Lecture 2.ipynb

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   "source": [
    "using Plots, ComplexPhasePortrait, ApproxFun"
   ]
  },
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   "metadata": {},
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    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "$$\n",
    "\n",
    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "Dr. Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "\n",
    "\n",
    "# Lecture 2: Cauchy's theorem"
   ]
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    "References: \n",
    "\n",
    "1. M.J. Ablowitz & A.S. Fokas, Complex Variables: Introduction and Applications, Second Edition, Cambridge University Press, 2003\n",
    "2. R. Earl, Metric Spaces and Complex Analysis, https://courses.maths.ox.ac.uk/node/view_material/5392, 2015\n",
    "\n",
    "\n",
    "### Complex-differentiable functions \n",
    "\n",
    "\n",
    "**Definition (Complex-differentiable)** Let $D \\subset {\\mathbb C}$ be an open set.  A function $f : D \\rightarrow {\\mathbb C}$ is called _complex-differentiable_ at a point $z_0 \\in D$ if \n",
    "$$ \n",
    "   f'(z_0) =  \\lim_{z \\rightarrow z_0} {f(z) - f(z_0) \\over z - z_0}\n",
    "$$\n",
    "exists, for any angle of approach to $z_0$."
   ]
  },
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    "### Holomorphic functions\n",
    "\n",
    "**Definition (Holomorphic)** Let $D \\subset {\\mathbb C}$ be an open set.  A function $f : D \\rightarrow {\\mathbb C}$ is called _holomorphic_ in $D$ if it is complex-differentiable at all $z \\in D$.\n",
    "\n",
    "**Definition (Entire)** A function is _entire_ if it is holomorphic in ${\\mathbb C}$\n",
    "\n",
    "\n",
    "*Examples*\n",
    "\n",
    "1. $1$ is entire\n",
    "2. $z$ is entire\n",
    "3. $1/z$ is holomorphic in ${\\mathbb C} \\backslash \\{0\\}$\n",
    "4. $\\sin z$ is entire\n",
    "5. $\\csc z$ is holomorphic in ${\\mathbb C} \\backslash \\{\\ldots,-2\\pi,-\\pi,0,\\pi,2\\pi,\\ldots\\}$\n",
    "6. $\\sqrt z$ is holomorphic in ${\\mathbb C} \\backslash (-\\infty,0]$\n",
    "\n",
    "We can usually infer the domain where a function is holomorphic from a phase portrait, here we see that ${\\rm arcsinh}\\, z$ has cuts on $[\\I,\\I \\infty)$ and $[-\\I,-\\I \\infty)$, and a zero  (red–green–blue–red) at zero, hence we can infer that it is holomorphic in $\\C \\backslash ([\\I,\\I \\infty) \\cup [-\\I,-\\I \\infty))$."
