r5jyqgAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
            ],
            "text/plain": [
              "(x + 1)⋅(x + 2)⋅(x + 3)"
            ]
          },
          "execution_count": 33,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "(x+1)*(x+2)*(x+3)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "dkuJHJoUOnwu",
        "outputId": "68d59731-284d-42d4-e37a-0204afc41de5"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$x^{3} + 6 x^{2} + 11 x + 6$$"
            ],
            "text/plain": [
              " 3      2           \n",
              "x  + 6⋅x  + 11⋅x + 6"
            ]
          },
          "execution_count": 34,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expand((x+1)*(x+2)*(x+3))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yTlpWDEUOnwu"
      },
      "source": [
        "The `expand` function takes a number of keywords arguments which we can tell the functions what kind of expansions we want to have performed. For example, to expand trigonometric expressions, use the `trig=True` keyword argument:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "gnFmDNEnOnwu",
        "outputId": "32fac63c-eed7-4884-c1c9-9a348526f629"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\sin{\\left (a + b \\right )}$$"
            ],
            "text/plain": [
              "sin(a + b)"
            ]
          },
          "execution_count": 35,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "sin(a+b)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "leLtdIQeOnwu",
        "outputId": "9bd13ef1-180e-472b-c873-33328430731c"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\sin{\\left (a \\right )} \\cos{\\left (b \\right )} + \\sin{\\left (b \\right )} \\cos{\\left (a \\right )}$$"
            ],
            "text/plain": [
              "sin(a)⋅cos(b) + sin(b)⋅cos(a)"
            ]
          },
          "execution_count": 36,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expand(sin(a+b), trig=True)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EuhobLEpOnwv"
      },
      "source": [
        "See `help(expand)` for a detailed explanation of the various types of expansions the `expand` functions can perform."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qKUZ-f_eOnwv"
      },
      "source": [
        "The opposite of product expansion is of course factoring. To factor an expression in SymPy use the `factor` function:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "yVhBeIr5Onwv",
        "outputId": "95da42d5-362f-46f0-e1fa-768983825177"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
            ],
            "text/plain": [
              "(x + 1)⋅(x + 2)⋅(x + 3)"
            ]
          },
          "execution_count": 37,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "factor(x**3 + 6 * x**2 + 11*x + 6)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7kRmsxY3Onwv"
      },
      "source": [
        "### Simplify"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-sI2JjM_Onwv"
      },
      "source": [
        "The `simplify` tries to simplify an expression into a nice looking expression, using various techniques. More specific alternatives to the `simplify` functions also exists: `trigsimp`, `powsimp`, `logcombine`, etc.\n",
        "\n",
        "The basic usages of these functions are as follows:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ExvcTT2POnwv",
        "outputId": "99851ca7-233f-4535-dbc0-01b1c7b0f9e1"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAUBAMAAADGn0QzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQqzLsm4+cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACtklEQVQ4EY2Vv2tTURTHv+/l10vyUmOlDrqkRZ2jf4B94sNBUIKlkxReQVERMQ6lg5XGRReHgi6KQybFyYCKQ4U+QSwihdZF3Lo4G6SoQzCec38k75QX6h1uzj3nez/f3PN+AU4NI8akyY+PqMOt6opTG6VQiAOjqqhogFNPU0zMnAS2dCUV4ay+amnEqbT98J4CfqBKWfNPhc7fxFobH3QuFTEB76dCuI3kTq+pVxtzPQreqsXRpAD39aoSo7iCspJLhFUst/BLIfLiH1oHFNjhk6LNpzkUI1T+wl/hmkRYhxdtf0chDor90iHbpqITCYU5Q7bHDjjLNYmwDuAugRALJFm/c+i2bo90yDWoWKDJX9x6rls23I88HXOWHSRiqPjYAQhxgvZ33Cv5gLWQDm5AqUwN+IxjzUsU0zBnoOglAY5wSiIGisOXY8ANcBUox+WeVyXpbgcvolS+BbzDWnyXYhpDhxu0uscpiRgqyk8IGIFmH9k6K8theOZRGG5yrK70WJeiYhuIcZqTeB2G18LwnIpVC39wmEAIBR7HIASVgX1NnmnILg0cgN+6njgDdx/WIYGwZ9gPTLfYgY5Iwm2eaUiHUkQp7hLG6L7Rw3bJa4AYzzgpEdahH7MDIegyefEyCrECSAc3oCRf6QeFLpYopmEdvgO3hlc6gbCKh8DNKtwAb+imaE1jnHfvPkOuQ6lyA6U/ma5To5iGcXCur85FwEVOSYRVTMGj1hKCzrmxuH6hxtKhQ+7bzhd6XLYpRef0vy4tTHGZhnHI9vv9CDjPKYmwitLsTFsh5CM/6BLvNA/rvIrtZLuk1k7EPxJhHZSAEfK15Vd1Qc9pb773SUHqm08oGHE8uUXE5u2dEa5CYd7eeyBSPx+KY9BOXVCTC4PeA+HUknuS8aRZmPssWdLxf31F/wED/Kt4r5jyqgAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
            ],
            "text/plain": [
              "(x + 1)⋅(x + 2)⋅(x + 3)"
            ]
          },
          "execution_count": 38,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# simplify (sometimes) expands a product\n",
        "simplify((x+1)*(x+2)*(x+3))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "LV6-pTVNOnwv",
        "outputId": "50228da4-7dcd-4bf5-a8fc-41451a363af3"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=",
            "text/latex": [
              "$$1$$"
            ],
            "text/plain": [
              "1"
            ]
          },
          "execution_count": 39,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# simplify uses trigonometric identities\n",
        "simplify(sin(a)**2 + cos(a)**2)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "CoNdL_EVOnwv",
        "outputId": "cdf4d74b-a0ac-438e-948a-e29b34af585d"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAvBAMAAABJZWRJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWZE3YkiMrsGmOkjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABaElEQVQ4EWNgIA0IGeBTz+zqj1eegSF/VB5fABIKnxD9kga8+kec5H/8YDCGxwmoo3qwO47nAVSc5QCIwQnlwSn2A1AmcwGIYQrlwalWOGsbiIXgQsXXwOWNgCzGr3AuhMGzAC7APoGBoeWXiwBDW3YCw3XVaykJQCm2AAYG5jSfaUDDWYFMhqVAM9YwhDAwrEvg+QPk8zUwMFgzdCUsZ2DgADkQJK/LMN+A4REDww8gn0uAgWEzw34DGaD4Aog8A292/AGG5wwM34B8pgkMDAYMdUAWA+8HiDxz3YH7FxjeIeQhLJh8At8ChvuXDaDyIPMZeP+C9HMuABIlDAH8Gxjkb16AyoPcJ8z2gSEZ6r4ohgCmBwz7b8D0cwcwcH7k+8ADVMa6AaifxfsAj2u2WdmV/75ynxzAhjK7Jad2AKXYLwAJDIAI31MYciABRISA4wdDDd8BqBAkfjHk4ekDrhBNyQkoH5S+AKXTe9mWCL8rAAAAAElFTkSuQmCC",
