Lab_2_Complex_Integration_and_Contour_Integrals.ipynb

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      "cell_type": "markdown",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/majumderarnob/BRACU_MAT215-Complex_Variables_and_Laplace_Trasformation/blob/main/Lab_3_Complex_Integration_and_Contour_Integrals.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IdJLSkeaolfH"
      },
      "source": [
        "#**Complex Integration and Contour Integrals**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QO4E8lJZo5Ow"
      },
      "source": [
        "Make sure you remove and fill in any place that says `# YOUR CODE HERE`, as well as your `NAME`, `ID`, and `LAB_SECTION` below:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Lr466DA0o-sJ"
      },
      "outputs": [],
      "source": [
        "NAME = \"ARNOB MAJUMDER\"\n",
        "ID = \"20301089\"\n",
        "LAB_SECTION = \"09\""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qBmrpF2DpBXt"
      },
      "source": [
        "##Integration using sympy"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6y14FDCRrq84"
      },
      "source": [
        "We will use [`scipy.integrate.quad`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html#scipy-integrate-quad) to compute a definite integral. Also a [`lambda`](https://www.w3schools.com/python/python_lambda.asp) function will be used, which can take any number of arguments, but can only have one expression."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VIE0Rr2ssi5N"
      },
      "source": [
        "Single integrals\n",
        "\n",
        "$$\n",
        "\\int_{0}^{1} x^{3} \\sin (3 x) e^{3x} d x\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2vP3sLQesh1Q",
        "outputId": "bce04fb7-ba90-4782-8a0c-728c9a47795a"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1.472127601108333\n",
            "1.6343899579366694e-14\n"
          ]
        }
      ],
      "source": [
        "import sympy as sp\n",
        "import numpy as np\n",
        "from scipy.integrate import quad\n",
        "integrand = lambda x: x**3 * np.sin(3*x) * np.exp(3*x)\n",
        "integral, integral_error = quad(integrand, 0, 1)\n",
        "print (integral)\n",
        "print (integral_error)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "2m1lxfbrr4En",
        "outputId": "4c8c8d71-de46-4094-af19-99357723128d",
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      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1.472127601108333\n",
            "1.6343899579366694e-14\n",
            "1.472127601108333\n"
          ]
        }
      ],
      "source": [
        "import sympy as sp\n",
        "import numpy as np\n",
        "from scipy.integrate import quad\n",
        "expression = lambda x: x**3 * np.sin(3*x) * np.exp(3*x)\n",
        "integral, integral_error = quad(expression, 0, 1)\n",
        "print(integral)\n",
        "print(integral_error)\n",
        "integral = quad(expression, 0, 1)[0]\n",
        "print(integral)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T5vRHFhXtMba"
      },
      "source": [
        "Double Integrals\n",
        "\n",
        "$$\n",
        "\\int_{0}^{1} \\int_{-x}^{x^{2}} \\cos \\left(x+y^{2}\\right) d y d x\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "35XEDEAPtSLA",
        "outputId": "4527c451-c8a2-45ad-f78e-bc6e034a83ba"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.5131189124115851\n",
            "1.2222205528786866e-14\n"
          ]
        }
      ],
      "source": [
        "from scipy.integrate import dblquad\n",
        "integrand = lambda y, x: np.cos(x+y**2)\n",
        "lwr_y = lambda x: -x\n",
        "upr_y = lambda x: x**2\n",
        "integral, integral_error = dblquad(integrand, 0, 1, lwr_y, upr_y)\n",
        "print (integral)\n",
        "print (integral_error)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "jJS7ypPMr4Ex",
