In Stein's Complex Variables I'm having trouble verifying the conjecture in $(2.)$  $(1.)$  If $f$ is a complex-valued, on $\partial D$ and if it's absolute value $|f(z)|$ is bounded by a constant $M$ for all z on $\partial D$.  $$\left|\int_{\partial D} f(z)dz \right| \leq \max:=|f(z)|\int_{a}^{b} \gamma'(t)$$  $$\left|\int_{\partial D} f(z)\right | \leq \max_{z \, \in \, \partial D}\left|f(z) \right |\int_{a}^{b} \gamma'(t)$$  $$$$  $(2.)$  $$\int_{-\infty}^{\infty} \frac{e^{ix}-1}{x}dx = \frac{1}{2i}\int_{-\infty}^{\infty}\frac{e^{ix}-1}{x}dx = \frac{\pi}{2}$$  My initial task began within the given hint in $(1.)$, I focused on the integral on the RHS side, since it would be easier to define and transform the integral on our Semicircular Contour denoted by $\partial D$.  Lemma$(1.)$:  From $(2.)$ I observed that the integral now in $\mathbb{C}$ becomes the following:  $$\oint_{\partial D}\frac{e^{iz}-1}{z}dz$$  Remark: $\partial{D}$ denotes the boundary of our contour which can be depicted in the following image :   Lemma$(2)$  From Lemma(), our integral can be parametrized via the definition in $(1.3)$ as follows:  $$\frac{1}{2i}\oint_{\partial D}\frac{e^{rei\theta}-1}{re^{\theta}}iRe^{i\theta}d\theta, \, (0 \leq \theta \leq \pi)$$  $$\left|\frac{1}{2i}\oint_{\partial D} \frac{e^{ir^{ei\theta}}-1}{re^{i\theta}}\right | \leq \max_{z \, \in \, \partial D}\left|\frac{e^{ire^{i\theta}-1}}{re^{i\theta}} \right |\int_{a}^{b} \gamma'\left(\frac{e^{ire^{i\theta}-1}}{re^{i\theta}}\right)$$  One can construct the upper and lower bounds via the reverse triangle inequality as follows: $$|R(e^{ie\theta})-1|| \geq ||r(e^{ie\theta})-1|| ||R(e^{ie\theta})-1||$$ $$|R(e^{iei\theta})-1| \geq ||r(e^{(rei\theta)}-1|| R(e^{iei\theta})-1||$$     From this point in the problem i'm having troubles, on what's the next move to make would one take the $$\lim_{r \rightarrow \infty}\max_{\partial D}$$ and then calculate residues and apply the residue theorem ?
So I integrate the holomorphic function $\frac{e^{iz} -1}{iz}$ over the half disk in the upper half plane, let me name it $\Gamma$. By using Cauchy's theorem, it equals 0.  $$0 = \int_{\Gamma} \frac{e^{iz} -1}{iz} = \int_{0}^{\pi}(e^{iRe^{i\theta}}-1)d\theta + \int_{-R}^{R} \frac{e^{it}-1}{it}dt$$  ($Re^{[0, \pi]}$ and $t: [-R, R]$ being the respective parametrizations.)  By splitting the left integral in two, I get $\int_{0}^{\pi} e^{iRe^{i\theta}}d\theta \rightarrow 0$ by dominated convergence, and $\int_{0}^{\pi} -1d\theta = -\pi$. By making $R \rightarrow \infty$, I'm left with  $$\pi = \int_{-\infty}^{\infty} \frac{e^{it} - 1}{it}dt = \int_{-\infty}^{\infty}\frac{\sin t}{t}dt + \int_{-\infty}^{\infty}\frac{\cos t -1}{it}dt.$$  How do I get rid of the integral on the right? I welcome all feedback.
Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$? I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!  Thanks! My daughter bought a ticket for a flight with a layover in Emirates (from Dallas, Texas to India). She had a stop of 48 hours in Dubai. When she was checking in in Dubai for Delhi, she was informed that her visa for India has expired in 2012. Did Emirates not have an obligation to check the visa for India before selling a ticket? Why do first years cross the lake to get to the castle? What I get:  host:~ user$ cat example.txt some texthost:~ stas$   What I want to get:  host:~ user$ cat example.txt some text host:~ stas$   Is there a way I can make cat behave like this?  I'm using bash on Mac OS X. I am running vsftpd as ftp server on my linux (rasbian), I log in to the machine as a root user.  I would like to be still locked to using only /var/www, how can I configure vsftpd conf to accomplish it? Table of Contents for set of PDF files How to make Table of Contents for a set of PDF files?  Here's my code!  \documentclass{book} \usepackage{pdfpages} \usepackage[pdfauthor={&lt;somename&gt;},pdftitle={&lt;sometitle&gt;},pdftex]{hyperref} \usepackage{fancyhdr} \usepackage{tocloft} \usepackage[margin=0.5in,bottom=0.75in,top=0.75in]{geometry}   \pagestyle{fancy} \renewcommand{\headrulewidth}{0.4pt} \fancyhead[L]{X Y Z} \fancyhead[R]{} \begin{document}  \tableofcontents \newpage  \clearpage \phantomsection \includepdfset{pages=-,pagecommand=\thispagestyle{fancy}} \addcontentsline{toc}{chapter}{SAMPLE PDF 1\\ \textit{description}} \includepdf[pages=-]{template1.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 2\\ \textit{description}} \includepdf[pages=-]{template2.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 3\\ \textit{description}} \includepdf[pages=-]{template3.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 4\\ \textit{description}} \includepdf[pages=-]{template4.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 5\\ \textit{description}} \includepdf[pages=-]{template5.pdf}  \end{document}   Here's the output.    My problem is that the page number is not aligned with the title, it is aligned with the description. Also I'd like to put dots.   My desired output looks like this: According to :     By looking for trades that take place in between the bid and ask, you   can tell when a strong trend is about to come to an end. This is   because these trades are often placed by large traders who take a   small loss in order to make sure that they get out of the stock in   time.   A trader when buying needs to buy at the ask price and when selling needs to sell at the bid price. So how can a trade happen 'in between' the bid and ask? Is there evidence of dark matter in our galaxy? Is there evidence of dark matter in our galaxy?  