{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations, DifferentialEquations, LinearAlgebra\n",
    "gr();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# M3M6: Methods of Mathematical Physics 2018\n",
    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "$$\n",
    "\n",
    "Dr. Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "<br>\n",
    "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
    "\n",
    "# Solution Sheet 1\n",
    "\n",
    "\n",
    "----------\n",
    "\n",
    "## Problem 1.1\n",
    "\n",
    "### 1.\n",
    "Use fundamental theorem of algebra: a polynomial is a constant times a product of terms like $z - \\lambda_k$, where $\\lambda_k$ are the roots. In this case, the roots are $a$ times the quartic-root of $-1$, hence  this gives us:\n",
    "\\begin{align*}\n",
    "z^4 + a^4 = (z-a \\E^{\\I \\pi /4})(z-a \\E^{3\\I \\pi /4})(z-a \\E^{5\\I \\pi /4})(z-a \\E^{7\\I \\pi /4})  \n",
    "\\end{align*}\n",
    "We are only interested in the root $a \\E^{\\I \\pi /4}$, thus we simplify the expression\n",
    "Therefore \n",
    "$$\n",
    "{z^3 \\sin z \\over z^4 + a^4}  = {z^3 \\sin z \\over (z-a \\E^{3\\I \\pi /4})(z-a \\E^{5\\I \\pi /4})(z-a \\E^{7\\I \\pi /4})} {1 \\over z-a \\E^{\\I \\pi /4}} = {a^3\\E^{3\\I \\pi /4} \\sin(a \\E^{\\I \\pi /4}) \\over a^3 (\\E^{\\I \\pi / 4} - \\E^{3\\I \\pi /4})(\\E^{\\I \\pi / 4} - \\E^{5\\I \\pi /4})(\\E^{\\I \\pi / 4} - \\E^{7\\I \\pi /4})} {1 \\over z-a \\E^{\\I \\pi /4}} + O(1)\n",
    "$$\n",
    "Therefore, \n",
    "$$\n",
    "\\Res_{z = a \\E^{\\I \\pi /4}} {z^3 \\sin z \\over z^4 + a^4} = { \\E^{3\\I \\pi /4} \\sin(a \\E^{\\I \\pi /4}) \\over \\E^{3\\I \\pi /4} (1 - \\E^{\\I \\pi \\over 2})(1 - \\E^{\\I \\pi })(1 - \\E^{3\\I \\pi/2 })}= { \\sin(a \\E^{\\I \\pi /4}) \\over (1 - \\I) (2) (1+ \\I)}= { \\sin(a \\E^{\\I \\pi /4}) \\over 4}\n",
    "$$\n",
    "\n",
    "Let's check our work: we compare the numerically calculated residue to the formula we have derived:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.5378838853348212 + 0.07544036746694016im, 0.5378838853348215 + 0.0754403674669402im)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 2.0\n",
    "γ = Circle(a*exp(im*π/4), 0.1)\n",
    "f = Fun(z -> z^3*sin(z)/(z^4+a^4), γ)\n",
    "sum(f)/(2π*im), sin(a*exp(im*π/4))/4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. \n",
    "\n",
    "We have \n",
    "$$\n",
    "(z^2-1)^2 = (z-1)^2(z+1)^2\n",
    "$$\n",
    "Thus this is a slightly more challenging since it has a double pole. But we can expand using Geometric series:\n",
    "$$\n",
    "    {z+1 \\over (z^2-1)^2} = {1 \\over (z-1)^2} {1 \\over 2 - (1-z)} = {1 \\over (z-1)^2} {1 \\over 2} (1 + (1-z)/2 +O(1-z)^2) = {1 \\over 2(z-1)^2} - {1 \\over 4 (z-1)}  + O(1)\n",
    "$$\n",
    "Thus the residue is the negative-first Laurent coefficient, namely $-{1 \\over 4}$.\n",
    "\n",
    "We again check our work:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.2500000000000023 - 1.1916485920484316e-16im"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "γ = Circle(1, 0.1)\n",
    "f = Fun(z -> (z+1)/(z^2-1)^2, γ)\n",
    "sum(f)/(2π*im) # almost equals -1/4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3. \n",
    "\n",
    "$$ {z^2 \\E^z \\over z^3 - a^3} = {z^2 \\E^z \\over (z-a) (z^2 + az + a^2)}\n",
    "$$\n",
    "We thus need only evaluate the extra term at $z=a$:\n",
    "$$\n",
    "\\Res_{z = a} {z^2 \\E^z \\over z^3 - a^3} = {\\E^a \\over 3}\n",
    "$$\n",
    "\n",
    "Let's check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.4630186996435506 + 4.738982805534393e-16im, 2.46301869964355)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 2.0\n",
    "\n",
    "γ = Circle(a, 0.1)\n",
    "f = Fun(z -> z^2*exp(z)/(z^3-a^3), γ)\n",
    "sum(f)/(2π*im), exp(a)/3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1.2\n",
    "\n",
    "### 1.\n",
    "Change of variables $z = \\E^{\\I \\theta}$, $\\dz = \\I \\E^{\\I \\theta} \\D\\theta = \\I z \\D\\theta$, $\\cos \\theta = {z + z^{-1} \\over 2}$ gives\n",
    "$$\n",
