{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, ComplexPhasePortrait, ApproxFun\n",
    "gr();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "$$\n",
    "\n",
    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "Dr. Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "\n",
    "# Lecture 3: Cauchy's integral formula and Taylor series\n",
    "\n",
    "\n",
    "This lecture we cover\n",
    "\n",
    "1. Deformation of contours\n",
    "2. Cauchy's integral formula\n",
    "    - Application: Numerical differentiation\n",
    "3. Taylor series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Deformation of contours\n",
    "\n",
    "**Definition (Domain)** A _domain_ is a non-empty, open and connected set $D \\subset {\\mathbb C}$.\n",
    "\n",
    "**Definition (Homotopic)** Two closed contours $\\gamma_1 : [a,b] \\rightarrow D$ and $\\gamma_2 \\rightarrow D$ in a domain $D$ are _homotopic_ if they can be continuously deformed to one-another while remaining in $D$. \n",
    "\n",
    "**Theorem (Deformation of contours)** Let $f(z)$ be holomorphic in a domain $D$. Let $\\gamma_1$ and $\\gamma_2$ be two homotopic contours. Then\n",
    "$$\n",
    "    \\int_{\\gamma_1} f(z) dz = \\int_{\\gamma_2} f(z) dz\n",
    "$$   \n",
    "\n",
    "**Definition (Simply connected)** A domain is _simply connected_ if every closed contour is homotoic to a point.\n",
    "\n",
    "**Corollary (Deformation of contours on simply connected domains)** Let $f(z)$ be holomorphic in a simply-connected domain $D$. If $\\gamma_1$ and $\\gamma_2$ are two contours in $D$ with the same endpoints, then\n",
    "\n",
    "$$\n",
    "    \\int_{\\gamma_1} f(z) dz = \\int_{\\gamma_2} f(z) dz\n",
    "$$   \n",
    "\n",
    "_Demonstration_ We can test this experimentally: in the following, the integral (implemented as `sum`) over two different contours returns the same vaule (up to numerical precision)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2.1779795225909058 + 0.8414709848078968im, -2.1779795225909053 + 0.8414709848078966im)"
      ]
     },
     "execution_count": 2,
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     "output_type": "execute_result"
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   ],
   "source": [
    "f = Fun( z -> exp(z), Arc(0.,1.,(0,π/2)))  # Holomorphic!\n",
    "f̃ = Fun( z -> exp(z), Segment(1,im))  # Holomorphic!\n",
    "\n",
    "sum(f)  , sum(f̃)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a depection of the two contours:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "plot(domain(f); ratio=1.0, legend=false, arrow=true)\n",
    "plot!(domain(f̃), arrow=true)"
   ]
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    "## Orientation of contours\n",
    "\n",
    "Contours are _oriented_: there is a notion of \"left\" and \"right\" inherited from $[a,b]$. For closed contours, there is as notion of positive/negative orientation:\n",
    "\n",
    "**Definition (Positive/negative orientation)** Let $\\gamma$ be a simple closed contour and $z$ in the interior of $\\gamma$.  We say that $\\gamma$ is _positively oriented_ if \n",
    "$${1 \\over 2 \\pi i} \\oint_\\gamma {d\\zeta \\over \\zeta - z} = 1$$\n",
    "It is _negatively oriented_ if the reversed contour $\\gamma_{\\rm reversed}(t) = \\gamma(b+a-t)$ for $t \\in [a,b]$ is positively oriented, or equivalentally\n",
    "$${1 \\over 2 \\pi i} \\oint_\\gamma {d\\zeta \\over \\zeta - z} = -1$$\n",
    "\n",
    "\n",
    "_Demonstration_ Here we see numerically that $1/z$ is positively oriented:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0 + 1.2545118101832475e-16im"
      ]
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     "execution_count": 4,
     "metadata": {},