            "text/latex": [
              "$$\\frac{1}{\\tan{\\left (x \\right )}}$$"
            ],
            "text/plain": [
              "  1   \n",
              "──────\n",
              "tan(x)"
            ]
          },
          "execution_count": 40,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "simplify(cos(x)/sin(x))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YADWfKuROnwv"
      },
      "source": [
        "### apart and together"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RhXSBoIwOnwv"
      },
      "source": [
        "To manipulate symbolic expressions of fractions, we can use the `apart` and `together` functions:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "vBwph1moOnww"
      },
      "outputs": [],
      "source": [
        "f1 = 1/((a+1)*(a+2))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "3eDtW7STOnww",
        "outputId": "2e850f8e-3e4c-483c-d289-c671cdb040f5"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$$"
            ],
            "text/plain": [
              "       1       \n",
              "───────────────\n",
              "(a + 1)⋅(a + 2)"
            ]
          },
          "execution_count": 42,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f1"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "zHljZqsPOnww",
        "outputId": "77de28e0-ebac-4700-f242-0c9ddec2efc9"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$- \\frac{1}{a + 2} + \\frac{1}{a + 1}$$"
            ],
            "text/plain": [
              "    1       1  \n",
              "- ───── + ─────\n",
              "  a + 2   a + 1"
            ]
          },
          "execution_count": 43,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "apart(f1)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "1iMOAZAEOnww"
      },
      "outputs": [],
      "source": [
        "f2 = 1/(a+2) + 1/(a+3)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "NrmjED5mOnww",
        "outputId": "aaf3fd11-fac8-4bca-99fd-8a24e64ab219"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$"
            ],
            "text/plain": [
              "  1       1  \n",
              "───── + ─────\n",
              "a + 3   a + 2"
            ]
          },
          "execution_count": 45,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "pd41VHCAOnww",
        "outputId": "f724c733-9fba-4f59-c745-fdba303fc6bc"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12VGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACs0lEQVRIDe2WzWsTQRjGn02y+dpuSCMo9LQkYC9KK3pS0OhfsKdCvSQIXrRgiGIugrnWi5YiWAsSPNSDivVqhKQFUU9G/wHr1UPpwdDi1/rOzs42231NFkoPgnPYd/eZ3zz77uzMvguEW6F0KiwCxpvEcU4Pa3oVt5thGRnHqTEyIyVtZO4N6GZXXhg31wbUYaeZFpLfBgDlkBzQhp8a/f06kH+2D23y2H15Jz+HzfWyVCIc79Ywnze2JKkc4ra2E2GsRFagP0JsOuhAV6ejOsQsxPrIdolPtduvltvtqhw6lZdx5HEdyEyjUpageoqr4NcJY2daGK/U8NnrUg7LwJTN4Ix0AVir9HDGlDkrBwt4ytCMpD1pHG0Z3dxKSnYqhwJyCwzOSIbjOC197sbhd0EHbfJSxIfYa6py2KtHv9ajvsLolv/JA5oBWj77aweU179oq238LesvsmMkcIgz0Iony0jKPcUDjes9BbzkHGaQ2oG+6naxQAHmbw/IWYMO6mNwGVgCbokuHvjQw3cPyAa2v3J4CFwrY1Y48MB8U/8BCUwIym/KoWMLB6NJHTwA8RQSoOLAFEkSX9ig0gMIAIX6e1tEqFsAF2uQwFmSw0USSG/TFIipFEB6AR0K1HyHI4/J0gWegy+S2SrhLRpDAGY3UKJAzXdA6oEHUGSLZJHwxBYdCBDF6hmFYBVdsn2AKZIYs2iE77AI3futUTmMA+d6EqAkmSKJT9DySLfIhgB9G6ZIh5pycGzh4AI0UUyRTFgYy/szqf9CfPVjwGGRXpYH1OmthovkRKN+hSaoRqMIwGt8rYrz3Rw2Yf70gDuUZLhIdhyHFq1RpjEEYObt+bkmxV2HdLFEggvwi9al5XIcCQQ3TrBIMjuLAXBC3o85ert7JMB+QFy/eN4NI4GRH7HhwB+u8vuFvy0h3AAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
            ],
            "text/plain": [
              "    2⋅a + 5    \n",
              "───────────────\n",
              "(a + 2)⋅(a + 3)"
            ]
          },
          "execution_count": 46,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "together(f2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bUeenkvEOnww"
      },
      "source": [
        "Simplify usually combines fractions but does not factor:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Veg7gETvOnww",
        "outputId": "e9473083-20e6-4b99-af9c-3e1039dbf6ca"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12VGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACs0lEQVRIDe2WzWsTQRjGn02y+dpuSCMo9LQkYC9KK3pS0OhfsKdCvSQIXrRgiGIugrnWi5YiWAsSPNSDivVqhKQFUU9G/wHr1UPpwdDi1/rOzs42231NFkoPgnPYd/eZ3zz77uzMvguEW6F0KiwCxpvEcU4Pa3oVt5thGRnHqTEyIyVtZO4N6GZXXhg31wbUYaeZFpLfBgDlkBzQhp8a/f06kH+2D23y2H15Jz+HzfWyVCIc79Ywnze2JKkc4ra2E2GsRFagP0JsOuhAV6ejOsQsxPrIdolPtduvltvtqhw6lZdx5HEdyEyjUpageoqr4NcJY2daGK/U8NnrUg7LwJTN4Ix0AVir9HDGlDkrBwt4ytCMpD1pHG0Z3dxKSnYqhwJyCwzOSIbjOC197sbhd0EHbfJSxIfYa6py2KtHv9ajvsLolv/JA5oBWj77aweU179oq238LesvsmMkcIgz0Iony0jKPcUDjes9BbzkHGaQ2oG+6naxQAHmbw/IWYMO6mNwGVgCbokuHvjQw3cPyAa2v3J4CFwrY1Y48MB8U/8BCUwIym/KoWMLB6NJHTwA8RQSoOLAFEkSX9ig0gMIAIX6e1tEqFsAF2uQwFmSw0USSG/TFIipFEB6AR0K1HyHI4/J0gWegy+S2SrhLRpDAGY3UKJAzXdA6oEHUGSLZJHwxBYdCBDF6hmFYBVdsn2AKZIYs2iE77AI3futUTmMA+d6EqAkmSKJT9DySLfIhgB9G6ZIh5pycGzh4AI0UUyRTFgYy/szqf9CfPVjwGGRXpYH1OmthovkRKN+hSaoRqMIwGt8rYrz3Rw2Yf70gDuUZLhIdhyHFq1RpjEEYObt+bkmxV2HdLFEggvwi9al5XIcCQQ3TrBIMjuLAXBC3o85ert7JMB+QFy/eN4NI4GRH7HhwB+u8vuFvy0h3AAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
            ],
            "text/plain": [
              "    2⋅a + 5    \n",
              "───────────────\n",
              "(a + 2)⋅(a + 3)"
            ]
          },
          "execution_count": 47,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "simplify(f2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lW5mA0YdOnww"
      },
      "source": [
        "## Calculus"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QXrxIwq8Onwx"
      },
      "source": [
