{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, ComplexPhasePortrait, ApproxFun\n",
    "gr();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "Dr. Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "\n",
    "\n",
    "# Lecture 5: Residue Theorem\n",
    "\n",
    "\n",
    "This lecture we cover\n",
    "\n",
    "1. Contour integrals and Laurent coefficients\n",
    "2. Isolated singularities\n",
    "    - Residue at a point\n",
    "2. Contour integrals in domains with multiple holes\n",
    "    - The residue theorem\n",
    "3. Calculated integrals\n",
    "    - Application: Trigonometric integrals with rational functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contour integrals and Laurent coefficients\n",
    "\n",
    "In this course, we will _always_ think of Laurent series living on a circle $ \\gamma_r(z_0) = \\{z : |z-z_0| = r \\}$. That is,\n",
    "$$\n",
    "    f(z) \\approx \\sum_{k=-\\infty}^\\infty f_k (z-z_0)^k\n",
    "$$\n",
    "\n",
    "for $z \\in \\gamma_r(z_0)$.  \n",
    "\n",
    "**Proposition (Residue on a circle)** \n",
    "Suppose the Laurent series is absolutely summable on $\\gamma_r$. Then \n",
    "$$\n",
    "\\oint_{\\gamma_r} f(z) dz = 2 \\pi i f_{-1}\n",
    "$$\n",
    "We refer to $f_{-1}$ as the _residue over $\\gamma_r$_.\n",
    "\n",
    "*Example* For all $0 < r < \\infty$, \n",
    "\n",
    "$$\n",
    "\\oint_{\\gamma_r} {1 \\over z} dz = 2 \\pi i\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-2..2, -2..2, z-> 1/z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 6.283185307179586im"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "γ = Circle(0.0, 1.0)\n",
    "z = Fun(γ)\n",
    "sum(1/z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Example_ This works for functions not analytic:\n",
    "$$\n",
    "    \\oint_{\\gamma_1} (\\sqrt{z-1}\\sqrt{z+1})^3 dz \n",
    "    $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> (sqrt(z-1)*sqrt(z+1))^3\n",
    "\n",
    "phaseplot(-2..2, -2..2, f)\n",
    "plot!(Circle(1.1); color=:black, label=\"contour\", linewidth=1.5, arrow=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sum(Fun(f, Laurent(Circle(1.1)))) = -1.2751605303035503e-16 + 2.3561944901923444im\n",
      "(2π) * im * f₋₁ = -1.1592368457305e-16 + 2.141994991083949im\n"
     ]
    }
   ],
   "source": [
    "@show sum(Fun(f, Laurent(Circle(1.1))))  # integral over circle\n",
    "f₋₁ = Fun(f, Laurent(Circle(1.1))).coefficients[2] # numerical Laurent coefficient\n",
    "@show 2π*im*f₋₁;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "When $f$ is holomorphic in a neighbourhood of the circle, we can extend it to an annulus (like Taylor series and disks):\n",
    "\n",
    "**Proposition (Laurent series in an annulus)**\n",
    "Suppose $f$ is holomorphic in an open annulus $A_{\\rho R}(z_0) = \\{z : \\rho  < | z - z_0| < R\\}$. Then the Laurent series converges uniformly in  any closed annulus inside $A_{\\rho R}$\n",
    "\n",
    "**Proof** _Exercise_.  Hint: use the decay in the Laurent coefficients $f_k$ from last lecture.\n",
    "\n",
    "\n",
    "_Proposition (Residue on a circle)_ holds true regardless of the radius.\n",
    "\n",
    "## Isolated singularities\n",
    "\n",
    "**Definition (isolated singularity)** $f$ has an  _isolated singularity at_ $z_0$ if it is holomorphic in an open annulus with inner radius 0: \n",
    "$$\n",
    "A_{0R}(z_0) = \\{z : 0 < |z - z_0| < R \\}.\n",
    "$$\n",
    "\n",
    "\n",
    "**Definition (Removable singularity)** $f$ has a _removable singularity at_ $z_0$ if it has an isolated singularity at $z_0$ and all negative terms in the Laurent series in $A_{0R}(z_0)$ are zero:\n",
    "$$\n",
    "f(z) = f_0 + f_1 (z-z_0) + f_2 (z-z_0)^2 + \\cdots\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NaN"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> (exp(z)-1)/z\n",