    "\\int_0^{2 \\pi} {\\D\\theta \\over 5-4 \\cos \\theta}  = - \\I \\oint {\\dz \\over 5z - 2z^2 - 2}  = \\I \\oint {\\dz \\over (z-2)(2z-1)} = -\\pi \\Res_{z=1/2} {1 \\over (z-2)(z-1/2)} = {2 \\over 3} \\pi \n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.881784197001252e-16"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "θ = Fun(0 .. 2π)\n",
    "sum(1/(5-4cos(θ))) -2π/3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. \n",
    "Use $\\cos 2 \\theta = { \\E^{2 \\I \\theta} + \\E^{-2 \\I \\theta} \\over 2} = {z^2 + z^{-2} \\over 2}$ to get\n",
    "$$\n",
    "\\int_0^{2 \\pi} {\\cos 2 \\theta \\D\\theta \\over 5+4 \\cos \\theta}  = - {\\I \\over 4} \\oint { (z^4 + 1)\\dz \\over z^2 (z+1/2)(z+2)}  = {\\pi \\over 2} \\left( \\Res_{z=-1/2} + \\Res_{z=0}\\right) {z^4+1 \\over z^2(z+2)(z+1/2)} = {\\pi \\over 6}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1666666666666669"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "θ = Fun(0 .. 2π)\n",
    "sum(cos(2θ)/(5-4cos(θ)))/π"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.  \n",
    "\n",
    "Because the integrand is analytic  and $O(z^{-2})$ in the upper half plane, we can use the residue theorem in the upper half plane using\n",
    "$$\n",
    " {1 \\over (z^2+1) (z^2 + 4)} = {1 \\over (z+\\I)(z-\\I) (z+2\\I)(z-2\\I)}\n",
    "$$\n",
    "\n",
    "This has two poles in the upper half plane:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-3..3, -3..4, z-> 1/((z^2+1)*(z^2+4)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "\t\\int_{-\\infty}^\\infty    {1 \\over (x^2+1)(x^2+4)} \\dx &=\n",
    "    2 \\pi \\I \\left( \\Res_{z = \\I} + \\Res_{z = 2\\I}\\right){1 \\over (z^2+1)(z^2+4)} \\\\\n",
    "    & = 2 \\pi \\I \\left( {1 \\over 2 \\I 3 \\I (-\\I)} + {1 \\over 3 \\I  \\I 4\\I} \\right) = \\pi/ 6\n",
    "\\end{align*}\n",
    "\n",
    "We can check the result numerically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.523598775598299"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Fun( Line())\n",
    "sum(1/((x^2+1)*(x^2+4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5235987755982988"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π/6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. \n",
    "\n",
    "Again, decays like $O(z^{-2})$ in upper half plane so we can use residue calculus. This integrand has poles at $z = \\I$ and $z = 3 \\I$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "\" transform=\"translate(149, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-4..4, -4..4, z-> (z^2 - z + 2) / (z^4 + 10z^2 +9))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The residues are $(-1-\\I)/16$ and $(3-7\\I)/48$ giving the answer\n",
    "$$\n",
    "5\\pi \\over 12\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.\n",
    "$$\n",
    "\t\\int_{-\\infty}^\\infty    {1 \\over x + \\I } \\dx\n",
    "$$\n",
    "\n",
    "Trick question: it's undefined because the integral doesn't decay fast enough. But what if I had asked for \n",
    "$$\n",
    "\t\\dashint_{-\\infty}^\\infty    {1 \\over x + \\I } \\dx?\n",
    "$$\n",
    "\n",
    "We can't use residue theorem since it doesn't decay fast enough, but we can use, with a contour $C_R = \\{ R \\E^{\\I theta} : 0 \\leq \\theta \\leq \\pi \\}$\n",
    "$$\n",
    "\\oint_{[-M, M] \\cup C_R} {1 \\over z + \\I}   = 0\n",
    "$$\n",
    "Further, by direct substitution, we have \n",
    "$$\n",
    "\\int_{C_R} {1 \\over z + \\I}\\dz = \\I \\int_0^\\pi  R {\\E^{\\I \\theta} \\over R \\E^{\\I \\theta} + \\I} \\D \\theta \n",
    "$$\n",
    "Letting $R \\rightarrow \\infty$, the integrand tends to one uniformly hence  \n",
    "$$\n",
    " \\int_{C_R} {1 \\over z + \\I}\\dz  \\rightarrow \\I \\int_0^\\pi  \\D \\theta  = \\I \\pi.\n",
    "$$\n",
    "Therefore, we have\n",
    "$$\n",
    "\t\\dashint_{-\\infty}^\\infty    {1 \\over x + \\I } \\dx = \\lim_{M\\rightarrow \\infty} \\int_{-M}^M {1 \\over x + \\I}  \\dx =  - \\I \\pi .\n",
    "    $$\n",
    "    \n",
    "    Indeed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8.881784197001252e-16 - 3.13959265425659im"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Fun(-1000 .. 1000)\n",
    "sum(1/(x+im))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty    {\\sin 2x \\over x^2 + x + 1} \\dx = \\Im \\int_{-\\infty}^\\infty    {\\E^{ 2 \\I x} \\over x^2 + x + 1} \\dx \n",