        "In addition to algebraic manipulations, the other main use of CAS is to do calculus, like derivatives and integrals of algebraic expressions."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OILQwyn3Onwx"
      },
      "source": [
        "### Differentiation"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XoQqgwqyOnwx"
      },
      "source": [
        "Differentiation is usually simple. Use the `diff` function. The first argument is the expression to take the derivative of, and the second argument is the symbol by which to take the derivative:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ttZjuzs8Onwx",
        "outputId": "7c65c0ff-2e89-4f29-aea1-a9923587b6cf"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left(x + \\pi\\right)^{2}$$"
            ],
            "text/plain": [
              "       2\n",
              "(x + π) "
            ]
          },
          "execution_count": 48,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "y"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "HhF9wdxXOnwx",
        "outputId": "5e7fba54-784b-4de7-da75-6d7545cc0958"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$4 \\left(x + \\pi\\right)^{3}$$"
            ],
            "text/plain": [
              "         3\n",
              "4⋅(x + π) "
            ]
          },
          "execution_count": 49,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "diff(y**2, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FFbXHRbQOnwx"
      },
      "source": [
        "For higher order derivatives we can do:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "zk4Rj0iEOnwx",
        "outputId": "d60c32a6-cca3-4d53-df07-40c5627bb409"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$12 \\left(x + \\pi\\right)^{2}$$"
            ],
            "text/plain": [
              "          2\n",
              "12⋅(x + π) "
            ]
          },
          "execution_count": 50,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "diff(y**2, x, x)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "IeGuYIJNOnwx",
        "outputId": "faacc356-bab1-4763-952d-23d3e5628d43"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$12 \\left(x + \\pi\\right)^{2}$$"
            ],
            "text/plain": [
              "          2\n",
              "12⋅(x + π) "
            ]
          },
          "execution_count": 51,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "diff(y**2, x, 2) # same as above"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xgNG86XbOnwx"
      },
      "source": [
        "To calculate the derivative of a multivariate expression, we can do:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ljBKhjA2Onwy"
      },
      "outputs": [],
      "source": [
        "x, y, z = symbols(\"x,y,z\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "pgKUGpyeOnwy"
      },
      "outputs": [],
      "source": [
        "f = sin(x*y) + cos(y*z)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zFmW4BQmOnwy"
      },
      "source": [
        "$\\frac{d^3f}{dxdy^2}$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "fBKqVRnOOnwy",
        "outputId": "b3b5023f-59b3-425b-9873-9b2440742a92"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAAUBAMAAABSee3BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnarIkSJZlS7me8N5bApAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADlElEQVRIDYVWXYhMYRh+zszsnvnbcVKrlJiGcOFiIzY/F5twpz0ulqRhifyUGrkg7DYlF27sUBsuZCKJUiPJjdjkpySmDUVkibhRu7V+1t943u/8zJzv7PLUO/u+z/d8z3vOe77ZM0AIB0JMkDDLjfXFxuIfecA1WQAkQjAHQlSQOBQom3WPnld3XUHsQ12puW7nioSOiO6mC+YECCMbKGF042zRoWJj9SXNdTJXJHRc0QmtTlaDRL9bpmyVNFvIDAUVUmmuCYoldLTrhFZHC0Fiulu6zTNtaB4NKqTSXI0hQEKD2aYRerlXIyJFh3CbR0b85uYKXxpyncslCQdG1/YjMsGmKq9J5a3PYL5l8eJMBVO7TiG24UKWgjyjde3+9baRt9FroYUbBG5zZtERKPGhr1j+eeVam1TI9TVJCQdTcd6ezzRdBpy8dAtNw0BvJfYx1o60PQ1oo+Ahr6eUXBjNJlJD2MSvTJYkUW/eVwKUeDawzDa/cTHkeo6khIPjOGvdYBqtACo3K4NIDwA5pEajJcRGj1mwKVjKo2IlRlKFgy1ZPGJTuSKi3nwRKyW+DzwGfrAMuV4jeQ2TFwhmwcI81kCmCCc38AGTbBi/SXbyisaiv9aJ4BPvHJEBqjI2vnDAwyQTudzMd7lcN1PnOSgxm9+BaEKu6CQp4UFEjszZEB9BZwfiQt8uAr+ME7UqczaHXBXQZ8V4rlVzVv6dy6GAErM5R+M1b3TVm4sRIQPijPnBqW6ki3fnv/cg+Z00x86L7uDHIJqGgHgbU8JrnqpiH6DE9ea6Kw5zh4SDqzxca5jK0XDydDe4m88cqUwV8e/XgV2seeBS1iY0WVylJHTgVgN8PEpcbx5ylcMmoRD/mR42y0wT7OPkLQNxOSu9ZZyMvUekdMvCA9ZnOO/KE/BF8Qp93NFSIkm4d24uPv2cs1Biztwde8j1BXdIKBjb1uQvS8YpurmxM/+NhLF7fQE9p3fg6IVLRdYc1pSu1pdl/ul/UwAiHVBwm0dqtRqbi3hVbfOq2tabf7rHcZ3FPRIa2us1n3oYjf9eB7ns/cdzm4c3KEZzNXl9EjqueMQWzCh6ecNf/8XSnDXkAPa7awanMDE014leLGnPZAnujWsmD14QLadtPpUs0/9Dc53oleq/9o/kO8Y19X5MxLuect13HVfrk5qr3IB3E75GksAPnsCKU6gvhc9f9LN/JwHXJKfL+AuGGQ+nZudVwQAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$- x \\left(x y \\cos{\\left (x y \\right )} + 2 \\sin{\\left (x y \\right )}\\right)$$"
            ],
            "text/plain": [
              "-x⋅(x⋅y⋅cos(x⋅y) + 2⋅sin(x⋅y))"
            ]
          },
          "execution_count": 54,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "diff(f, x, 1, y, 2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7Z0EgX5vOnwy"
      },
      "source": [
        "## Integration"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "P1bHgL7COnwy"
      },
      "source": [
        "Integration is done in a similar fashion:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "pFLrtYgEOnwy",
        "outputId": "6c42c39b-3435-46e0-c015-03db98c2b28e"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
            ],
            "text/plain": [
              "sin(x⋅y) + cos(y⋅z)"
            ]
          },
          "execution_count": 55,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "P8fRIN1qOnwy",
        "outputId": "b7713426-8473-4856-f06d-6970e921e42e"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$x \\cos{\\left (y z \\right )} + \\begin{cases} 0 & \\text{for}\\: y = 0 \\\\- \\frac{1}{y} \\cos{\\left (x y \\right )} & \\text{otherwise} \\end{cases}$$"
            ],
            "text/plain": [
              "             ⎛⎧    0       for y = 0⎞\n",
