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    "using ApproxFun, Plots, ComplexPhasePortrait, ApproxFun,  SingularIntegralEquations,\n",
    "        SpecialFunctions\n",
    "\n",
    "using SingularIntegralEquations.HypergeometricFunctions\n",
    "gr();"
   ]
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    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\ds{\\D s}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\H{{\\mathbb H}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\FF{{\\cal F}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\Ei{{\\rm Ei}\\,}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqand{\\qquad\\hbox{and}\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\acos{\\,{\\rm acos}\\,}\n",
    "\\def\\erfc{\\,{\\rm erfc}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n",
    "\\def\\br[#1]{\\left[#1\\right]}\n",
    "\\def\\norm#1{\\left\\|#1\\right\\|}\n",
    "\\def\\half{{1 \\over 2}}\n",
    "\\def\\fL{f_{\\rm L}}\n",
    "\\def\\fR{f_{\\rm R}}\n",
    "\\def\\HF{{}_2F_1}\n",
    "\\def\\questionequals{= \\!\\!\\!\\!\\!\\!{\\scriptstyle ? \\atop }\\,\\,\\,}\n",
    "$$\n",
    "\n",
    "Dr Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "Office Hours: 3-4pm Mondays, 11-12am Thursdays, Huxley 6M40\n",
    "<br>\n",
    "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
    "\n",
    "# Chapter 6: Special functions\n",
    "\n",
    "A _special function_ is a function that can't be expressed in closed form in terms of classical functions, like $\\cos$, $\\sin$.   We've seen a few special functions so far:\n",
    "\\begin{align*}\n",
    "\\Ei z &= \\int_{-\\infty}^z {\\E^\\zeta \\over \\zeta} \\D \\zeta  \\\\\n",
    "\\erfc z &= {2 \\over \\sqrt \\pi} \\int_z^\\infty \\E^{-\\zeta^2} \\D \\zeta \\\\\n",
    "\\Gamma(\\alpha, z) &= \\int_z^\\infty \\zeta^{\\alpha-1} \\E^{-\\zeta} \\D\\zeta.\n",
    "\\end{align*}\n",
    "But we've also seen special functions in the form of orthogonal polynomials:\n",
    "1. $P_n^{(a,b)}(x)$ are orthogonal w.r.t. $(1-x)^a(1+x)^b$\n",
    "2. $L_n^{(a)}(x)$ are orthogonal w.r.t. $x^a \\E^{-x}$\n",
    "3. $H_n(x)$ are orthogonal w.r.t. $\\E^{-x^2}$\n"
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    "\n",
    "# Lecture 26:  Analyticity of solutions of ordinary differential equations\n",
    "\n",
    "\n",
    "Most special functions solve simple ODEs involving very low order rational functions. For example, these three special functions satisfy second ODEs:\n",
    "1. $u(z) = \\E^{-z} \\Ei z$ satisfies\n",
    "\\begin{align*}\n",
    "{\\D u \\over \\dz} + u &= {1 \\over z}\\qquad\\Rightarrow \\\\\n",
    "z {\\D^2 u \\over \\dz^2} + (z+1) {\\D u \\over \\dz} + u &= 0\n",
    "\\end{align*}\n",
    "2. $u(z) = {\\sqrt \\pi \\over 2} \\E^{z^2} \\erfc z$ satisfies\n",
    "\\begin{align*}\n",
    "{\\D u \\over \\dz} - 2 z u   &=  1 \\qquad\\Rightarrow \\\\\n",
    "{\\D^2 u \\over \\dz^2} -2z {\\D u \\over \\dz} -2 u &= 0\n",
    "\\end{align*}\n",
    "3. $u(z) = \\E^{z} \\Gamma(\\alpha, z)$ satisfies \n",
    "\\begin{align*}\n",
    "{\\D u \\over \\dz} - u &= z^{\\alpha-1} \\qquad\\Rightarrow \\\\\n",
    "z {\\D^2 u \\over \\dz^2} + (1- \\alpha -z) {\\D u \\over \\dz} + (\\alpha-1)u &= 0\n",
    "\\end{align*}\n",
    "\n",
    "4. Laguerre satisfies\n",
    "$$\n",
    "x {\\D^2 L_n^{(a)} \\over \\dx^2} + (a+1-x) {\\D L_n^{(a)} \\over \\dx} + n L_n^{(a)} = 0\n",
    "$$\n",
    "5. Hermite satisfies\n",
    "$$\n",
    " {\\D^2 H_n \\over \\dx^2} -2x{\\D H_n \\over \\dx} + 2n H_n = 0\n",
    "$$\n",
    "\n",
    "A natural question becomes what is the relationship between the singularities of the variable coefficients and the singularities of the solutions?\n",
    "\n",
    "1. General properties of ODEs in the complex plane\n",
    "   - Solving an ODE on a contour\n",
    "   - Radius of convergence\n",
    "   - Analytic continuation"
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    "\n",
    "## ODEs on contours\n",
    "\n",