    "$$\n",
    "This can be deformed in the upper half plane with a pole at ${-1 + \\I \\sqrt{3} \\over 2}$, using residue calculus gives us\n",
    "$$\n",
    "   -{2 \\pi \\over \\sqrt 3} {\\sin 1 \\over \\E^\\sqrt{3}}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5400548830723747"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Fun(-100 .. 100)\n",
    "sum(sin(2x)/(x^2+x+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5400553569742235"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-2π/sqrt(3) * sin(1)/exp(sqrt(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "\n",
    "### 7.\n",
    "$$\n",
    "\t\\int_{-\\infty}^\\infty    {\\cos x \\over x^2 + 4} \\dx = \\Re \t\\int_{-\\infty}^\\infty    {\\E^{\\I x}\\over x^2 + 4} \\dx\n",
    "$$\n",
    "and residue calculus gives ${\\pi \\over 2 \\E^2}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.21254026836702827"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = 200\n",
    "x = Fun(-M .. M)\n",
    "sum(cos(x)/(x^2+4)) # converges if we make M even bigger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.21258416579381814"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π/(2*exp(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### 8.\n",
    "$$\n",
    "\t\\int_{-\\infty}^\\infty    {x \\sin x \\over x^2 + 1} \\dx = { \\pi \\over \\E}\n",
    "    $$\n",
    "using Residue calculus. You need to appeal to Jordan's lemma to argue that it can still be done even with only $O(x^{-1})$ decay."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1560943440671996"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = 2000\n",
    "x = Fun(-M .. M)\n",
    "sum(x*sin(x)/(x^2+1)) # Converges if we make M even bigger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1557273497909217"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π/ℯ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 9. \n",
    "$$\n",
    "\\int_{-\\infty}^\\infty {\\cos a x - \\cos b x \\over x^2}  \\dx \\qqwhere a,b > 0\n",
    "$$\n",
    "\n",
    "We have for $f(x) = {\\E^{\\I a x} - \\E^{\\I b x} \\over x^2}$\n",
    "$$\n",
    "\\Re f(x) = {\\cos a x - \\cos b x \\over x^2}\n",
    "$$\n",
    "\n",
    "Note that, since $\\cos x = 1 + x^2/2 + O(x^4)$, the integrand is fine near zero:\n",
    "$$\n",
    "{\\cos a x - \\cos b x \\over x^2} = {(a -b)  \\over 2} + O(x^2)\n",
    "$$\n",
    "But $f(x)$ has a pole:\n",
    "$$\n",
    "     {\\E^{\\I a x} - \\E^{\\I b x} \\over x^2}  = {\\I (a-b) \\over x} + O(1)\n",
    "$$\n",
    "To rectify this, we need to be a bit more careful. First note that\n",
    "$$\n",
    "\\int_{\\infty}^\\infty {\\cos a x - \\cos b x \\over x^2} \\dx = \\lim_{\\epsilon \\rightarrow 0} \\left(\\int_{-\\infty}^{-\\epsilon} + \\int_\\epsilon^\\infty \\right){\\cos a x - \\cos b x \\over x^2} \\dx = \\Re \\dashint_{-\\infty}^\\infty f(x) \\dx\n",
    "$$\n",
    "Then we construct a contour avoiding zero as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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       "  1190.54,1356.45 1188.34,1359.01 1186.23,1361.61 1184.21,1364.23 1182.29,1366.89 1180.45,1369.56 1178.7,1372.24 1177.04,1374.92 1175.47,1377.6 1173.97,1380.28 \n",
       "  1172.55,1382.94 1171.22,1385.59 1169.95,1388.22 1168.76,1390.82 1167.64,1393.4 1166.59,1395.94 1165.6,1398.45 1164.68,1400.92 1163.81,1403.35 1163,1405.74 \n",
       "  1162.24,1408.09 1161.54,1410.39 1160.88,1412.64 1160.27,1414.85 1159.71,1417 1159.18,1419.11 1158.7,1421.16 1158.25,1423.16 1157.84,1425.11 1157.46,1427.01 \n",
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       "  1810,316.617 1834.5,339.467 1858.14,362.932 1880.91,386.939 1902.82,411.419 1923.86,436.301 1944.04,461.518 1963.37,487.005 1981.85,512.7 1999.49,538.543 \n",
       "  2016.31,564.475 2032.32,590.444 2047.55,616.398 2062.01,642.288 2075.72,668.07 2088.7,693.701 2100.98,719.142 2112.58,744.358 2123.51,769.314 2133.82,793.98 \n",
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      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = 10\n",
    "ε = 1.0\n",
