{ "cells": [ { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations\n", "gr();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M3M6: Methods of Mathematical Physics\n", "\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", "Dr. Sheehan Olver\n", "<br>\n", "s.olver@imperial.ac.uk\n", "\n", "Office Hours: 3-4pm Mondays, Huxley 6M40\n", "<br>\n", "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n", "\n", "\n", "\n", "\n", "# Lecture 9: Cauchy transforms\n", "\n", "1. Cauchy transforms on the interval\n", " - Plemelj's theorem\n", "2. Hilbert transform\n", " \n", "Consider a function $\\phi(z)$ analytic in $\\bar\\C \\backslash [-1,1]$ such that $\\phi(\\infty) = 0$. Surrounding it by a positively oriented simple closed contour $\\Gamma$ (e.g., an ellipse) we have from Cauchy's exterior integral formula\n", "$$\n", "\\phi(z) = -{1 \\over 2 \\pi \\I} \\oint_\\Gamma {\\phi(\\zeta) \\over \\zeta - z} \\D\\zeta\n", "$$\n", "Assuming the limits \n", "$$\n", "\\phi^\\pm(x) = \\lim_{\\epsilon \\rightarrow 0} \\phi(x \\pm \\I \\epsilon)\n", "$$\n", "are \"nice\", we can deform $\\Gamma$ to be on the contour itself, giving \n", "$$\n", "\\phi(z) = {1 \\over 2 \\pi \\I} \\int_{-1}^1 {\\phi^+(x) - \\phi^-(x) \\over x - z} \\dx\n", "$$\n", "This lecture studies the relationship between the subtractive jump $\\phi^+(x) - \\phi^-(x)$ and $\\phi(z)$. \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Cauchy transform on the interval\n", "\n", "\n", "We can consistently construct a function with a prescribed subtractive jump using the _Cauchy transform_, which really is just Cauchy's integral formula but where we know nothing about $f(\\zeta)$: we don't assume it is analytic.\n", "\n", "**Definition (Cauchy transform)** \n", "$$\n", "\\CC_\\gamma f(z) := {1 \\over 2 \\pi \\I} \\int_\\gamma {f(\\zeta) \\over \\zeta - z} \\D\\zeta\n", "$$\n", "\n", "\n", "We focus on the case of an interval $[a,b]$:\n", "$$\n", "\\CC_{[a,b]} f(z) := {1 \\over 2 \\pi \\I} \\int_a^b {f(x) \\over x - z} \\dx\n", "$$\n", "Here is a phase portrait of the Cauchy transform of a simple function:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "400×400 Array{RGB{Float64},2} with eltype RGB{Float64}:\n", " RGB{Float64}(1.0,0.0,0.771285) … RGB{Float64}(1.0,0.691152,0.0)\n", " RGB{Float64}(1.0,0.0,0.771285) RGB{Float64}(1.0,0.701169,0.0)\n", " RGB{Float64}(1.0,0.0,0.781302) RGB{Float64}(1.0,0.701169,0.0)\n", " RGB{Float64}(1.0,0.0,0.781302) RGB{Float64}(1.0,0.701169,0.0)\n", " RGB{Float64}(1.0,0.0,0.781302) RGB{Float64}(1.0,0.701169,0.0)\n", " RGB{Float64}(1.0,0.0,0.781302) … RGB{Float64}(1.0,0.701169,0.0)\n", " RGB{Float64}(1.0,0.0,0.781302) RGB{Float64}(1.0,0.711185,0.0)\n", " RGB{Float64}(1.0,0.0,0.791319) RGB{Float64}(1.0,0.711185,0.0)\n", " RGB{Float64}(1.0,0.0,0.791319) RGB{Float64}(1.0,0.711185,0.0)\n", " RGB{Float64}(1.0,0.0,0.791319) RGB{Float64}(1.0,0.711185,0.0)\n", " RGB{Float64}(1.0,0.0,0.791319) … RGB{Float64}(1.0,0.721202,0.0)\n", " RGB{Float64}(1.0,0.0,0.801336) RGB{Float64}(1.0,0.721202,0.0)\n", " RGB{Float64}(1.0,0.0,0.801336) RGB{Float64}(1.0,0.721202,0.0)\n", " ⋮ ⋱ \n", " RGB{Float64}(0.0,0.193656,1.0) RGB{Float64}(0.0,1.0,0.27379) \n", " RGB{Float64}(0.0,0.203673,1.0) RGB{Float64}(0.0,1.0,0.27379) \n", " RGB{Float64}(0.0,0.203673,1.0) … RGB{Float64}(0.0,1.0,0.283806)\n", " RGB{Float64}(0.0,0.203673,1.0) RGB{Float64}(0.0,1.0,0.283806)\n", " RGB{Float64}(0.0,0.203673,1.0) RGB{Float64}(0.0,1.0,0.283806)\n", " RGB{Float64}(0.0,0.213689,1.0) RGB{Float64}(0.0,1.0,0.283806)\n", " RGB{Float64}(0.0,0.213689,1.0) RGB{Float64}(0.0,1.0,0.293823)\n", " RGB{Float64}(0.0,0.213689,1.0) … RGB{Float64}(0.0,1.0,0.293823)\n", " RGB{Float64}(0.0,0.213689,1.0) RGB{Float64}(0.0,1.0,0.293823)\n", " RGB{Float64}(0.0,0.213689,1.0) RGB{Float64}(0.0,1.0,0.293823)\n", " RGB{Float64}(0.0,0.223706,1.0) RGB{Float64}(0.0,1.0,0.293823)\n", " RGB{Float64}(0.0,0.223706,1.0) RGB{Float64}(0.0,1.0,0.30384) " ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = Fun(-1 .. 1)\n", "f = exp(x)*sqrt(1-x^2)\n", "portrait(-3..3, -3..3, z -> cauchy(f,z))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Remark** As an aside, these integrals are computationally difficult because of the singularity in the integrand, hence standard integration methods become slow as $z$ approaches the interval. There are other specialised routines (as implemented in `cauchy(f,z)`) that are much more efficient:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 0.000018 seconds (13 allocations: 640 bytes)\n", " 0.013608 seconds (287 allocations: 12.823 MiB)\n" ] }, { "data": { "text/plain": [ "0.5493442175548764 - 0.2181261671282511im" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = 0.1 +0.001im\n", "@time cauchy(f, z )\n", "@time sum(f/(x-z))/(2π*im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It turns out that the Cauchy transform has a very simple subtractive jump. Here we denote\n", "$$\n", " \\CC_{[a,b]}^+ f(x) = \\lim_{\\epsilon \\rightarrow 0} \\CC_{[a,b]} f( x+ \\I \\epsilon)\\\\\n", " \\CC_{[a,b]}^- f(x) = \\lim_{\\epsilon \\rightarrow 0} \\CC_{[a,b]} f( x- \\I \\epsilon)\n", "$$\n", "\n", "\n", "**Theorem (Plemelj on the interval I)**\n", "Suppose $(b-x)^\\alpha (x-a)^\\beta f(x)$ is differentiable on $[a,b]$, for $\\alpha, \\beta < 1$. Then the Cauchy transform has the following properties:\n", "1. $\\CC_{[a,b]} f(z)$ is analytic in $\\bar \\C \\backslash [a,b]$\n", "3. $\\CC_{[a,b]} f(\\infty) = 0$\n", "4. It has the subtractive jump:\n", "$$\n", "\\CC_{[a,b]}^+ f(x) - \\CC_{[a,b]}^- f(x) = f(x) \\qqfor a < x < b\n", "$$\n", "2. $\\CC_{[a,b]} f(z)$ has weaker than pole singularities at $a$ and $b$\n", "\n", "_Demonstration_ We can evaluate the Cauchy transform using `cauchy`, including the limit from above and below. Here we see numerically that we recover $f$ from taking the difference:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "cauchy(f, 0.1+0.0im)-cauchy(f, 0.1-0.0im) , f(0.1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Sketch of Proof** \n", "We show the proof for $[-1,1]$.\n", "\n", "1. From the dominated convergence theorem, we know that $\\CC f(z)$ is complex-differentiable off $[-1,1]$:\n", "$$\n", "{\\D \\over \\D z} {\\cal C} f(z) = {1 \\over 2 \\pi \\I}\\int_{-1}^1 {\\D \\over \\D z} {f(x) \\over x - z} \\dx = {1 \\over 2 \\pi \\I}\\int_{-1}^1 {f(x) \\over (x - z)^2} \\dx\n", "$$\n", "We know it is analytic at $\\infty$ because\n", "$$\n", "{\\cal C} f(z^{-1}) = z {1 \\over 2 \\pi \\I}\\int_{-1}^1 {f(x) \\over z x - 1} \\dx\n", "$$\n", "is differentiable at zero. \n", "2. ${\\cal C} f(\\infty) = 0$ follows from uniform convergence of $1 \\over z - x$ to zero as $z \\rightarrow \\infty$.