              "             ⎜⎪                     ⎟\n",
              "x⋅cos(y⋅z) + ⎜⎨-cos(x⋅y)            ⎟\n",
              "             ⎜⎪──────────  otherwise⎟\n",
              "             ⎝⎩    y                ⎠"
            ]
          },
          "execution_count": 56,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "integrate(f, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x7PZVuJpOnwy"
      },
      "source": [
        "By providing limits for the integration variable we can evaluate definite integrals:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "noiqfjOeOnwz",
        "outputId": "6c9cd0af-84e7-4843-faea-b6a2e7f2cfd4"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAUBAMAAADo9qfkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12VGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABuElEQVQoFXWTP2hTURTGf695r3lNkxgUh2Z6tEMHHVIKTqUEurkkINihQlNBLBU0Zqngkk0MDgotpNWh3bqoFToUOjRmKYKF4CBEBLN1aIb+kxYR43dNW/NK8sE57/u+c7jv3Hfvgw6wai2FaAys/uFki3VOr5wzQ6bgNsETn3cqPvjMCbgHCz6vKaKezwyWWIRHSZ/5T4Q0VQucl2ym2zbGW7oMXTfpfRqcwVyF8c9PsLfrq/LKijvXseYpFO0a9Eu7x0p9FfuVvUygdAuWpEcV2SrhPWekPCC+qwhllDaIHIWy2EfP05Sk3+rDVa4RSDhYCelvCrOqc6CUqsBx6PCjKEV5vOFSCcY0GSkIe3q6v5V+5OHQedowhhpxf5JK4mZETeMXrNj/FQ+uEjUHoFfTtcon+Gr6KGB7hGNmRiI9Hu7Jd3iogtlMIMMNIh7d4rvEZ3MzIn01Httz9GaraYakc4quhPuHmzv1FfFBNhuNXyLO3a0Yl2eneVbfyUsXjHe/vI8XfW30mqItmkeoKZuwltp2yTSX4gGT+dO6LkUnaNB3jJxVdc06QRf3RTl5VjX764CLv8JfPcJj8HGPsdwAAAAASUVORK5CYII=",
            "text/latex": [
              "$$2 \\cos{\\left (y z \\right )}$$"
            ],
            "text/plain": [
              "2⋅cos(y⋅z)"
            ]
          },
          "execution_count": 57,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "integrate(f, (x, -1, 1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qP6T-lRjOnwz"
      },
      "source": [
        "and also improper integrals"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Ogh6AgHSOnwz",
        "outputId": "348b21b6-afc3-49b9-fdab-2cad26d873a2"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVBAMAAABI7vhRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbvuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVQYGWNggADG/2DwAcplYHaAsSC0KCqXIR2VzzEBlc9WgMrvROUyrEblg7TrAx3wDSrMtIGBa12RtKIAkO8CxJwMDM8ZFjI9ADKZjgIJJSBmuMDcACQl9B0YGEC28xkAVQFB/wUG7gVAmm0DfwOQYmD7yMAJYvQ38DsAKQbGbwy7QLQmA88CEM1g3zADRN1mYF4AohneL08A0zCC9WgDjAmm2WGuhIkGwhhAGgDwdic2xV4k0wAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\sqrt{\\pi}$$"
            ],
            "text/plain": [
              "  ___\n",
              "╲╱ π "
            ]
          },
          "execution_count": 58,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "integrate(exp(-x**2), (x, -oo, oo))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "b66HvLkNOnwz"
      },
      "source": [
        "Remember, `oo` is the SymPy notation for inifinity."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8PkfvFkJOnwz"
      },
      "source": [
        "### Sums and products"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JE1uOUMVOnwz"
      },
      "source": [
        "We can evaluate sums using the function `Sum`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "XfvYel6mOnwz"
      },
      "outputs": [],
      "source": [
        "n = Symbol(\"n\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "C3SffYItOnw0",
        "outputId": "3359b90d-b85d-4027-aa3e-7fcf76ac10da"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\sum_{n=1}^{10} \\frac{1}{n^{2}}$$"
            ],
            "text/plain": [
              "  10    \n",
              " ____   \n",
              " ╲      \n",
              "  ╲   1 \n",
              "   ╲  ──\n",
              "   ╱   2\n",
              "  ╱   n \n",
              " ╱      \n",
              " ‾‾‾‾   \n",
              "n = 1   "
            ]
          },
          "execution_count": 60,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Sum(1/n**2, (n, 1, 10))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "g0Mzm608Onw2",
        "outputId": "6cc4edaa-8296-486e-e4db-58a39db0778d"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdMokiu0RRNjIpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACUElEQVQ4Ea2TP2gTUQDGf+k1TZo/7eHoYvwDXaqeRShODaVB0aEVmqn+CYoUVEiGDuJicNAOgg6Kk5jVyQ5FxCAJgpMOtzg4NSg4C61WQRu/995drt09wpfL+77fl/fevYPoylzNnddtbgFvab3Li0q7vQbtG63c46V22wqF9ls/zttv2TZDQkPqsPWG+n0VMBoy1SxM8rDf75eZ7qUbWd1tWeE9qV8xsC/A2k4SmpOLm7Yws37TfF8JmYUKF2AYPlEM010oW6Hu89sBXmVehcZ2ktAw4gqF6yrcD9mGDprtG8b+aCivT8sKX3qeTAfUA2fbjPnziI79aCifCb2/UG/BWI1iycTg20DMkgeF1nYZVyhayXiGq9da8EBDE5phD4owfvvVJSUwMSenNPNBobVdZnhAxz7FoLCDV1NhB+404TXM3GLI14r1y8mHJ0EMaMnWdpmEjn0hfNZ+qTBTS3V8vO8q3CStMzBjTCvkj8WAKTR2lBnQsW+Yy939ppB71zstRhpa8iOyO7BsTCtwKNi1ZGNHmYSO9/C0du7rmi0EnQkdSEYbZPUUnplCIx9hw08Kre0yu+nooUzoP07Mzc0fqYlc0RLLOoklM0NvSyNW+sGeQmu7zG46KlyApwKHQu52czp/LzWQ0yaVyP3QuJWj8LybzNDaLpPQbg9nGnqtspMCx0OWg+kmHFCGc0z5pEyhlVVSP+NN10Nxts0kNOnZ7TOMliksVRTJb2w181Vz+N75knz1rN6eg7qzMla92HMAi8dXms52mQGt7P+9/gFEfua1t0ciqgAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$1.54976773116654$$"
            ],
            "text/plain": [
              "1.54976773116654"
            ]
          },
          "execution_count": 61,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Sum(1/n**2, (n,1, 10)).evalf()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "CySBrCKaOnw3",
        "outputId": "24e070a6-7d6b-429d-fca1-f8050cf77f37"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$1.64493406684823$$"
            ],
            "text/plain": [
              "1.64493406684823"
            ]
          },
          "execution_count": 62,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Sum(1/n**2, (n, 1, oo)).evalf()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MXeqZLaVOnw3"
      },
      "source": [
        "Products work much the same way:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "TUbCEUrGOnw3",