    "\n",
    "plot(Segment(-M, -ε) ∪ Segment(ε, M);label=\"[-M,e] U [e,M]\")\n",
    "plot!(Arc(0.,ε, (π,0.)); label=\"C_e\")\n",
    "plot!(Arc(0., M, (0,π)); label = \"C_M\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that $\\oint_\\gamma f(z) \\dz = 0$, \n",
    "$$\\int_{C_\\epsilon} {\\E^{\\I a z } - \\E^{\\I b z} \\over z^2} \\dz =\n",
    " \\int_\\pi^0 { (b-a) \\E^{\\I \\theta}  + O(\\epsilon) \\over \\E^{ \\I \\theta}} \\D\\theta \\rightarrow (a-b) \\pi\n",
    "$$\n",
    "Also, as the integrand is $O(z^{-2})$ the integral over $C_M$ vanishes as $M \\rightarrow \\infty$. We therefore get\n",
    "$$\n",
    "    \\dashint f(x)\\dx = (b-a)\\pi\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.703235784846408"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ε =0.001\n",
    "M = 600.0\n",
    "x = Fun(Segment(-M , -ε) ∪ Segment(ε, M))\n",
    "a = 2.3; b = 3.8\n",
    "sum((cos(a*x) - cos(b*x))/x^2) # Converges if we make M bigger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.71238898038469"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π*(b-a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 10.\n",
    "Use binomial formula\n",
    "\\begin{align*}\n",
    "\\int_0^{2 \\pi} (\\cos \\theta)^n \\D \\theta &= {1 \\over 2^n \\I} \\oint (z+z^{-1})^n {\\dz \\over z} \\\\\n",
    " &=  {1 \\over 2^n \\I} \\sum_{k=0}^n {n! \\over k! (n-k)!} \\oint z^k z^{k-n} {\\dz \\over z} \\\\\n",
    " &=  {1 \\over 2^n \\I} \\sum_{k=0}^n {n! \\over k! (n-k)!} \\oint z^{2k-n-1}\\dz\n",
    "\\end{align*}\n",
    "\n",
    "We only have a residue of $2k-n-1 = -1$, that is, if $2k = n$. If $n$ is odd, this can't happen (duh! the integral is symmetric with respect to $\\theta$). If it's even, then we have\n",
    "$$\n",
    "\\int_0^{2 \\pi} (\\cos \\theta)^n \\D \\theta =  {\\pi \\over 2^{n-1} }  {n! \\over 2 (n/2)!} \n",
    " $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.356194490192353"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "θ = Fun(0 .. 2π)\n",
    "n = 4;\n",
    "sum(cos(θ)^n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.356194490192345"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π*factorial(1.0n)/(2^(n-1)*2*factorial(n/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2.1 \n",
    "\n",
    "By integrating around a rectangular contour with vertices at $\\pm R$ and $\\pi \\I \\pm R$ and letting $R \\rightarrow \\infty$, show that:\n",
    "\t$$\\int_0^\\infty \\sech x \\dx = {\\pi \\over 2}$$\n",
    "where $\\sech x = {2 \\over \\E^{-x} + \\E^x}$.\n",
    "\n",
    "\n",
    "Recall $\\sech x = {2 \\over \\E^{-x} + \\E^x} $. This shows that $\\sech(-x) = \\sech x$\n",
    "But we also have \n",
    "$\n",
    "\\sech(x + \\I \\pi) =  {2 \\over \\E^{-x-\\I \\pi} + \\E^{x+ \\I \\pi}} =  {2 \\over -\\E^{-x} - \\E^{x}} = -\\sech x\n",
    "$\n",
    "Thus we have\n",
    "$$\n",
    "    4\\int_0^\\infty \\sech x \\dx = \\left[ \\int_{-\\infty}^\\infty + \\int_{\\infty+\\I\\pi}^{-\\infty + \\I \\pi} \\right] \\sech z \\dz\n",
    "$$\n",
    "\n",
    "We can approximate this using\n",
    "$$\n",
    "\\left[ \\int_{-R}^R + \\int_R^{R+\\I \\pi} + \\int_{R+\\I \\pi}^{-R+\\I\\pi} + \\int_{-R+\\I \\pi} \\right ] \\sech z \\dz = 2 \\pi \\I \\Res_{z = {\\I \\pi \\over 2}} \\sech z  = 2 \\pi \n",
    "$$\n",
    "since, for $z_0 = {\\I \\pi \\over 2}$, we have\n",
    "$$\n",
    "\\sech z = {1 \\over \\cos \\I z} = {1 \\over - \\I \\sin \\I z_0 (z- z_0) + O(z-z_0)^2}  \n",
    "= -{\\I \\over (z- z_0)} + O(1)\n",
    "$$\n",
    "\n",
    "Finally, we need to show that the limit as $R \\rightarrow \\infty$ tends to the right value. In this case, it follows since \n",
    "$$\n",
    " \\left|\\int_R^{R+\\I \\pi} \\sech z \\dz \\right| \\leq {2 \\pi \\E^{-R} \\over 1 - \\E^{-2R}} \\rightarrow 0\n",
    "$$\n",
    "(and by symmetry for $\\int_{-R+\\I \\pi}^{-R}$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2.2 \n",
    "\n",
    "Show that the Fourier transform of $\\sech x$ satisfies\n",
    "\t$$\n",
    "\t\\int_{-\\infty}^\\infty \\E^{\\I k x } \\sech x \\dx = \\pi \\sech {\\pi k \\over 2}\n",
    "\t$$\n",
    "    \n",
    "Define \n",