\n", "3. For the constant function, which is analytic, this follows by considering a contour $\\gamma_x^+$ perturbed above $x$ and $\\gamma_x^-$ perturbed below $x$, see plots below. Therefore, by Cauchy integral formula we have\n", "$$\n", "\\CC^+ 1 (x) - \\CC^- 1 (x) = {1 \\over 2 \\pi \\I}\\int_{\\gamma_x^+} {1 \\over x - z} \\dx - {1 \\over 2 \\pi \\I}\\int_{\\gamma_x^-} {1 \\over x - z} \\dx = {1 \\over 2 \\pi \\I} \\oint {1 \\over x -z } \\dx = 1.\n", "$$\n", "For other functions, we consider, for $z = x + \\I \\epsilon$,\n", "$$\n", " {\\cal C} f(z) = {1 \\over 2 \\pi \\I} \\int_{-1}^1 {f(t) - f(x) \\over t - z} \\dt + f(x) \\CC 1(z)\n", "$$\n", "For $\\epsilon = 0$, the first integral exists because the singularity at $t = x$ is removable: \n", "$$\n", "\\lim_{t \\rightarrow x} {f(t) - f(x) \\over t - x} = f'(x)\n", "$$\n", "We leave it as an excercise (or see [Trogdon & Olver 2015, Lemma 2.7]) to show that $\\int_{-1}^1 {f(t) - f(x) \\over t - z} \\dt$ converges to $\\int_{-1}^1 {f(t) - f(x) \\over t - x} \\dt$ as $z \\rightarrow x$. It follows that\n", "$$\n", "{\\cal C}^\\pm f(x) = {1 \\over 2 \\pi \\I} \\int_{-1}^1 {f(t) - f(x) \\over t - x} \\dt + f(x) \\CC^\\pm 1(x)\n", "$$\n", "and in particular\n", "$$\n", "{\\cal C}^+ f(x) - {\\cal C}^- f(x) = f(x) (\\CC^+ 1(x) - \\CC^- 1(x)) = f(x)\n", "$$\n", "4. We show that it has a weaker than pole singularity at $+1$, with $-1$ following by the same argument. First note that $f$ is absolutely integrable. If we assume we approach $1$ at an angle of $ -\\pi + \\delta \\leq \\theta \\leq \\pi - \\delta$, the uniform convergence of $(z - 1) \\CC f(z)$ to zero follows from observing that ${z-1 \\over z - t}$ can be made arbitrarily small in a larger and larger interval. This is easiest to see for real $x > 1$, where for\n", " $1 \\leq x \\leq 1 + \\epsilon^2$ we have\n", "$$\n", "\\left|{x -1 \\over x-t }\\right| \\leq \\epsilon\n", "$$\n", "for all $t \\leq 1 + \\epsilon^2 - \\epsilon$, or more generously, $t \\leq 1 - \\epsilon$. Therefore,\n", "$$\n", "| (x-1) {\\cal C} f(x) | \\leq {1 \\over 2 \\pi}\\int_{-1}^{1-\\epsilon} |f(t) | \\left|{x -1 \\over x-t }\\right| \\dt + \\int_{1-\\epsilon}^1 |f(t) | \\dt \\leq\n", "\\epsilon \\int_{-1}^1 |f(t)| \\dt + \\int_{1-\\epsilon}^1 |f(t) | \\dt\n", "$$\n", "Both terms tends to zero as $\\epsilon \\rightarrow 0$, hence so does $| (x-1) {\\cal C} f(x) |$. To extend this to the interval itself (that is, $\\delta = 0$), we use the stronger requirement that $(1-x)^\\alpha(1+x)^\\beta f(x)$ is differentiable. For $\\alpha = \\beta = 0$, this follows from the expression in condition (3) and the fact that (found via direct integration)\n", "$$\n", " \\CC 1(z) = {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I}\n", "$$ \n", "has only logarithmic singularities, and $f(x)$ is bounded.\n", "\n", "⬛️\n", "\n", "\n", "Here is a plot of ${x - 1 \\over x- t}$ showing that it is small on an increasing portion of the interval as $x \\rightarrow 1$ from the right:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "x = 1 + 0.01\n", "tt = linspace(-1.,1.,1000)\n", "plot(tt, abs.((x - 1) ./ (x .- tt)); legend=false)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is a plot of $\\gamma_x^\\pm$:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "x = 0.4\n", "r = 0.1\n", "tt = linspace(π,0.,100)\n", "plot([-1.; x + r*cos.(tt);1.0], [0.; r*sin.(tt); 0.0];ylims=(-0.5,0.5),label=\"g_x^+\")\n", "plot!([-1.; x + r*cos.(tt);1.0], [0.; -r*sin.(tt); 0.0];ylims=(-0.5,0.5),label=\"g_x^-\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use the results of the previous results to show that it is in fact unique.