    "f(0.0) "
   ]
  },
  {
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     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-1..1, -1..1, f) # no singularity appears because"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proposition (Removing a removable singularity)** If  $f$ has a removable singularity at $z_0$, then\n",
    "$$\n",
    "\\tilde f(z) = \\begin{cases} f_0 & z = z_0 \\\\\n",
    "                                f(z) & 0 < |z-z_0| < R\n",
    "                                \\end{cases}\n",
    "$$\n",
    "is analytic in the disk $B_R(z_0) = \\{ z : |z-z_0| < R \\}$, with a convergent Taylor series. Hence the name."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f̃ = z -> z ≈ 0 ? 1 : f(z)\n",
    "phaseplot(-1..1, -1..1, f̃)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition (simple pole)** $f$ has a  _simple pole at_ $z_0$ if it is holomorphic in \n",
    "$$\n",
    "  A_{0R}(z_0) = \\{z : 0 < |z - z_0| < R \\}\n",
    "$$ \n",
    "with only one negative term in the Laurent series in $A_{0R}(z_0)$:\n",
    "$$\n",
    "  f(z) = {f_{-1} \\over z - z_0}  + f_0 + f_1 (z - z_0) + \\cdots\n",
    "$$\n",
    "where $f_{-1} \\neq 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-1..1, -1..1, z -> exp(z)/z)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition (higher order pole)** $f$ has a  _pole of order $N$ at_ ${z_0}$ if it is holomorphic in \n",
    "$$\n",
    " A_{0R}(z_0) = \\{z : 0 < |z - z_0| < R \\}\n",
    " $$\n",
    "with only $N$ negative coefficients in the Laurent series:\n",
    "$$\n",
    " f(z) = {f_{-N} \\over (z - z_0)^N}  + {f_{1-N} \\over (z - z_0)^{N-1}} +  \\cdots + {f_{-1} \\over z-z_0} + f_0 + f_1 (z-z_0) + \\cdots\n",
    "$$\n",
    "where $f_{-N} \\neq 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-1..1, -1..1, z -> exp(z)/z^3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**Definition (essential singularity)** $f$ has an _essential singularity at_ $z_0$ if it is holomorphic in $A_{0R}(z_0)$ and has an infinite number of negative Laurent coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
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       "\" transform=\"translate(190, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-1..1, -1..1, z -> exp(1/z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-8.078182973046723e-16 + 6.283185307179586im, -8.078182973046723e-16 + 6.283185307179586im)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(z -> exp(1/z), Circle())), \n",
    "2π*im*Fun(z -> exp(1/z), Laurent(Circle())).coefficients[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Residue at a point\n",
    "\n",
    "**Definition (Residue at a point)** Suppose $f$ has an isolated singularity at $z_0$, and is analytic in the annulus $A_{0R}(z_0)$ for some $R > 0$. Then we define the _residue at_ $z_0$ as\n",
    "$$\n",
    "{\\underset{z = z_0}{\\rm Res}}\\, f(z) = f_{-1}\n",
    "$$\n",
    "where $f_{-1}$ is the first negative coefficent of the Laurent series in $A_{0R}(z_0)$. \n",
    "\n",
    "**Proposition (Residue of ratio of analytic functions with simple pole)** Suppose\n",
    "$$\n",
    "f(z) = {A(z) \\over B(z)}\n",
    "$$\n",
    "and $A$, $B$ are analytic/holomorphic in a disk of radius $R$ around $z_0$ and that $B$ has only a single zero at $z_0$:\n",
    "\\begin{align*}\n",
    "A(z) = A_0 + A_1(z-z_0) + \\cdots \\cr\n",
    "B(z) = B_1(z-z_0) + \\cdots\n",
    "\\end{align*}\n",
    "Then ${\\underset{z = z_0}{\\rm Res}}\\, f(z) = {A_0 \\over B_1}$\n",
    "\n",
    "**Exercise (Residue of ratio of analytic functions with higher order  poles)** What is the residue at $z_0$ if $B$ has a higher order zero: $B(z) = B_N (z-z_0)^N + \\cdots$?"