    "$$f(z) = \\E^{\\I k z } \\sech z = {2 \\E^{(1+\\I k) z} \\over \\E^{2 z } + 1}$$\n",
    "In this case, we have the symmetry\n",
    "$$\n",
    "f(x + \\I \\pi) = - \\E^{- k \\pi} \\E^{\\I k x} \\sech x = - \\E^{-k \\pi} f(x)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00032444937189257726 + 0.0003756543878221788im"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = 2.0\n",
    "f = z -> exp(im*k*z)*sech(z)\n",
    "\n",
    "-exp(-k*π)f(2.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0003244493718925772 + 0.00037565438782217884im"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(2.0+im*π)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, we have\n",
    "$$\n",
    "(1 + \\E^{-k \\pi}) \\int_{-\\infty}^\\infty f(x) \\dx =\\left(\\int_{-\\infty}^\\infty +  \\int_{\\infty+\\I \\pi}^{-\\infty + \\I \\pi}\\right) f(z) \\dz \n",
    "$$\n",
    "By similar logic as above, we can show that the integral over the rectangular contour converges to this.\n",
    "\n",
    "Again, the only pole inside is at $z = {\\I \\pi \\over 2}$, where the residue is $-\\I \\E^{-\\pi k \\over 2}$. Thus we have\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty f(x) \\dx = {2 \\pi \\E^{-\\pi k \\over 2} \\over 1 + \\E^{-k \\pi}} = \\pi \\sech{\\pi k \\over 2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3.1\n",
    "\n",
    "We have \n",
    "$$\n",
    "\\rho(A) \\subseteq B(1,3) \\cup B(2,3) \\cup B(4,1)\n",
    "$$\n",
    "where $B(z_0,r)$ is the ball of radius $r$ around $z_0$.\n",
    "\n",
    "Here's a depiction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
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   ],
   "source": [
    "drawdisk!(z0, R) = plot!(θ-> real(z0) + R[1]*cos(θ), θ-> imag(z0) + R[1]*sin(θ), 0, 2π, fill=(0,:red), α = 0.2, legend=false)\n",
    "\n",
    "A = [1 2 -1; -2 2 1; 0 1 4]\n",
    "\n",
    "λ = eigvals(A)\n",
    "p = plot()\n",
    "drawdisk!(1,3)\n",
    "drawdisk!(2,3)\n",
    "drawdisk!(4,1)\n",
    "scatter!(complex.(λ); label=\"eigenvalues\")\n",
    "scatter!(complex.(diag(A)); label=\"diagonals\")\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3.2 \n",
    "\n",
    "We get \n",
    "$$\n",
    "\\rho(A) \\subseteq B(1,2) \\cup B(2,3) \\cup B(4,2)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
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     "execution_count": 27,
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   ],
   "source": [
    "λ = eigvals(A)\n",
    "p = plot()\n",
    "drawdisk!(1,2)\n",
    "drawdisk!(2,3)\n",
    "drawdisk!(4,2)\n",
    "scatter!(complex.(λ); label=\"eigenvalues\")\n",
    "scatter!(complex.(diag(A)); label=\"diagonals\")\n",
    "p"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3.3\n",
    "\n",
    "Because the spectrum live in the intersection of the two estimates, the sharpest bound is\n",
    "$$\n",
    "\\rho(A) \\subseteq B(2,3) \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
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       "  272.039,754.238 273.03,774.978 275.012,795.686 277.983,816.343 281.939,836.927 286.877,857.419 292.792,877.797 299.678,898.042 307.527,918.133 316.333,938.051 \n",
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       "  1290.3,743.863 1321.68,743.863 1353.01,743.863 1384.26,743.863 1415.39,743.863 1446.36,743.863 1477.16,743.863 1507.75,743.863 1538.1,743.863 1568.18,743.863 \n",
       "  1597.96,743.863 1627.41,743.863 1656.5,743.863 1685.2,743.863 1713.48,743.863 1741.32,743.863 1768.69,743.863 1795.55,743.863 1821.89,743.863 1847.67,743.863 \n",
       "  1872.88,743.863 1897.48,743.863 1921.45,743.863 1944.77,743.863 1967.41,743.863 1989.36,743.863 2010.58,743.863 2031.07,743.863 2050.79,743.863 2069.73,743.863 \n",
       "  2087.87,743.863 2105.19,743.863 2121.67,743.863 2137.31,743.863 2152.07,743.863 2165.95,743.863 2178.94,743.863 2191.02,743.863 2202.17,743.863 2212.4,743.863 \n",
       "  2221.68,743.863 2230.01,743.863 2237.37,743.863 2243.78,743.863 2249.2,743.863 2253.65,743.863 2257.11,743.863 2259.59,743.863 2261.08,743.863 2261.57,743.863 \n",