\n", "\n", "\n", "**Theorem (Plemelj on the interval II)** Suppose $\\phi(z)$ satsfies the following properties:\n", "1. $\\phi(z)$ is analytic in $\\bar \\C \\backslash [a,b]$\n", "2. $\\phi(z)$ has weaker than pole singularities at $a$ and $b$\n", "3. $\\phi(\\infty) = 0$\n", "4. It has the subtractive jump:\n", "$$\n", "\\phi^+(x) - \\phi^-(x) = f(x) \\qqfor a < x < b\n", "$$\n", "where $(b-x)^\\alpha (x-a)^\\beta f(x)$ is differentiable in $[a,b]$ for $\\alpha,\\beta < 1$. \n", "\n", "Then $\\phi(z) = {\\cal C}_{[a,b]} f(z)$.\n", "\n", "**Sketch of Proof**\n", "Consider \n", "$$A(z) = \\phi(z) - {\\cal C}_{[a,b]} f(z)$$\n", "This is continuous (hence analytic) on $(a,b)$ as\n", "$$A^+(x) - A^-(x) = \\phi^+(x) -\\phi^-(x) - {\\cal C}_{[a,b]}^+ f(x) + {\\cal C}_{[a,b]}^- f(x) =f(x) - f(x) = 0$$\n", "Also, $A$ has weaker than pole singularities at $a$ and $b$, hence is analytic there as well: it's entire. Only entire functions that are bounded are constant, since it vanishes at $\\infty$ the constant must be zero.\n", "\n", "⬛️\n", "\n", "\n", "_Example 1_ We can use this theorem to prove the following relationships (using $\\diamond$ for the dummy variable):\n", "\n", "$$\n", "{1 \\over \\sqrt{z-1} \\sqrt{z+1}} = -2 \\I {\\cal C}\\left[{1 \\over \\sqrt{1-\\diamond^2}}\\right](z) = -{1 \\over \\pi}\\int_{-1}^1 {\\dx \\over \\sqrt{1-x^2} (x-z)}\n", "$$\n", "(1) follows because the jumps cancel. (2 and 3) are immediate. (4) follows from a simple calculation. Here we show that it has the correct jump:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "x = Fun()\n", "z = 2 +2im\n", "1/(sqrt(z-1)sqrt(z+1)),-2im*cauchy(1/sqrt(1-x^2),z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "_Example 2_ Now consider a problem of reducing \n", "$$\n", "\\phi(z) = \\sqrt{z-1} \\sqrt{z+1}\n", "$$ \n", "to its behaviour near its singularities. It has two singularities: it blows up at $\\infty$ and has a branch cut on $[-1,1]$\n", "\n", "\n", "We can subtract out the singularity at infinity first to determine\n", "\n", "$$\\phi(z) = z + 2 \\I {\\cal C}[\\sqrt{1-\\diamond^2}](z)$$\n", "\n", "Note this works because, as $z \\rightarrow \\infty$, we have \n", "$$\n", "\\phi(z) = z (\\sqrt{1-{1/z}}\\sqrt{1 + {1/z}}) = z (1 + O({1/z}))(1+O(1/z)) = z + O(1/z)\n", "$$\n", "hence $\\phi(z) - z$ vanishes at the origin. This is an example of summing over the behaviour at each singularity to recover the function (in this case, $\\phi$ has a singularity along the cut $[-1,1]$ and polynomial growth at $\\infty$). \n", "\n", "Because $\\phi(z)-z$ decays, we can now deploy Plemelj II to determine:\n", "$$\n", "\\phi(z) -z = \\CC[\\phi_+-\\phi_-](z)\n", "$$\n", "where\n", "$$\n", "\\phi_+(x) - \\phi_-(x) = 2\\I \\sqrt{1-x^2}\n", "$$\n", "Here we see that our derived expression matches $phi(z)$:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "sqrt(z-1)sqrt(z+1), z +2im*cauchy(sqrt(1-x^2),z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "_Example 3_ Finally, we have the following (also verifiable using indefinite integration):\n", "\n", "$${\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I} = {\\cal C}[1](z) = {1 \\over 2 \\pi \\I} \\int_{-1}^1 {\\dx \\over x -z} $$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "(log(z-1)-log(z+1))/(2π*im),cauchy(Fun(one(x)),z)" ] } ], "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 }