        "outputId": "f36c16bf-4ebd-4aa3-bfe2-43d3ab8b4dce"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs2Zq91U7yJ2iWZEELuNX9C8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABXElEQVQ4EWNgAAEhBgbOlQvATBRC2JWBYRdDCooYhJPKwBDDwH8BUyaVgfcjA3sCNgnGHwysG7BK/GVgnYBV4gcDO1YJoB38B7DpwOGqdAaGydj80RzvycBlZYtp0iAUYf+PAB+R3cdx5P+/DhDY/19NAFmCgeH9FzA///8FVHGG9VCJr2jiQ1zilEfnFEiCQfOgwbsJvD/AfkWVYEwIYuCAhCaqBC/vHwbuD1h0MLB9YGBRwCbBrMDAvwEc0KhGMbAvYMi/sACkBU2i/wDDIm4BLBJHGRhatoHE0XWAxQZe4j0k3axHS1ccwv//v2lg4JX6/1+tAeFUBgZw2v3AwAdKwShpF1kRPdjAkg4rAJV02AGwpMMOkCW4rSUnw1WhSJzfwN7AogQE2gwMyBK9U5GKT2QJBk+G41iNYihkMGYEG6WJahTHB4aYRJgWYEkHBzwODLM2QHmgkg4GAOSmp5CzUioBAAAAAElFTkSuQmCC",
            "text/latex": [
              "$$\\prod_{n=1}^{10} n$$"
            ],
            "text/plain": [
              "  10   \n",
              "┬───┬  \n",
              "│   │ n\n",
              "│   │  \n",
              "n = 1  "
            ]
          },
          "execution_count": 63,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Product(n, (n, 1, 10)) # 10!"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NuPaX8AaOnw3"
      },
      "source": [
        "## Limits"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ww9t_9djOnw3"
      },
      "source": [
        "Limits can be evaluated using the `limit` function. For example,"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "QZlUhRtFOnw3",
        "outputId": "dd7997ee-403f-462d-dd32-6df7c2a0748e"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=",
            "text/latex": [
              "$$1$$"
            ],
            "text/plain": [
              "1"
            ]
          },
          "execution_count": 64,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "limit(sin(x)/x, x, 0)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kkSrZy59Onw3"
      },
      "source": [
        "We can use 'limit' to check the result of derivation using the `diff` function:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "2Lc-h1RiOnw3",
        "outputId": "c968d87d-bd44-4b72-8409-24a84e3f5fd5"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
            ],
            "text/plain": [
              "sin(x⋅y) + cos(y⋅z)"
            ]
          },
          "execution_count": 65,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "1l4io66QOnw3",
        "outputId": "f3917916-0625-4f78-ce24-dac72ba60f45"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$y \\cos{\\left (x y \\right )}$$"
            ],
            "text/plain": [
              "y⋅cos(x⋅y)"
            ]
          },
          "execution_count": 66,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "diff(f, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9dFl4A4NOnw4"
      },
      "source": [
        "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Xu4vktX5Onw4"
      },
      "outputs": [],
      "source": [
        "h = Symbol(\"h\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ZiAjMElWOnw4",
        "outputId": "8bdfca06-bca6-4bf2-f42f-0ffca69539ea"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKrZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABlUlEQVQoFXWTPUvDUBSG31jbfDUxFBQEh9BKXYRmEkXEiBRRUDo46RJaEXEKgrMZHDo6uAiKXXS0/gMFN0EIRZyLCK5qQWg76IlJ9V6jBxLOc86T+8G9Af6Lfp/t3LPwO1/kCmmbQx72ORQ8DjnQLA5R5pEl3WYJGOSRpRUWKJdqEHImigYgTFRcDGW3oY41PGrl6MmML+XNSEhYSIklFKhcdNVV9QmyOQw0iU/oa0d71L1I0DwsJDwcUasKsas7ULtnBkziZyBlpF5FOxLEJoykiTcaoUXtKxd419ujlGKNapAO0BPUF6BuqF1AIR17NaAtbH1YoQoMmJSFQqBOoq/EjNqag9YhgRZA01zSKxSUJnAM+ZoKVUBMWlA6O8AdMW1LNAroMyKBtoUp1H1qFX1sqOuQnF0Dh8QVmtk9xXJPSDjASHnWppZwk7cxv3mL88ZDjfiCOtnMDI0SClKwFlrMH8EebCDQ6aU9IdhFLL6vSyTQddF92Yx5QSFYcBChEFxCJTv9VYm9elc7FGQ7JvwUYj/MJ3CTXM0O35peAAAAAElFTkSuQmCC",
            "text/latex": [
              "$$y \\cos{\\left (x y \\right )}$$"
            ],
            "text/plain": [
              "y⋅cos(x⋅y)"
            ]
          },
          "execution_count": 68,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "limit((f.subs(x, x+h) - f)/h, h, 0)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XLZhS5y7Onw4"
      },
      "source": [
        "OK!"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tCkJKiNqOnw4"
      },
      "source": [
        "We can change the direction from which we approach the limiting point using the `dir` keyword argument:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "-TOBzIWrOnw4",
        "outputId": "dad0e872-132f-41af-d3f3-547e0df6f100"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGBbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGdgSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\infty$$"
            ],
            "text/plain": [
              "∞"
            ]
          },
          "execution_count": 69,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "limit(1/x, x, 0, dir=\"+\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "AoPB1xhIOnw4",
        "outputId": "397a8ae5-5588-448d-8e5c-0b9f19f2d1c9"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZUIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0AV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAmwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElFTkSuQmCC",
            "text/latex": [
              "$$-\\infty$$"
            ],
            "text/plain": [
              "-∞"
            ]
          },
          "execution_count": 70,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "limit(1/x, x, 0, dir=\"-\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VmkpxvstOnw4"
      },
      "source": [
        "## Series"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZEo991RxOnw4"
      },
      "source": [
        "Series expansion is also one of the most useful features of a CAS. In SymPy we can perform a series expansion of an expression using the `series` function:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "H0ABVvK1Onw4",
        "outputId": "cb778d27-aa60-45c3-fc07-62e9c73410bc"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
            ],