   ]
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   "source": [
    "## Contour integrals on domains with multiple holes\n",
    "\n",
    "\n",
    "Consider the following example:\n",
    "\n",
    "$$ \n",
    "    {\\sqrt{z-1}\\sqrt{z+1} \\over z^2 + 4}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We still have the contour integral over a circle, and so _Proposition (Residue on a circle)_ still holds true for $r > 2$. But we can also deform the contour into three contours:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
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     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> sqrt(z-1)sqrt(z+1)/(z^2+4)\n",
    "\n",
    "Γ = Circle(1.1) ∪ Circle(2.0im,0.1) ∪ Circle(-2.0im,0.1)\n",
    "phaseplot(-2..2, -3..3, f)\n",
    "plot!(Γ; color=:black, label=:contour, arrow=true, linewidth=1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4.782717259505325e-16 - 9.336183501133093e-16im, 2.555681169620637e-16 + 1.1200207106231122e-15im)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(f, Circle(2.1))), sum(Fun(f, Γ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus we can sum over three residues."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Residue theorem\n",
    "\n",
    "**Theorem (Cauchy's Residue Theorem)** Let $f$ be holomprohic inside and on a simple closed, positively oriented contour $\\gamma$ except at isolated points $z_1, \\ldots, z_r$ inside $\\gamma$. Then\n",
    "\n",
    "$$\\oint_\\gamma f(z) dz = 2 \\pi i \\sum_{j=1}^r {\\underset{z = z_j}{\\rm Res}}\\, f(z)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculating integrals\n",
    "\n",
    "We can use the Residue theorem to calculate \"hard\" integrals.\n",
    "\n",
    "First, two trivial examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> 1/(z*(z+2))\n",
    "phaseplot(-3..3, -3..3, f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.874234295592502e-19 - 9.742139082662117e-17im"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(f, Circle(3.0)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> exp(z)/(z*(z+2))\n",
    "phaseplot(-3..3, -3..3, f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-5.073166565789438e-16 + 2.716424322002157im"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(f, Circle(3.0)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 2.716424322002157im"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2π*im*(1/2 - exp(-2)/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.313334476762615e-16 + 1.7833804925887144im"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(z -> exp(z)/(z^2*(z+2)), Circle(3.0)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 1.783380492588715im"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*pi*im * (1/4 + exp(-2)/4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application: Integrals on the real line of rational functions\n",
    "\n",
    "We can calculate integrals of the form \n",
    "$$\\int_0^{2 \\pi} R(\\cos \\theta, \\sin \\theta) d \\theta$$\n",
    "where $R(x,y)$ is rational by doing the change of variables $z = e^{i \\theta}$ to reduce it to\n",
    "$$\\oint_{\\gamma_1} R\\left({z + z^{-1} \\over 2}, {z - z^{-1} \\over 2 i} \\right) {d z \\over i z}$$\n",
    "\n",
    "\n",
    "*Example* Consider\n",
    "\n",
    "$$\\int_0^{2\\pi} {d \\theta \\over 1 - 2\\rho \\cos \\theta + \\rho^2}$$\n",
    "\n",
    "for $0 < \\rho < 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 22,
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   "source": [
    "ρ = 0.5\n",
    "plot(Fun(θ -> 1/(1-2ρ*cos(θ) + ρ^2), 0 .. 2π))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
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       "\" transform=\"translate(149, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-2..2, -2..2, z ->  1/(1-ρ*(z+(z^(-1))) + ρ^2) * 1/(im*z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8.377580409572783, 8.377580409572781)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Fun(θ -> 1/(1-2ρ*cos(θ) + ρ^2), 0 .. 2π)), 2π /(1-ρ^2)"
   ]
  }
 ],
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   "name": "julia-1.0"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.0.1"
  }
 },
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