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       "  1872.88,1265 1897.48,1252.1 1921.45,1238.69 1944.77,1224.79 1967.41,1210.41 1989.36,1195.57 2010.58,1180.27 2031.07,1164.54 2050.79,1148.39 2069.73,1131.84 \n",
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      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "λ = eigvals(A)\n",
    "p = plot()\n",
    "drawdisk!(2,3)\n",
    "scatter!(complex.(λ); label=\"eigenvalues\")\n",
    "scatter!(complex.(diag(A)); label=\"diagonals\")\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus we can take $2 + 3 \\E^{\\I \\theta}$ as the contour."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4.1\n",
    "\n",
    "Note that in the scalar case $u'' = a u$ we have the solution\n",
    "$$\n",
    "u(t) = u_0 \\cosh \\sqrt{a} t + v_0 {\\sinh \\sqrt{a} t \\over \\sqrt a}\n",
    "$$\n",
    "Write \n",
    "$$A = Q \\begin{pmatrix} \\lambda_1 \\\\ & \\ddots \\\\\n",
    "                        && \\lambda_n \\end{pmatrix}  Q^\\top $$\n",
    "where $\\lambda_k > 0$ \n",
    "and then the solution has the form, where $\\gamma$ is a contour surrounding the eigenvalues and to the right of zero: \n",
    "\\begin{align*}\n",
    "    \\vc u(t) = Q \\begin{pmatrix} \\cosh \\sqrt{\\lambda_1} t \\\\ & \\ddots \\\\\n",
    "                        && \\cosh \\sqrt{\\lambda_n} t  \\end{pmatrix}  Q^\\top \\vc u_0 + \n",
    "                        Q \\begin{pmatrix} {\\sinh \\sqrt{\\lambda_1} t \\over \\sqrt{\\lambda_1}}\\\\ & \\ddots \\\\\n",
    "                        && {\\sinh \\sqrt{\\lambda_n} t \\over \\sqrt{\\lambda_n}} \\end{pmatrix}  Q^\\top \\vc v_0 \\\\\n",
    "              = {1 \\over 2 \\pi \\I} Q \\begin{pmatrix}  \\oint_\\gamma {\\cosh \\sqrt{z} t  \\dz \\over z - \\lambda_1}\\\\ & \\ddots \\\\\n",
    "                        &&  \\oint_\\gamma {\\cosh \\sqrt{z} t  \\dz \\over z - \\lambda_n}\\end{pmatrix}  Q^\\top \\vc u_0 + \n",
    "                        {1 \\over 2 \\pi \\I} Q \\begin{pmatrix} \\oint_\\gamma {\\sinh \\sqrt{z} t \\over \\sqrt z} {\\dz   \\over z - \\lambda_1}  \\\\ & \\ddots \\\\\n",
    "                        && \\oint_\\gamma {\\sinh \\sqrt{z} t \\over \\sqrt z} {\\dz   \\over z - \\lambda_n}  \\end{pmatrix}  Q^\\top \\vc v_0 \\\\\n",
    "= {1 \\over 2 \\pi \\I} \\oint_\\gamma \\cosh \\sqrt{z} t   Q \\begin{pmatrix}  (z - \\lambda_1)^{-1}\\\\ & \\ddots \\\\\n",
    "                        &&    (z - \\lambda_n)^{-1}\\end{pmatrix}  Q^\\top \\vc u_0 \\dz +  {1 \\over 2 \\pi \\I} \\oint_\\gamma {\\sinh \\sqrt{z} t \\over \\sqrt z}   Q \\begin{pmatrix}  (z - \\lambda_1)^{-1}\\\\ & \\ddots \\\\\n",
    "                        &&    (z - \\lambda_n)^{-1}\\end{pmatrix}  Q^\\top \\vc v_0 \\dz \\\\\n",
    "= {1 \\over 2 \\pi \\I} \\oint_\\gamma \\cosh \\sqrt{z} t  (z I - A)^{-1} \\vc u_0 \\dz +  {1 \\over 2 \\pi \\I} \\oint_\\gamma {\\sinh \\sqrt{z} t \\over \\sqrt z}   (z I - A)^{-1} \\vc v_0 \\dz                         \n",
    "\\end{align*}\n",
    "\n",
    "Here we verify the formulae numerically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.323277889766459e-14"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 5\n",
    "A = randn(n,n) \n",
    "A = A+ A' + 10I\n",
    "\n",
    "λ, Q = eigen(A)\n",
    "\n",
    "norm(A - Q*Diagonal(λ)*Q')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Time-stepping solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "u₀ = randn(n)\n",
    "v₀ = randn(n)\n",
    "uv = solve(ODEProblem((uv,_,t) -> [uv[n+1:end]; A*uv[1:n]], [u₀; v₀], (0.,2.)); reltol=1E-10);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Float64,1}:\n",
       "   431.3487825666393\n",
       "   842.6721648972994\n",
       "  -510.6730776173628\n",
       " -1275.0672015889802\n",
       "  -852.8144701538115"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = 2.0\n",
    "uv(t)[1:n]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solution via diagonalization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Float64,1}:\n",
       "   431.34878253723247\n",
       "   842.672164825094  \n",
       "  -510.67307756879427\n",
       " -1275.0672015062562 \n",
