From owner-chemistry@ccl.net Fri Nov 4 10:22:00 2011 From: "Jozsef Csontos jcsontos.lists##gmail.com" To: CCL Subject: CCL: Question Szabo/Ostlund Ex.1.14 Message-Id: <-45824-111104062730-30024-AmL0+hymCLtlUNXMoHsyGA]~[server.ccl.net> X-Original-From: Jozsef Csontos Content-Type: multipart/alternative; boundary="------------080804090000070601020807" Date: Fri, 04 Nov 2011 11:27:18 +0100 MIME-Version: 1.0 Sent to CCL by: Jozsef Csontos [jcsontos.lists.++.gmail.com] This is a multi-part message in MIME format. --------------080804090000070601020807 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Dear Martin, Please compile the followings with latex, \documentclass[a4paper,10pt]{article} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{fontenc} \usepackage{graphicx} \usepackage[dvips]{hyperref} \author{JCsontos} \title{1.14 Szabo-Ostlund} \date{11/04/11} \begin{document} \begin{align*} \int_{-\infty}^{\ \infty}{a(x) \cdot \delta(x)}\ dx & = \int_{-\infty}^{-\epsilon}{a(x) \cdot 0}\ dx + \int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \delta_{\epsilon}(x)}\ dx + \int_{\ \epsilon}^{\ \infty}{a(x)\cdot 0}\ dx \\ & = \int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \delta_{\epsilon}(x)}\ dx \\ & = \int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \frac{1}{2\epsilon}}\ dx \\ & = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_{-\epsilon}^{\ \epsilon}{a(x)}\ dx\\ & = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} {[\epsilon-(-\epsilon)]}{a(\xi)}, \ \xi \in (-\epsilon, \epsilon) ;\ \textbf{mean value theorem for integration} \end{align*} \begin{align*} \text{if $\epsilon \to 0$ then $\xi \to 0$ and } \lim_{\epsilon \to 0} \frac{1}{2\epsilon} {[\epsilon-(-\epsilon)]}{a(\xi)} & = \frac{2\epsilon}{2\epsilon}{a(0)} \end{align*} \end{document} Best, Jozsef On 11/03/2011 04:56 PM, Martin Hediger ma.hed-x-bluewin.ch wrote: > Sent to CCL by: "Martin Hediger" [ma.hed^^bluewin.ch] > Dear List > I also posted this question to the google.gamess user group, but was encouraged to do so here. I'm new to the CCL and looking forward to interesting discussions. > > I was trying to solve ex. 1.14 from Szabo/Ostlund. If > > d(x) = \lim_{eps->\infinity} d_eps(x), > > where > > d_eps(x) = 1/(2eps) when -eps<= x<= eps and d_eps(x)=0 else, > > we are supposed to show that > the value of a function a(0) = \int dx a(x) d(x) > > My Ansatz was to insert the definition into the integral, but then I'm > not exactly seeing how to treat the limit function. > How does one show this? > > Thank you for any suggestions. > Martin> > > --------------080804090000070601020807 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: 8bit Dear Martin,

Please compile the followings with latex,

\documentclass[a4paper,10pt]{article}

\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{fontenc}
\usepackage{graphicx}

\usepackage[dvips]{hyperref}

\author{JCsontos}
\title{1.14 Szabo-Ostlund}
\date{11/04/11}

\begin{document}
\begin{align*}
\int_{-\infty}^{\ \infty}{a(x) \cdot \delta(x)}\ dx & =
\int_{-\infty}^{-\epsilon}{a(x) \cdot 0}\ dx +
\int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \delta_{\epsilon}(x)}\ dx +
\int_{\ \epsilon}^{\ \infty}{a(x)\cdot 0}\ dx \\
& = \int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \delta_{\epsilon}(x)}\ dx \\
& = \int_{-\epsilon}^{\ \epsilon}{a(x) \cdot \frac{1}{2\epsilon}}\ dx \\
& = \lim_{\epsilon \to 0} \frac{1}{2\epsilon} \int_{-\epsilon}^{\ \epsilon}{a(x)}\ dx\\
& = \lim_{\epsilon \to 0} \frac{1}{2\epsilon}
{[\epsilon-(-\epsilon)]}{a(\xi)}, \ \xi \in (-\epsilon, \epsilon)
;\ \textbf{mean value theorem for integration}
\end{align*}

\begin{align*}
\text{if $\epsilon \to 0$ then $\xi \to 0$ and }
\lim_{\epsilon \to 0} \frac{1}{2\epsilon}
{[\epsilon-(-\epsilon)]}{a(\xi)} & =
\frac{2\epsilon}{2\epsilon}{a(0)}
\end{align*}

\end{document}

Best,
Jozsef

On 11/03/2011 04:56 PM, Martin Hediger ma.hed-x-bluewin.ch wrote:

Sent to CCL by: "Martin  Hediger" [ma.hed^^bluewin.ch]
Dear List
I also posted this question to the google.gamess user group, but was encouraged to do so here. I'm new to the CCL and looking forward to interesting discussions.

I was trying to solve ex. 1.14 from Szabo/Ostlund. If

d(x) = \lim_{eps->\infinity} d_eps(x),

where

d_eps(x) = 1/(2eps) when -eps <= x <= eps and d_eps(x)=0 else,

we are supposed to show that
the value of a function a(0) = \int dx a(x) d(x)

My Ansatz was to insert the definition into the integral, but then I'm
not exactly seeing how to treat the limit function.
How does one show this?

Thank you for any suggestions.
MartinE-mail to subscribers: CHEMISTRY++ccl.net or use:
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