    "Consider the solution of a first order ODE\n",
    "$$\n",
    "{\\D u\\over \\dz} = a(z) u\\qqand u(z_0) = c\n",
    "$$\n",
    "which we can write as\n",
    "$$\n",
    "u(z) = c \\E^{\\int_{z_0}^z a(\\zeta) \\D \\zeta}\n",
    "$$\n",
    "That is, we can think of the solution as living on a contour, corresponding to the contour of integration of the integral. \n",
    "\n",
    "Alternatively, we can think of the ODE as living on a contour  $\\gamma : (a,b) \\rightarrow \\C$, in the first order case we do the change of variables $v(t) = u(\\gamma(t))$, the ODE is reduced to\n",
    "$$\n",
    "{\\D v \\over \\dt} = \\gamma'(t) u'(\\gamma(t)) = \\gamma'(t) a(\\gamma(t)) u(\\gamma(t)) = \\gamma'(t) a(\\gamma(t)) v\n",
    "$$\n",
    "Thus provided we can choose the contour to avoid the singularities of $a(z)$, we can define the solution, but the value of $u(z)$ can depend on the choice of contour.\n",
    "\n",
    "\n",
    "Normally, the contour is taken as a straight line, so that poles in $a(z)$ can induce branch cuts in $u(z)$.  \n",
    "\n",
    "**Example** consider\n",
    "$$\n",
    "{\\D u\\over \\dz} =  u \\qqand u(0) = 1\n",
    "$$\n",
    "with solution $u(z) = \\E^z$. Consider a contour like $\\gamma(t) = (1+\\I )t$. Then we have for $v(t) = u(\\gamma(t)) = \\E^{(1+\\I )t}$ that $v$ satisfies the ODE\n",
    "$$\n",
    "{\\D v \\over \\dt}  = (1+\\I t) v \\qqand v(0) = 1\n",
    "$$\n",
    "\n",
    "\n",
    "**Example** Now consider an ODE with a pole:\n",
    "$$\n",
    "{\\D u\\over \\dz} =  {k u \\over z} \\qqand u(1) = 1\n",
    "$$\n",
    "with solution $u(z) = z^k$. Consider two different choices of contours: $\\gamma_1(t) = \\E^{\\I t}$ and $ \\gamma_2(t) = \\E^{-\\I t}$ for $0 \\leq t \\leq 2\\pi$. For $v_1(t) = u(\\gamma_1(t))$ we have the ODE:\n",
    "$$\n",
    "{\\D v_1\\over \\dt} =  \\I k  v_1 \\qqand v(0) = 1\n",
    "$$\n",
    "with solution $v_1(t) = \\E^{\\I k t}$ (and similarly $v_2(t) = \\E^{- \\I k t}$).  Hence we have \n",
    "\\begin{align*}\n",
    "u(1) &= u(\\E^{2 \\I \\pi}) \\questionequals v_1(2\\pi) = \\E^{2 \\pi \\I k} \\\\\n",
    "u(1) &= u(\\E^{-2 \\I \\pi}) \\questionequals v_2(2\\pi) = \\E^{-2 \\pi \\I k} \n",
    "\\end{align*}.\n",
    "When $k$ is not an integer, each of these is a different number.\n"
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    "\n",
    "\n",
    "### Radius of convergence of ODEs\n",
    "\n",
    "This non-uniqueness means we think of solving an ODE in terms along a contour. In what sense is $u(z)$ analytic? Well, we can deduce that the radius of convergence of the solution $z_0$ is dictated by the radius of convergence of $a(z)$, that is, the closest singularity. \n",
    "\n",
    "\n",
    "**Theorem** Suppose $a(z)$ is analytic in a disk of radius $R$. Then $u(z)$ is also analytic in a disk of radius $R$.\n",
    "\n",
    "**Sketch of proof**\n",
    "We will show this using Taylor series (using operator notation). Note that if we represent (here we take $z_0 = 0$):\n",
    "$$\n",
    "u(z) = u_0 + u_1 z+ u_2 z^2 + \\cdots = (1,z,z^2,\\ldots) \\begin{pmatrix} u_0\\\\u_1\\\\u_2\\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "The derivative operator has a very nice simple form:\n",
    "$$\n",
    "u'(z) =  (1,z,z^2,\\ldots) \\begin{pmatrix} 0 & 1 \\\\ && 2 \\\\ &&&3 \\\\ &&&&\\ddots \\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\u_2 \\\\\n",
    "\\vdots \\end{pmatrix}\n",
    "$$\n",
    "On the other hand, multiplication by $z$ has the following operator form:\n",
    "$$\n",
    "z u(z) = \\begin{pmatrix} 0 \\\\ 1 \\\\ & 1 \\\\ &&1 \\\\ &&&\\ddots \\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\u_2 \\\\\n",
    "\\vdots \\end{pmatrix}\n",
    "$$\n",
    "Each time we multiply by $z$, this expression gets shift down. Thus multiplication by \n",
    "$$\n",
    "a(z) =  a_0 + a_1 z+ a_2 z^2 + \\cdots\n",
    "$$\n",
    "has the form\n",
    "$$\n",
    "a(z) u(z) = \\begin{pmatrix} a_0 \\\\ a_1 & a_0 \\\\ a_2 & a_1 & a_0 \\\\ a_3 & a_2 & a_1 & a_0 \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\u_2 \\\\\n",