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   ],
   "source": [
    "sum(Fun(z -> 1/z, Circle()))/(2π*im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
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   "source": [
    "## Cauchy's integral formula\n",
    "\n",
    "Cauchy's integral formula allows us to recover a function from knowlerdge of its values on a surrounding contour:\n",
    "\n",
    "**Theorem (Cauchy integral formula)** Suppose $f$ is holomorphic inside and on a positively oriented, simple, closed contour $\\gamma$. Then\n",
    "$$\n",
    "f(z) = {1 \\over 2 \\pi i} \\oint_\\gamma {f(\\zeta) \\over \\zeta - z} d \\zeta$$\n",
    "\n",
    "_Demonstration_ This shows numerically that we can calculate $\\E^{0.1}$ by integrating over a circle that surrounds $z = 0.1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 - 0.0im"
      ]
     },
     "execution_count": 5,
     "metadata": {},
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   ],
   "source": [
    "γ = Circle()\n",
    "ζ = Fun(γ)\n",
    "z = 0.1\n",
    "sum(exp(ζ)/(ζ-z)) /(2π*im)  - exp(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Not only do we know $f$, we also know that $f$ is infinitely-differentiable, and we know all its values:\n",
    "\n",
    "**Corollary (Cauchy integral formula for derivatives)** Suppose $f$ is holomorphic inside and on a positively oriented, simple, closed contour $\\gamma$. Then $f$ is infinitely-differentiable at $z$ and \n",
    "$$\n",
    "f^{(k)}(z) = {k! \\over 2 \\pi i} \\oint_\\gamma {f(\\zeta) \\over (\\zeta - z)^{k+1}} d \\zeta$$\n",
    "\n",
    "_Demonstration_ Here we show that we can recover $\\E^z$ using any of the derivatives:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7801382590221237e-10 - 0.0im"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k=10\n",
    "factorial(1.0k)*sum(exp(ζ)/(ζ-z)^(k+1))/(2π*im)  - exp(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.218847493575595e-15 - 0.0im"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = 0.1\n",
    "k = 5\n",
    "factorial(1.0k)*sum(exp(ζ)/(ζ-z)^(k+1))/(2π*im) - exp(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Taylor series\n",
    "\n",
    "\n",
    "**Theorem (Taylor)** Suppose $f$ is holomorphic in a ball $B(z_0,r)$. Then inside this ball we have\n",
    "$$\n",
    "    f(z) = \\sum_{k=0}^\\infty {f^{(k)}(z_0) \\over k!} (z-z_0)^k\n",
    "$$    \n",
    "\n",
    "\n",
    "_Examples_\n",
    "1. $\\E^z$\n",
    "2. $1/(1-z)$\n",
    "3. $\\sec z$\n",
    "4. $\\sqrt z$\n",
    "\n",
    "here we plot the $n$-term Taylor approximation of $\\E^z$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expⁿ = (n,z) -> sum(z^k/factorial(1.0k) for k=0:n)\n",
    "phaseplot(-20..20, -20..20, z -> expⁿ.(10,z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now the $n$-term Taylor approximation to $1/(1-z)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geometricⁿ = (n,z) -> sum(z^k for k=0:n)\n",
    "\n",
    "phaseplot(-2..2, -2..2, z -> geometricⁿ.(20,z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here the $n$-term Taylor approximation of $\\sqrt z$ near $z_0$:"
   ]
  },
  {
   "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": [
    "function sqrtⁿ(n,z,z₀) \n",
    "    ret = sqrt(z₀)\n",
    "    c = 0.5/ret*(z-z₀)\n",
    "    for k=1:n\n",
    "        ret += c\n",
    "        c *= -(2k-1)/(2*(k+1)*z₀)*(z-z₀)\n",
    "    end\n",
    "    ret\n",
    "end\n",
    "\n",
    "\n",
    "z₀ = 0.3\n",
    "n = 40\n",
    "phaseplot(-2..2, -2..2, z -> sqrtⁿ.(n,z,z₀))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.0.1",
   "language": "julia",
   "name": "julia-1.0"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.0.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}