How can we measure this, say, how many percent of the center of our galaxy is dark matter?  I did not find the answer in the question Language Options -> Spell Checking in Windows 10 version 1803 I have installed Apache(LAMP) web server on my ubuntu machine and I want to do the above mentioned thing.I have tried creating symbolic links in the /var/www directory but it only shows files but not directories inside the directory linked.I want to redirect /var/www to my home folder so that I can access all the files and directories contained therein.Please help me. "Dare" with and without "to" To my surprise, there's a missing question about this particularly interesting verb, dare. All I know about it is the fact it can be in two forms, as an auxiliary (without to: "I dare not mention their names") and main (with to: "Did anyone dare to admit it?") verb and the difference has something to do with agreeing with subject. Can somebody explain in a greater detail? Horizontal enumeration in multiple columns Consider the following minimal example:  \documentclass{article}  \setlength{\parindent}{0mm}  \usepackage{paralist} \usepackage{tabto}  \begin{document}  \NumTabs{3} \begin{inparaenum} \item text \tab\item text \tab\item text \tab\item text \tab\item text \tab\item text    \end{inparaenum}  \end{document}   Which results in:    It does exactly what I want, but I was wondering whether it was possible to create an environment, say tabbedEnum so that we can eliminate the use of the \tab, in particular, that the first case doesn't need a tab but all others do. Something of the form:   \begin{tabbedEnum}[3] \item text \item text \item text \item text \item text \item text \end{tabbedEnum} they were question banned; it turned out it (in part) it was because of deleted questions.   was deleted by the Community user.  I'm aware it will delete old posts that have a negative score, but for a minute or two, I was really puzzled as to why it was unilaterally deleted without some notation left.  Can we have the Community user leave a notation in a post's history as to why the post was deleted?    Something like the following would suffice:     Post deleted automatically due to having negative votes and no activity for three months How can I add a second hard drive to a previously configured UEFI/ACHI Windows 8 machine? Recently purchased a new Windows 8 PC.  It came with one hard drive.  I want to a second hard drive to it.  This second hard drive is my data hard drive from my previous computer.  However, I have run into issues when the system accesses it.  The drive is found in the BIOS. But is not seen by Explorer or Disk Management.  I have added the drive to the next available SATA slot:  SATA 2.  The machine is a UEFI/ACHI based machine.  In my reading I have found people documenting the following:  1) adding multiple partitioned hard drives (like mine is) to UEFI based machines is not  possible 2) I have seen it suggested that you can only add blank hard drives to UEFI based machines.  However, in doing so, I did not have success.  I tried to add it as a hard drive with unallocated space and then as a hard drive with a single simple partition.  Both attempts failed.  My ultimate question:  What is the proper procedure for adding a second hard drive to a UEFI/ACHI machine?  I do not want to reinstall the OS and start from scratch as I have seen suggested elsewhere.  There has to be a way to accomplish this without all that hassle.  Thanks in advance for your help. Understanding etoolbox \ifdefempty and \ifdefvoid I am trying to understand the  \ifdefempty and \ifdefvoid macros. The documentation says that \ifdefempty     Expands to true if the control sequence is defined and is a   parameterless macro whose replacement text is empty, and to false   otherwise.   Similarly \ifdefvoid     Expands to true if the control sequence is undefined, if is a macro   whose meaning is \relax, or a parameterless macro whose replacement   text is empty, and to false otherwise.   From these descriptions I thought that \def\myempty{}, \def\myrelax{\relax}, and \def\myemptyempty{\myempty} would all be both "empty" and "void", but \def\myempty{} is the only one that is either "empty" or "void". What am I missing?  \documentclass{minimal} \usepackage{etoolbox}  \begin{document}  \def\myempty{} \def\myrelax{\relax} \def\myemptyempty{\myempty}   \ifdefempty{\myempty}{Empty}{Not empty}  \ifdefempty{\myrelax}{Empty}{Not empty}  \ifdefempty{\myemptyempty}{Empty}{Not empty}  \ifdefvoid{\myempty}{Void}{Not void}  \ifdefvoid{\myrelax}{Void}{Not void}  \ifdefvoid{\myemptyempty}{Void}{Not void}  \end{document} Last activity name still missing from migrated question According to this post  this should be fixed. But it happened again.  See Image below (link to )  There was also another migrated question () in the list where the name was missing, but it was edited on meta. It seems that panel shadow is somehow hardcoded in KDE 4 and devs do not have any plans to change it and/or make it customizable. So, what I'm actually asking is some kind of hack (if there is any) which will remove the shadow of the panel in KDE 4.x completely (but without need to disable all desktop effects in the process).  Possible application: When a user wants a completely transparent panel, no top/bottom borders at all, the shadow is completely out of place and destroys the experience. Sublime Text 3 System Package Location? If a derivative of a continuous function has a limit, must it agree with that limit? Combine Two SQL queries in one statement