    "\\vdots \\end{pmatrix}\n",
    "$$\n",
    "Thus the ODE $u'(z) - a(z) u(z)= 0$ and $u(0) = c$ becomes:\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\\\ -a_0 & 1 \\\\ -a_1 & -a_0 & 2 \\\\ -a_2 & -a_1 & -a_0 & 3 \\\\ -a_3 & -a_2 & -a_1 & -a_0 & 4 \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots & \\ddots \\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\u_2 \\\\\n",
    "\\vdots \n",
    "\\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "This is solvable via forward substitution.\n",
    "\n",
    "Assume that the radius of convergence of $a$ is $R$, that is, for every $r < R$ we have $|a_k| \\leq {C(r) \\over r^k}$ for some constant $C$. The worst case in the growth of $u_k$ is in the case every $a_k$ is positive, therefore, we have\n",
    "$$\n",
    "\\left| \\begin{pmatrix} u_0\\\\u_1\\\\u_2 \\\\\n",
    "\\vdots \n",
    "\\end{pmatrix} \\right| \\leq \\begin{pmatrix} 1 \\\\ -C & 1 \\\\ -C r^{-1} & -C & 2 \\\\ -C r^{-2} & -C r^{-1} & -C & 3 \\\\ -C r^{-3} & -C r^{-2} & -C r^{-1} & -C & 4 \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots & \\ddots \\end{pmatrix}^{-1}\\begin{pmatrix} 1 \\\\ 0 \\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "That is, we can bound $|u_k| < w_k$ where $w_k$ solves\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\\\ -C & 1 \\\\ -C r^{-1} & -C & 2 \\\\ -C r^{-2} & -C r^{-1} & -C & 3 \\\\ -C r^{-3} & -C r^{-2} & -C r^{-1} & -C & 4 \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots & \\ddots \\end{pmatrix}\\begin{pmatrix}w_0 \\\\ w_1 \\\\\\vdots \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "This can be observed exactly when we note that this is the ODE with $\\tilde a(z)$ defined as\n",
    "$$\n",
    "\\tilde a(z) = C\\sum_{k=0} r^{-k} z^k = {C r \\over r-z}\n",
    "$$\n",
    "This motivates multiplying the equation by $z-r$, or in coefficient space, by:\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 \\\\\n",
    "-1 & r \\\\\n",
    "&-1 & r \\\\\n",
    "&&\\ddots & \\ddots\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "which simplifies things:\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\\\ -1-Cr & r \\\\  & -1-Cr & 2r \\\\  & & -2-Cr & 3r \\\\  & &  & -3-Cr & 4r \\\\  &&&&\\ddots & \\ddots \\end{pmatrix}\\begin{pmatrix}w_0 \\\\ w_1 \\\\\\vdots \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ -1 \\\\0 \\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "Therefore we have \n",
    "$$\n",
    "    w_k = r^{-1}(1 + C r/k) w_{k-1} = r^{-2}(1 + C r/k)(1 + C r/(k-1)) w_{k-2} = \\cdots = r^{-k}(1 + C r/k) \\cdots (1 + C r) w_1\n",
    "$$    \n",
    "With a bit of work, it can be shown that the product is uniformly bounded, giving us $O(r^{-k})$ decay.\n",
    "\n",
    "⬛️\n",
    "\n",
    "\n",
    "**Remark** This proof can be adapted to the vector-valued case, which gives the equivalent result for\n",
    "$$\n",
    "u''(z) + a(z) u'(z) + b(z) u(z) = 0\n",
    "$$\n",
    "that the radius of convergence is the smaller of radius convergence.\n",
    "\n",
    "\n",
    "### Analytic continuation\n",
    "\n",
    "\n",
    "We know $u$ is analytic around $z_0$ with a non-zero radius. But given a curve, we can re-expand around another point inside the radius of convergence of the first point, to get analyticity in another circle:"
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      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "γ = Arc(0.,1., (0,π))\n",
    "\n",
    "p = plot(γ; label=\"contour\")\n",
    "scatter!([0.],[0.]; label=\"singularity of a\")\n",
    "r = 0.5\n",
    "for k = linspace(0.,π,10)\n",
    "    plot!(Circle(exp(im*k),r); color=:green)\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this sense, provided $a$ is analytic in a neighbourhood of $\\gamma$, $u$ can be analytically continued along $\\gamma$. Note as soon as this analytic continuation wraps back to itself we have no guarantee."
   ]
  }
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