            "text/plain": [
              "         2    3    4     5        \n",
              "        x    x    x     x     ⎛ 6⎞\n",
              "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
              "        2    6    24   120        "
            ]
          },
          "execution_count": 71,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "series(exp(x), x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NptDMTz4Onw4"
      },
      "source": [
        "By default it expands the expression around $x=0$, but we can expand around any value of $x$ by explicitly include a value in the function call:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "xq9KrZIQOnw5",
        "outputId": "ea5f219f-8478-4cf4-9565-48f0d535c6ea"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\mathcal{O}\\left(\\left(x - 1\\right)^{6}; x\\rightarrow1\\right)$$"
            ],
            "text/plain": [
              "                         2            3            4            5             \n",
              "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       6\n",
              "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + O⎝(x - 1) \n",
              "                    2            6            24          120                 \n",
              "\n",
              "        \n",
              "       ⎞\n",
              "; x → 1⎠\n",
              "        "
            ]
          },
          "execution_count": 72,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "series(exp(x), x, 1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sABoWpu2Onw5"
      },
      "source": [
        "And we can explicitly define to which order the series expansion should be carried out:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "dOuVFwI1Onw5",
        "outputId": "a3089c01-3ee9-4cd8-d361-9425b06e894d"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\frac{e}{720} \\left(x - 1\\right)^{6} + \\frac{e}{5040} \\left(x - 1\\right)^{7} + \\frac{e}{40320} \\left(x - 1\\right)^{8} + \\frac{e}{362880} \\left(x - 1\\right)^{9} + \\mathcal{O}\\left(\\left(x - 1\\right)^{10}; x\\rightarrow1\\right)$$"
            ],
            "text/plain": [
              "                         2            3            4            5            6\n",
              "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1) \n",
              "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + ──────────\n",
              "                    2            6            24          120          720    \n",
              "\n",
              "            7            8            9                      \n",
              "   ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       10       ⎞\n",
              " + ────────── + ────────── + ────────── + O⎝(x - 1)  ; x → 1⎠\n",
              "      5040        40320        362880                        "
            ]
          },
          "execution_count": 73,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "series(exp(x), x, 1, 10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NI6B87z8Onw5"
      },
      "source": [
        "The series expansion includes the order of the approximation, which is very useful for keeping track of the order of validity when we do calculations with series expansions of different order:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "tXwfWSZrOnw5",
        "outputId": "70b7a2aa-a4ad-413e-d6ac-c2fe821fdd18"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$"
            ],
            "text/plain": [
              "     2    4        \n",
              "    x    x     ⎛ 5⎞\n",
              "1 - ── + ── + O⎝x ⎠\n",
              "    2    24        "
            ]
          },
          "execution_count": 74,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "s1 = cos(x).series(x, 0, 5)\n",
        "s1"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "K_m8mZdKOnw5",
        "outputId": "6ba92cc3-5765-4e6f-b953-2f885a38b5bc"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
            ],
            "text/plain": [
              "     ⎛ 2⎞\n",
              "x + O⎝x ⎠"
            ]
          },
          "execution_count": 75,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "s2 = sin(x).series(x, 0, 2)\n",
        "s2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "3bvbezIMOnw5",
        "outputId": "b8d35275-a3f2-4688-9dad-9f808aa5fc39"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
            ],
            "text/plain": [
              "     ⎛ 2⎞\n",
              "x + O⎝x ⎠"
            ]
          },
          "execution_count": 76,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expand(s1 * s2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "k9Cou7q0Onw5"
      },
      "source": [
        "If we want to get rid of the order information we can use the `removeO` method:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rHIwXGKwOnw5",
        "outputId": "aaa8d430-a827-4901-dc36-e93b1ef9fb21"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$"
            ],
            "text/plain": [
              " 5    3    \n",
              "x    x     \n",
              "── - ── + x\n",
              "24   2     "
            ]
          },
          "execution_count": 77,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expand(s1.removeO() * s2.removeO())"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KI2XyxrKOnw6"
      },
      "source": [
        "But note that this is not the correct expansion of $\\cos(x)\\sin(x)$ to $5$th order:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "VNlb75TiOnw6",
        "outputId": "fe2cff04-bf9c-437d-f2c1-9e29fc5c6b34"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$"
            ],
            "text/plain": [
              "       3      5        \n",
              "    2⋅x    2⋅x     ⎛ 6⎞\n",
              "x - ──── + ──── + O⎝x ⎠\n",
              "     3      15         "
            ]
