Integration closed contours
Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$
Gauss's mean value theorem problem Suppose that  $f(z)$ is analytic and $f(z)\neq0$  on the disc $|z|&lt;1$ .  Show that for $0&lt;r&lt;1$,  $$ \exp\left[\frac{1}{2\pi}\int_{0}^{2\pi}{\log}|f(re^{i\theta})|\,d\theta\right] = |f(0)| $$  Any other theorems needed to solve this problem? I don't have any idea just with "Gauss's MVT"  Thank you in advanced. What set of books makes a complete starter set for 4e? I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. An annulus in $\mathbb R^2$ is path connected Let $A = \{(x,y) \in\mathbb{R}^2: a \leq (x-c)^2+(y-d)^2 \leq b\}$ for given $a,b,c, d$ real numbers. I want to show that $A$ is path-connected.  How can I do that?  I know that every open subset of $\mathbb R^2$ that is connected is path connected. But this is obviously not open so I cannot use that. Then I thought of multiple cases. If we take arbitrary $x$ and $y$ and draw the line between them and they do not intersect with the circle centred at $(c,d)$ then we can obviously draw a line between the points which is still in the set, so we can then define the function. I am stuck on the other case. Identify a short story about post-apocalyptic household robots Media: Short story collection When I read it: 2011 but the story itself was much older (unsure how much) so this information only serves to clarify that it is by no means new. Plot summary: A futuristic home with household robots that help around the house (cooking, cleaning etc.), we learn about their routines and behaviour and later on it's revealed that nobody is home, in fact everyone is dead, and a description of the outside seems to imply post-nuclear apocalypse. Setting: I seem to remember snow or winter What it isn't: with Google I stumbled upon  by Ray Bradbury, and  is very similar, and the beginning made me think "Bingo!", but I do not remember there being a fire in the story I read, and I also remember a view from a briefly opened door, or window, of a snowy setting, so while it may be that I am confusing elements of another story, or my imagination, with There Will Come Soft Rains, I would still like to run this through here as a question, because I'm not convinced. Additional info: The short story collection had other short stories that I will include as I remember them, details will be fuzzy:   Meteor, by John Wyndham , by Charlotte Perkins Gilman A short story about a man who is driving a car and it is revealed in the end that he was dead the whole time , by Charles Dickens What regulations should a European driver know when driving in the US? There are the obvious ones. The US has the keep-your-lane system, where as in Europe you are expected to keep to the right (left for UK) as long as possible.  On my latest travel to the US, I was confronted with the possibility to go through a red light if you turn right, unless it is explicitly not allowed.  What should a european driver know when driving in the US? What are the do's and don'ts compared to driving in Europe. Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$ Why cannot a set be its own element? The double cone is not a surface. Count down in timeout is cut off The time out countdown which is flying in is missing half of the number.  Note the 33 here:        Can this be fixed please? So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?  When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold.  Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...).  So. Bottom line. Can I be burned by wind in some perfect scenario? Integration of exponential and square root function Redirect all requests to subdirectory except a few ip addresses How do I download a YouTube video? How do i know which wheels will fit my bike? My overall upgrade went well, but I am left with a critically lingering issue of mysql-server not being able to install itself, and nothing I am trying gets it to work.  This is the error I see when trying to install/reinstall it:  Setting up mysql-server-5.7 (5.7.11-0ubuntu6) ... Job for mysql.service failed because the control process exited with error code. See "systemctl status mysql.service" and "journalctl -xe" for details. invoke-rc.d: initscript mysql, action "start" failed. dpkg: error processing package mysql-server-5.7 (--configure):  subprocess installed post-installation script returned error exit status 1 dpkg: dependency problems prevent configuration of mysql-server:  mysql-server depends on mysql-server-5.7; however:   Package mysql-server-5.7 is not configured yet.  dpkg: error processing package mysql-server (--configure):  dependency problems - leaving unconfigured No apport report written because the error message indicates its a followup error from a previous failure.                    Processing triggers for systemd (229-4ubuntu4) ... Processing triggers for ureadahead (0.100.0-19) ... Errors were encountered while processing:  mysql-server-5.7  mysql-server E: Sub-process /usr/bin/dpkg returned an error code (1)   I've tried completely removing it, though doing so attempts to install MariaDB because of dependencies (?). Any suggestions on what I can do to fix this would be most welcome.  EDIT: Looks like I'm not the only one: I have a small utility that I use to download an MP3 file from a website on a schedule and then builds/updates a podcast XML file which I've added to iTunes.  The text processing that creates/updates the XML file is written in Python. However, I use wget inside a Windows .bat file to download the actual MP3 file. I would prefer to have the entire utility written in Python.  I struggled to find a way to actually download the file in Python, thus why I resorted to using wget.  So, how do I download the file using Python? I tried turn on in compizconfig but I does not work. Any solution or it is possible? Is Pikachu's gender ever confirmed? I was recently having a discussion about Pokémon, and it came up that none of us were sure whether Ash's Pikachu in the series was male or female. Some (myself included) always thought it was male, whereas others could've sworn it was female.  Was there ever any canon evidence to point in one direction or another? How to download a whole FTP site using command line? I need to download an FTP site (a web site source code actually) containing a very high quantity of relatively small files (downloading it with FileZilla took more than a day, but I believe it could be downloaded much faster if many files were downloaded at once). Unfortunately there is no SSH access to the site and no way to archive the files on the server side.  So at least the question is how to download a whole FTP site into a folder using command line tools. A better it'd be if the download could be parallelized by downloading many files simultaneously instead of one by one. With no good reason I had a look at my computer's storage going on the apple and About this Mac. Here is the display:    The pink bar doesn't make much sense. It obviously does not represent the 80.67GB of free storage. Two sections correspond to pink; "Photos" and "Backups". I obviously don't have 140.76TB of backups. And probably not on this computer that has only 320GB of storage.  What is going on here?