        "outputId": "54f24b40-b968-4794-d88c-0bcf3d1c23a3",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.5131189124115851\n",
            "1.2222205528786866e-14\n"
          ]
        }
      ],
      "source": [
        "from scipy.integrate import dblquad\n",
        "expression = lambda y, x: np.cos(x+y**2)\n",
        "lower_y = lambda x: -x\n",
        "upper_y = lambda x: x**2\n",
        "integral, integral_error = dblquad(expression, 0, 1, lower_y, upper_y)\n",
        "print(integral)\n",
        "print(integral_error)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ye1nfmVJuZt4"
      },
      "source": [
        "## Integration using Complex Numbers\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HbY_gH73vAW2"
      },
      "source": [
        "we define definite integrals of $w(t)$ over the domain $a ≤ t ≤ b$ as follows:\n",
        "\n",
        "\n",
        "$$\n",
        "\\int_{b}^{a} w(t) d t=\\int_{b}^{a} u(t) d t+i \\int_{b}^{a} v(t) d t\n",
        "$$\n",
        "Lets evaluate,\n",
        "$$\n",
        "\\int_{1}^{2}\\left(\\frac{1}{t}-i\\right)^{2} d t\n",
        "$$\n",
        "\n",
        "We will use `symbols` from `sympy`, discussed in last class, to evaluate in terms of $i$. Also here, we can apply integrals over a range, or in terms of $x$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "E4VcD7e2A5kg",
        "outputId": "a0c69d99-bc56-42b0-8afa-6b906fa369dc"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.5 + 2.0*I*log(2)\n",
            "-1.0*t - 2.0*I*log(t) - 1.0/t\n",
            "-0.5 - 2.0*I*log(2)\n"
          ]
        }
      ],
      "source": [
        "t = sp.symbols('t', real=True)\n",
        "z1= ((1/t)-1j)**2\n",
        "print(sp.integrate(z1,(t,2,1)))   #applying range\n",
        "z2=sp.integrate(z1,t)             # integrate in terms of x\n",
        "print(z2)\n",
        "print(z2.subs(t,2)-z2.subs(t,1))  #alternate method"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "z2f6Bq33r4E6",
        "outputId": "fd53fedf-9942-4510-9a8f-586e34723eba",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "<class 'complex'>\n"
          ]
        }
      ],
      "source": [
        "var = 2 + 1j\n",
        "print(type(var))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fsq_DvmhwcVb"
      },
      "source": [
        "##Task 1:\n",
        "\n",
        "Find the values of: \n",
        "\n",
        "$a. \n",
        "\\int_{0}^{1}\\left(\\frac{t^2}{2}-i\\right)^{3} d t\n",
        "$\n",
        "\n",
        "$b. \n",
        "\\int_{0}^{2}\\left(\\frac{z^3-4}{4}+2i\\right) d z\n",
        "$\n",
        "\n",
        "$c. \n",
        "\\int\\left(\\frac{x^2-4x+5y}{4x}\\right) d x\n",
        "$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "id": "kogjVvy1xjfp",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "ecce69e6-7b04-4aa4-a084-831f8b054873"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "-0.482142857142857 + 0.85*I\n"
          ]
        }
      ],
      "source": [
        "# YOUR CODE HERE\n",
        "#Task_1(a)\n",
        "t = sp.symbols('t', real = True)\n",
        "z1= ( (t ** 2) / 2 - 1j) ** 3\n",
        "print(sp.integrate(z1,(t, 0, 1)))"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Task_1(b)\n",
        "z= sp.symbols('z', real=True)\n",
        "z1 =(((z ** 3) - 4) / 4 + 2j)\n",
        "print(sp.integrate(z1,(z, 0, 2)))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6x27VX2-lnr_",
        "outputId": "26544637-95a5-4479-9620-50b46b449410"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "-1 + 4.0*I\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Task_1(c)\n",
        "x = sp.symbols('x')\n",
        "y= sp.symbols('y')\n",
        "f= (x**2 - 4*x + 5*y) / (4 * x)\n",
        "fInt=sp.integrate(f, x).simplify()\n",