          },
          "execution_count": 78,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "(cos(x)*sin(x)).series(x, 0, 6)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UUI1JbAMOnw6"
      },
      "source": [
        "## Linear algebra"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "r2KUA2NWOnw6"
      },
      "source": [
        "### Matrices"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6kj9M-a8Onw6"
      },
      "source": [
        "Matrices are defined using the `Matrix` class:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "_aVUYnkJOnw6"
      },
      "outputs": [],
      "source": [
        "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
        "b1, b2 = symbols(\"b1, b2\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "F__xlUDGOnw6",
        "outputId": "97f36283-792c-41df-fc2c-6db94b7e89d1"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$"
            ],
            "text/plain": [
              "⎡m₁₁  m₁₂⎤\n",
              "⎢        ⎥\n",
              "⎣m₂₁  m₂₂⎦"
            ]
          },
          "execution_count": 80,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A = Matrix([[m11, m12],[m21, m22]])\n",
        "A"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "1kvgApYEOnw6",
        "outputId": "c03ab73a-4831-4070-dae3-5b1c9d87aa69"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$"
            ],
            "text/plain": [
              "⎡b₁⎤\n",
              "⎢  ⎥\n",
              "⎣b₂⎦"
            ]
          },
          "execution_count": 81,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "b = Matrix([[b1], [b2]])\n",
        "b"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uYywtcj9Onw6"
      },
      "source": [
        "With `Matrix` class instances we can do the usual matrix algebra operations:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "C1KRuURPOnw6",
        "outputId": "dcaf7468-7249-40e1-b1a2-baffec4a67ee"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$"
            ],
            "text/plain": [
              "⎡    2                               ⎤\n",
              "⎢ m₁₁  + m₁₂⋅m₂₁    m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n",
              "⎢                                    ⎥\n",
              "⎢                                 2  ⎥\n",
              "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂   m₁₂⋅m₂₁ + m₂₂   ⎦"
            ]
          },
          "execution_count": 82,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A**2"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "-QPjAZQ_Onw7",
        "outputId": "11375d99-42a7-4e67-c9e5-3dc72e6927fb"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$"
            ],
            "text/plain": [
              "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n",
              "⎢               ⎥\n",
              "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦"
            ]
          },
          "execution_count": 83,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A * b"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cYJ3coaQOnw7"
      },
      "source": [
        "And calculate determinants and inverses, and the like:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Hp1ZpH3AOnw7",
        "outputId": "a4b4e764-d57c-47ae-8fa4-26e2b80855d8"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$m_{11} m_{22} - m_{12} m_{21}$$"
            ],
            "text/plain": [
              "m₁₁⋅m₂₂ - m₁₂⋅m₂₁"
            ]
          },
          "execution_count": 84,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A.det()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "0DEVqPd5Onw7",
        "outputId": "bdf6a750-0b3d-4c10-d91e-81e857003412"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left[\\begin{matrix}\\frac{1}{m_{11}} + \\frac{m_{12} m_{21}}{m_{11}^{2} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & - \\frac{m_{12}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)}\\\\- \\frac{m_{21}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & \\frac{1}{m_{22} - \\frac{m_{12} m_{21}}{m_{11}}}\\end{matrix}\\right]$$"
            ],
            "text/plain": [
              "⎡ 1          m₁₂⋅m₂₁                -m₁₂        ⎤\n",
              "⎢─── + ────────────────────  ───────────────────⎥\n",
              "⎢m₁₁      2 ⎛      m₁₂⋅m₂₁⎞      ⎛      m₁₂⋅m₂₁⎞⎥\n",
              "⎢      m₁₁ ⋅⎜m₂₂ - ───────⎟  m₁₁⋅⎜m₂₂ - ───────⎟⎥\n",
              "⎢           ⎝        m₁₁  ⎠      ⎝        m₁₁  ⎠⎥\n",
              "⎢                                               ⎥\n",
              "⎢          -m₂₁                       1         ⎥\n",
              "⎢   ───────────────────         ─────────────   ⎥\n",
              "⎢       ⎛      m₁₂⋅m₂₁⎞               m₁₂⋅m₂₁   ⎥\n",
              "⎢   m₁₁⋅⎜m₂₂ - ───────⎟         m₂₂ - ───────   ⎥\n",
              "⎣       ⎝        m₁₁  ⎠                 m₁₁     ⎦"
            ]
          },
          "execution_count": 85,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A.inv()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lJ4s6nRLOnw7"
      },
      "source": [
        "## Solving equations"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gWs8MkBlOnw7"
      },
      "source": [
        "For solving equations and systems of equations we can use the `solve` function:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "FLWu1wisOnw7",
        "outputId": "4774e29e-f37a-4099-e1da-189991e6c200"
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAdt3NMolEEFTvq5lmIsfapuIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB3SURBVCgVYxAyYSAEDqsxhBFSA5QPQ1bFMR2Ljs4CVFVVu55jqGJfvQ5NFQMjpioGBrlBrqpYCQSAgTuYXV9ngBb+6KHKvfLxGgYmdVRVu+ZZHUCNbbB8L6oqMA8lTYBFAohRxS5AjCoeLIqA7hJSwSaOIiakBgD0ZimSClhTrgAAAABJRU5ErkJggg==",
            "text/latex": [
              "$$\\left [ -1, \\quad 1\\right ]$$"
            ],
            "text/plain": [
              "[-1, 1]"