       "  -852.8144700629356 "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q*Diagonal(cosh.(sqrt.(λ) .* t))*Q'*u₀ + Q*Diagonal(sinh.(sqrt.(λ) .* t) ./ sqrt.(λ))*Q'*v₀"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solution via elliptic integrals. We chose the ellipse to surround all the spectrum of our particular $A$ with eigenvalues:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Complex{Float64},1}:\n",
       "  431.3489776293208 - 2.1947119676805203e-14im\n",
       "  842.6728232388965 - 8.751061717472864e-15im \n",
       "   -510.67355346183 + 2.8015565557209588e-14im\n",
       " -1275.067724375493 - 6.622136600059223e-14im \n",
       " -852.8154291281783 + 2.365862961710633e-14im "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "periodic_rule(n) = 2π/n*(0:(n-1)), 2π/n*ones(n)\n",
    "\n",
    "function ellipse_rule(n, a, b) \n",
    "    θ = periodic_rule(n)[1]\n",
    "    a*cos.(θ) + b*im*sin.(θ), 2π/n*(-a*sin.(θ) + im*b*cos.(θ))\n",
    "end\n",
    "\n",
    "function ellipse_f(f, A, n, z₀, a, b)\n",
    "    z,w = ellipse_rule(n,a,b)\n",
    "    z .+= z₀\n",
    "\n",
    "    ret = zero(A)\n",
    "    for j=1:n\n",
    "        ret += w[j]*f(z[j])*inv(z[j]*I - A)\n",
    "    end\n",
    "    ret/(2π*im)\n",
    "end\n",
    "\n",
    "\n",
    "n = 50\n",
    "ellipse_f(z -> cosh(sqrt(z)*t), A, n, 10.0, 7.0, 2.0)*u₀ +\n",
    "    ellipse_f(z -> sinh(sqrt(z)*t)/sqrt(z), A, n, 10.0, 7.0, 2.0)*v₀"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Problem 4.2\n",
    "\n",
    "I put the restriction in because of the $\\sqrt z$ term, which look like it is not analytic on $(-\\infty,0]$. However, this restriction was NOT necessary, since in fact $\\cosh \\sqrt z t$ and ${\\sinh \\sqrt z t \\over \\sqrt z}$ are entire:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-6..6, -3..3, z -> cosh(sqrt(z)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-3..3, -3..3, z -> sinh(sqrt(z))/sqrt(z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " This follows from Taylor series, though I prefer the following argument: we have\n",
    "$$\n",
    "\\cosh z = {\\E^{z} + \\E^{-z} \\over 2}\n",
    "$$\n",
    "hence $\\cosh{\\I t } = \\cos t = \\cosh{-\\I t}$. Therefore, on the possible branch cut $\\cosh \\sqrt z$ is continuous (hence analytic):\n",
    "$$\n",
    "\\cosh \\sqrt{x}_+ = \\cosh \\I \\sqrt{|x|} = \\cosh -\\I \\sqrt{|x|} = \\cosh\\sqrt{x}_-\n",
    "$$\n",
    "Similarly, \n",
    "$$\n",
    "\\sinh z = {\\E^z - \\E^{-z} \\over 2}\n",
    "$$\n",
    "implies $\\sinh{\\I t} = \\I \\sin t = - \\I \\sin(-t) = -\\sinh{-\\I t}$, which gives us continuity:\n",
    "$$\n",
    "{\\sinh \\sqrt{x}_+ \\over \\sqrt{x}_+} = {\\sinh \\I \\sqrt{|x|} \\over \\I \\sqrt{|x|}} = {\\sinh(-\\I \\sqrt{|x|)} \\over -\\I \\sqrt{|x|}} ={\\sinh \\sqrt{x}_- \\over \\sqrt{x}_-}\n",
    "$$\n",
    "furthermore, they are both bounded at zero, hence analytic there too.\n",
    "\n",
    "Here's a numerical example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.93500670014108e-15"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 5\n",
    "A = randn(n,n) \n",
    "λ, V = eigen(A)\n",
    "\n",
    "norm(A - V*Diagonal(λ)*inv(V))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "u₀ = randn(n)\n",
    "v₀ = randn(n)\n",
    "uv = solve(ODEProblem((uv,_,t) -> [uv[n+1:end]; A*uv[1:n]], [u₀; v₀], (0.,2.)); reltol=1E-10);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Float64,1}:\n",
       "  3.803344206258081 \n",
       "  0.9252585031733016\n",
       " -1.193805803087017 \n",
       "  5.72798765024654  \n",
       "  1.7188451287601578"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = 2.0\n",
    "uv(t)[1:n]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Complex{Float64},1}:\n",
       "   3.803344206066801 + 8.317464758586041e-16im \n",
       "  0.9252585029740441 + 2.2092966791530562e-16im\n",
       " -1.1938058030645906 + 9.308804664865466e-17im \n",
       "   5.727987650140507 + 7.402447551422406e-16im \n",
       "  1.7188451286631823 - 1.5268203762514324e-16im"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V*Diagonal(cosh.(sqrt.(λ) .* t))*inv(V)*u₀ + \n",