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-4..4, -4..4, z -> asinh(z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following example $\\sqrt{z-1} \\sqrt{z+1}$ is analytic in ${\\mathbb C}\\backslash [-1,1]$ and will be returned to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-4..4, -4..4, z -> sqrt(z-1)sqrt(z+1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contours\n",
    "\n",
    "**Definition (Contour)** A _contour_  is a continuous & piecewise-continuously differentiable function $\\gamma : [a,b] \\rightarrow {\\mathbb C}$.\n",
    "\n",
    "**Definition (Simple)** A _simple contour_ is a contour that is 1-to-1.\n",
    "\n",
    "**Definition (Closed)** A _closed contour_ is a contour such that $\\gamma(a) = \\gamma(b)$\n",
    "\n",
    "*Examples of contours*\n",
    "\n",
    "1. Line segment $[a,b]$ is a simple contour, with $\\gamma(t) =  t$\n",
    "1. Arc from $re^{ia}$ to $re^{ib}$ is a simple contour, with $\\gamma(t) =  re^{i t}$\n",
    "2. Circle of radius $r$ is a closed simple contour, with $\\gamma(t) = re^{i t}$ and $a = -\\pi$, $b = \\pi$\n",
    "3. $\\gamma(t) = \\cos (t+i)^2$ defines a contour that is not simple or closed\n",
    "4. $\\gamma(t) = e^{i t} + e^{2i t}$ for $[a,b] = [-\\pi,\\pi]$ defines a contour that is closed but not simple\n",
    "\n",
    "Here's an example of a closed contour that is not simple:"
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     "execution_count": 13,
     "metadata": {},
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    }
   ],
   "source": [
    "a,b = -π, π\n",
    "tt = range(a, stop=b, length=1000)\n",
    "\n",
    "γ = t -> exp(im*t) +exp(2im*t)\n",
    "\n",
    "plot(real.(γ.(tt)), imag.(γ.(tt)); ratio=1.0, legend=false, arrow=true)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contour integrals\n",
    "\n",
    "**Definition (Contour integral)** The _contour integral_ over $\\gamma$ is defined by\n",
    "$$\n",
    "\\int_\\gamma f(z) dz := \\int_a^b f(\\gamma(t)) \\gamma'(t) dt\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> real(exp(z)), Arc(0.,1.,(0,π/2)))  # Not holomorphic!\n",
    "\n",
    "plot(domain(f); legend=false, ratio=1.0, arrow=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.2485382363935424 + 1.949326343919058im"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(f)  # this means contour integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.2485382363935429 + 1.9493263439190578im"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = im*Fun(t-> f(exp(im*t))*exp(im*t), 0 .. π/2)\n",
    "sum( g )  # this is standard integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An important property of a contour is its _arclength_:\n",
    "\n",
    "**Definition (Arclength)** The _arclength_ of $\\gamma$ is defined as\n",
    "$$\n",
    "    {\\cal L}(\\gamma) := \\int_a^b |\\gamma'(t)| dt\n",
    "$$\n",
    "\n",
    "A very useful result is that we can use the maximum and arclength to bound integrals:\n",
    "\n",
    "**Proposition (ML)** Let $f : \\gamma \\rightarrow {\\mathbb C}$ and \n",
    "$$\n",
    "        M = \\sup_{z \\in \\gamma} |f(z)|\n",
    "$$        \n",
    "Then\n",
    "$$\n",
    "    \\left|\\int_\\gamma f(z) dz \\right| \\leq M {\\cal L}(\\gamma)\n",
    "    $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cauchy's theorem\n",
    "\n",
    "**Proposition** If $f(z)$ is holomorphic on $\\gamma$, then\n",
    "$$\\int_\\gamma f'(z) dz = f(\\gamma(b)) - f(\\gamma(a))$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2.1779795225909058 + 0.8414709848078968im, -2.177979522590906 + 0.8414709848078968im)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> exp(z), Arc(0.,1.,(0,π/2)))  # Not holomorphic!\n",
    "\n",
    "sum(f)  , f(im)-f(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2.1779795225909058 + 0.8414709848078968im, -2.177979522590906 + 0.8414709848078968im)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> exp(z), Arc(0.,1.,(0,π/2)))  # Holomorphic!\n",
    "\n",
    "sum(f)  , f(im)-f(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2.1779795225909053 + 0.8414709848078966im, -2.177979522590907 + 0.8414709848078968im)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> exp(z), Segment(1,im))  # Holomorphic!\n",
    "\n",
    "sum(f)  , f(im)-f(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem (Cauchy)** If $f$ is holomorphic inside and on a closed contour $\\gamma$, then \n",
    "$$\\oint_\\gamma f(z) dz = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.3420237696193494e-16 + 3.141592653589793im"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> real(exp(z)), Circle())  # Not holomorphic!\n",
    "\n",
    "sum(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9644937254112756e-17 - 1.4872544363724962e-16im"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun( z -> exp(z), Circle())  #  Holomorphic!\n",
    "\n",
    "sum(f)"
   ]
  }
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