        "print(fInt)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "FJoZQPBslvSx",
        "outputId": "0d6c5e41-c299-4c65-a6b2-4608494c5f22"
      },
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x**2/8 - x + 5*y*log(x)/4\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rZ6fTdUsyZoD"
      },
      "source": [
        "##Contour Integrals"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Tysd4bv2ykvW"
      },
      "source": [
        "Integrals of **complex valued functions** of a **complex variable** are defined on a **contour** rather than on intervals of the real line.\n",
        "\n",
        "A **contour** is an arc consisting of a finite number of connected smooth arcs; that is, a **contour** is a piecewise smooth arcs joined end to end.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MQKbBlpJzQs2"
      },
      "source": [
        "Let's use $f(z) = \\bar{z}$ and evaluate along two curves. Consider breaking up the contour as showed in the figure below,\n",
        "\n",
        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        "\n",
        "Now , to evaluate the integral $\\int_{C1} \\left(\\bar{z}\\right) d z $ over the contour given by the curve C1. We can write this as,\n",
        "\n",
        "$$A [0,0], B[1,0], C[1,1]$$\n",
        "\n",
        "\n",
        "$$\\int_{C1} \\left(\\bar{z}\\right) d z=\\int_{AB} \\left(\\bar{z}\\right) d z+\\int_{BC} \\left(\\bar{z}\\right) d z$$\n",
        "\n",
        "If $$z=x+iy$$ then $$dz =dx+i dy$$hence $$\\bar{z}dz =xdx+ydy+ixdy-iydx$$ \n",
        "\n",
        "For $AB$, we can take $y = 0$, hence $dy = 0$ and $ x\\epsilon [0,1] $. For $BC, x=1, dx=0$ and $y\\epsilon [0,1]$. For $AC, y=x,$ and $dy=dx$.Then we can subtitute in the above equation to find the contour integrals over $C1$ and $C2$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9U8p6CWl0Dft"
      },
      "source": [
        "##Task 2:\n",
        "\n",
        "a. Evaluate $\\int_{C1} \\left(\\bar{z}\\right) d z $ by finding the integral over $AB$ and $BC$.\n",
        "\n",
        "b. Also, evaluate $\\int_{C2} \\left(\\bar{z}\\right) d z $ by finding the integral over $AC$.\n",
        "\n",
        "c. Hence identify, if contour integrals are path dependant or not?\n",
        "\n",
        "\n",
        "[*Use your knowledge in theory to solve for the integrals in C1 and C2 manually, an then integrate using Python.*]\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "id": "fM52vDbNykPV",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "d66f1e3d-eec3-43eb-877e-3852d50552d0"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1\n"
          ]
        }
      ],
      "source": [
        "# YOUR CODE HERE\n",
        "#task_2(a)\n",
        "x= sp.symbols('x', real = True)\n",
        "y= sp.symbols('y')\n",
        "\n",
        "z1= x - 0j\n",
        "intZ_1 = sp.integrate(z1,(x, 0, 1))\n",
        "\n",
        "z2= y + 0j\n",
        "intZ_2= sp.integrate(z2,(y, 0, 1))\n",
        "c1= intZ_1 + intZ_2\n",
        "\n",
        "print(c1)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#task_2(b)\n",
        "x = sp.symbols('x', real= True)\n",
        "z1= 2*x \n",
        "c2 = sp.integrate(z1,(x,0,1))\n",
        "print(c2)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ks08g2rpzcAN",
        "outputId": "a7d7a278-0e0b-4ee0-ac2f-310fc60f7a17"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#task_2(c)\n",
        "print(\"Contour Intgrals are path independent as its values are same\")"
      ],
      "metadata": {
        "id": "xcInUwkA1Kgc"
      },
      "execution_count": null,
      "outputs": []
    }
  ],
  "metadata": {
    "colab": {
      "name": "Lab_3_Complex_Integration_and_Contour_Integrals.ipynb",
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "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": "3.9.7"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}