    "   V*Diagonal(sinh.(sqrt.(λ) .* t) ./ sqrt.(λ))*inv(V)*v₀"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the solution using an elliptic integral:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Complex{Float64},1}:\n",
       "  3.8033442060667997 - 1.0942835705045212e-16im\n",
       "  0.9252585029740352 + 2.0727119042774893e-16im\n",
       " -1.1938058030645888 - 2.8947259388740214e-16im\n",
       "  5.7279876501405145 - 1.9746496661590837e-16im\n",
       "  1.7188451286631805 + 8.925757831726994e-17im "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 100\n",
    "ellipse_f(z -> cosh(sqrt(z)*t), A, n, 0.0, 3.0, 3.0)*u₀ +\n",
    "    ellipse_f(z -> sinh(sqrt(z)*t)/sqrt(z), A, n, 0.0, 3.0, 3.0)*v₀ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5.1\n",
    "\n",
    "\n",
    "\n",
    "We have \n",
    "$$\n",
    "{1 \\over n} \\sum_{j=0}^{n-1} g(\\theta_j) = \\sum_{k=0}^\\infty g_k  {1 \\over n} \\sum_{j=0}^{n-1} \\E^{\\I k \\theta_j}\n",
    "$$\n",
    "Define the $n$-th root of unity as $\\omega = e^{2 \\pi \\I  \\over n}$ (that is $\\omega^n = 1$), and simplify \n",
    "$$\n",
    "\\sum_{j=0}^{n-1} \\E^{\\I k \\theta_j} =\\sum_{j=0}^{n-1} e^{{2 \\pi j \\I k \\over n}} =\\sum_{j=0}^{n-1} \\omega^{kj}= \\sum_{j=0}^{n-1} (\\omega^k)^j\n",
    "$$\n",
    "If $k$ is a multiple of $n$, then $\\omega^k = 1$, and this sum is equal to $n$. If $k$ is not a multiple of $n$, use Geometric series:\n",
    "$$\n",
    "\\sum_{j=0}^{n-1} (\\omega^k)^j = {\\omega^{nk} - 1 \\over \\omega^k -1} = {1^k - 1 \\over \\omega^k-1} = 0\n",
    "$$\n",
    "## Problem 5.2 \n",
    "\n",
    "\n",
    "From lecture 4, we have $|f_k| \\leq M_r r^{-|k|}$ for any $1 \\leq r < R$, where $M_r$ is the supremum of $f$ in an annulus $\\{ z : r^{-1} < |z| < r \\}$. Thus from the previous part we have (using geometric series)\n",
    "$$\n",
    "\\left| {1 \\over n} \\sum_{j=0}^{n-1} g(\\theta_j)  - { 1\\over 2 \\pi}\t \\int_0^{2\\pi} g(\\theta) \\D \\theta \\right| \\leq \\sum_{K=1}^\\infty |f_{Kn}| + |f_{-Kn}| \\leq 2M \\sum_{K=1}^\\infty r^{-Kn} = 2M_r {r^{-n} \\over 1 - r^{-n}}.\n",
    "$$\n",
    "This is an upper bound that  decays exponentially fast. \n",
    "\n",
    "## Problem 5.3\n",
    "\n",
    "Note that $f(z) = 2 z/(4z - z^2 - 1)$ satisfies $f(\\E^{\\I \\theta}) = g(\\theta)$. This has two poles at $2 \\pm \\sqrt 3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1102230246251565e-16 + 0.0im"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = z -> 2z/(4z-z^2-1)\n",
    "f(exp(0.1im)) - 1/(2-cos(0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a phase plot showing the location of poles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-5..5, -5..5, f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3.732050807568877, 0.2679491924311228, 0.2679491924311227)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2+sqrt(3), 2- sqrt(3), 1/(2+sqrt(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that $2+\\sqrt 3 = 1/(2-\\sqrt 3)$ so in the previous result, we take $R = 2+\\sqrt 3$. For any $1 \\leq r < 2+\\sqrt 3$ we have \n",
    "$$\n",
    "M_r = {2 \\over 4 - r - r^{-1}}\n",
    "$$\n",
    "\n",
    "Thus we get the upper bounds\n",
    "$$\n",
    "    {4 \\over 4 - r - r^{-1}} {r^{-n} \\over 1 - r^{-n}}\n",
    "$$\n",
    "\n",
    "Let's see how sharp it is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "periodic_rule(n) = 2π/n*(0:(n-1)), 2π/n*ones(n)\n",
    "\n",
    "g = θ -> 1/(2 - cos(θ))\n",
    "\n",
    "Q = sum(Fun(g, 0 .. 2π))\n",
    "\n",
    "\n",
    "err = Float64[ (\n",
    "            (θ, w) = periodic_rule(n);\n",
    "            sum(w.*g.(θ)) - Q\n",
    "        ) for n=1:30];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = length(err)\n",
    "\n",
    "scatter(abs.(err), yscale=:log10, label=\"true error\", title=\"Trapezium error bounds\", legend=:bottomleft)\n",
    "r = 1.1\n",
    "scatter!(4/(4-r-inv(r)) .* r.^(-(1:N)) ./ (1 .- r.^(-(1:N))), label = \"r = $r\")\n",
    "r = 3.0\n",
    "scatter!(4/(4-r-inv(r)) .* r.^(-(1:N)) ./ (1 .- r.^(-(1:N))), label = \"r = $r\")\n",
    "r = 2+sqrt(3)  - 0.01\n",
    "scatter!(4/(4-r-inv(r)) .* r.^(-(1:N)) ./ (1 .- r.^(-(1:N))), label = \"r = $r\")"
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