            ]
          },
          "execution_count": 86,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "solve(x**2 - 1, x)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "OpBfUBCeOnw7",
        "outputId": "4c0297d2-f81b-4d1e-98c4-f3d3763493cf"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left [ - i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\right ]$$"
            ],
            "text/plain": [
              "⎡        _____________         _____________        ___________       ________\n",
              "⎢       ╱         ___         ╱         ___        ╱       ___       ╱       _\n",
              "⎢      ╱    1   ╲╱ 5         ╱    1   ╲╱ 5        ╱  1   ╲╱ 5       ╱  1   ╲╱ \n",
              "⎢-ⅈ⋅  ╱   - ─ + ───── , ⅈ⋅  ╱   - ─ + ───── , -  ╱   ─ + ───── ,   ╱   ─ + ───\n",
              "⎣   ╲╱      2     2       ╲╱      2     2      ╲╱    2     2     ╲╱    2     2\n",
              "\n",
              "___⎤\n",
              "__ ⎥\n",
              "5  ⎥\n",
              "── ⎥\n",
              "   ⎦"
            ]
          },
          "execution_count": 87,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "solve(x**4 - x**2 - 1, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PyGQ0GTIOnw7"
      },
      "source": [
        "System of equations:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "a-4rhujWOnw7",
        "outputId": "70264928-4b6a-4ecd-f4f6-1386ea4bad31"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left \\{ x : 1, \\quad y : 0\\right \\}$$"
            ],
            "text/plain": [
              "{x: 1, y: 0}"
            ]
          },
          "execution_count": 88,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "solve([x + y - 1, x - y - 1], [x,y])"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TsPcjPxSOnw8"
      },
      "source": [
        "In terms of other symbolic expressions:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Nc9EiD-BOnw8",
        "outputId": "aad8f952-a9cb-4dbd-d6fd-cadedc1224a5"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/latex": [
              "$$\\left \\{ x : \\frac{a}{2} + \\frac{c}{2}, \\quad y : \\frac{a}{2} - \\frac{c}{2}\\right \\}$$"
            ],
            "text/plain": [
              "⎧   a   c     a   c⎫\n",
              "⎨x: ─ + ─, y: ─ - ─⎬\n",
              "⎩   2   2     2   2⎭"
            ]
          },
          "execution_count": 89,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "solve([x + y - a, x - y - c], [x,y])"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rduPcBEvOnw8"
      },
      "source": [
        "## Further reading"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nYhXOyBCOnw8"
      },
      "source": [
        "* http://sympy.org/en/index.html - The SymPy projects web page.\n",
        "* https://github.com/sympy/sympy - The source code of SymPy.\n",
        "* http://live.sympy.org - Online version of SymPy for testing and demonstrations."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "n12RmSKSOnw8"
      },
      "source": [
        "## Versions"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "xFPSY_oDOnw8",
        "outputId": "ee724448-5e2a-44dc-b694-edb54573b7b1"
      },
      "outputs": [
        {
          "data": {
            "application/json": {
              "Software versions": [
                {
                  "module": "Python",
                  "version": "2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]"
                },
                {
                  "module": "IPython",
                  "version": "3.2.1"
                },
                {
                  "module": "OS",
                  "version": "Darwin 14.1.0 x86_64 i386 64bit"
                },
                {
                  "module": "numpy",
                  "version": "1.9.2"
                },
                {
                  "module": "matplotlib",
                  "version": "1.4.3"
                },
                {
                  "module": "sympy",
                  "version": "0.7.6"
                }
              ]
            },
            "text/html": [
              "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]</td></tr><tr><td>IPython</td><td>3.2.1</td></tr><tr><td>OS</td><td>Darwin 14.1.0 x86_64 i386 64bit</td></tr><tr><td>numpy</td><td>1.9.2</td></tr><tr><td>matplotlib</td><td>1.4.3</td></tr><tr><td>sympy</td><td>0.7.6</td></tr><tr><td colspan='2'>Sat Aug 15 11:37:37 2015 JST</td></tr></table>"
            ],
            "text/latex": [
              "\\begin{tabular}{|l|l|}\\hline\n",
              "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
              "Python & 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)] \\\\ \\hline\n",
              "IPython & 3.2.1 \\\\ \\hline\n",
              "OS & Darwin 14.1.0 x86\\_64 i386 64bit \\\\ \\hline\n",
              "numpy & 1.9.2 \\\\ \\hline\n",
              "matplotlib & 1.4.3 \\\\ \\hline\n",
              "sympy & 0.7.6 \\\\ \\hline\n",
              "\\hline \\multicolumn{2}{|l|}{Sat Aug 15 11:37:37 2015 JST} \\\\ \\hline\n",
              "\\end{tabular}\n"
            ],
            "text/plain": [
              "Software versions\n",
              "Python 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]\n",
              "IPython 3.2.1\n",
              "OS Darwin 14.1.0 x86_64 i386 64bit\n",
              "numpy 1.9.2\n",
              "matplotlib 1.4.3\n",
              "sympy 0.7.6\n",
              "Sat Aug 15 11:37:37 2015 JST"
            ]
          },
          "execution_count": 90,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "%reload_ext version_information\n",
        "\n",
        "%version_information numpy, matplotlib, sympy"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "display_name": "scqubits-spc",
      "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": "3.11.7"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}