Problem set 9

Problem 1¶

Prove $\\frac{1}{4\\pi}\\Delta [\\log(x^2 + y^2)] = \\delta$ using three methods.

Use Cauchy kernel:

(1) Show that this is equivalent to $\\frac{1}{\\pi}\\frac{\\partial}{\\partial z}\\frac{\\partial}{\\partial \\bar{z}}[\\log|z|^2] = \\delta$ on $\\mathbb{C}$ and use the Cauchy kernel $\\frac{\\partial}{\\partial \\bar{z}}[\\frac{1}{\\pi}\\frac{1}{z}] = \\delta$ to prove it.

Directly verify $\\frac{1}{4\\pi}\\int\_{\\mathbb{R}^2} \\log(x^2+y^2) \\Delta \\varphi; dxdy = \\varphi(0)$ for every $\\varphi \\in \\mathscr{C}\_c^{\\infty}(\\mathbb{R}^2)$:

(2) Use the Laplacian operator on polar coordinate.

(3) Use the Gauss-Green formula from previous exercise.

Problem 3¶

Let $n$ be the invards normal of $\\partial D$. Define the normal derivative $\\partial_n u$ to be $n_1\\partial_x u + n_2 \\partial_y u$ (which is just the directional derivative to the direction $n$). Prove the Green identity $\\int (u\\Delta v - v\\Delta u)dxdy = -\\int\_{\\partial D} (u\\partial_n v - v\\partial_n u)ds$.

Problem 4¶

Prove that $e^{x}\\cos(e^x)$ is a tempered distribution.

Problem 5¶

Let $f\\in L^2(\\mathbb{R})$ which is supported on $[0,+\\infty)$.

(1) Let $\\mathcal Ff(z) = \\int\_{-\\infty}^{+\\infty}f(t)e^{-2\\pi it (x+iy)}dt$. Show that $\\mathcal Ff(z)$ is a holomorphic function on the lower half plane, i.e. on ${x+iy\\in \\mathbb{C}:y<0}$.

(2) Show that $\\int\_{-\\infty}^{+\\infty}|\\mathcal Ff(x+iy) - \\mathcal Ff(x)|^2 dx \\to 0$ when $y\\to 0^-$.

(3) Show that $\\int\_{-\\infty}^{+\\infty}|\\mathcal Ff(x+iy)|^2 \\leq |f|\_{L^2}^2$.

(3) For $x\\in \\mathbb{R}$, let ${x}$ be its fractional part, i.e. ${x} = x - [x]$ where $[x]$ is the largest integer $\\leq x$. Show that $F(z):=\\int\_{0}^{+\\infty}{t}e^{2\\pi i z t - \\sqrt{t}}dt$ defines a holomorphic function on the upper half plane which is $L^2$.