I need to solve this:  $$z^3 + \overline z = 0$$  how should I manage the 0?  I know that a complex number is in this form: z = a + ib so:  $$z^3 = \rho^3\lbrace \cos(3\theta) + i \sin (3\theta)\rbrace$$ $$\overline z = \rho\lbrace \cos(-\theta) + i \sin (-\theta)\rbrace$$ but how about the 0?  EDIT: ok, following some of your comments/answers this is what I have done:  $$z^3 = - \overline z$$ $$\rho^3\lbrace \cos(3\theta) + i \sin (3\theta)\rbrace = \rho\lbrace \cos(-\theta) + i \sin (-\theta)\rbrace$$  So $$ \begin{Bmatrix} \rho^3 = \rho\\  3\theta = -\theta + 2k\pi \end{Bmatrix}$$ $$ \begin{Bmatrix} \rho^3 = \rho\\  2\theta =  2k\pi \end{Bmatrix}$$  $$ \begin{Bmatrix} \rho = 0 or \rho = 1\\  \theta =  k\frac{\pi}{2} \end{Bmatrix}$$  is this the right way?
I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.  Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know which root was the extra ).  I noticed that $z=a+bi$ and there exists $\theta$ such that the trigonometric representation of $z$ is $\left ( \sqrt{a^2+b^2} \right )\left ( \cos \theta + i \sin \theta \right )$ .  It seems that $-\bar{z} = -\left ( \sqrt{a^2+b^2} \right )\left ( \cos (-\theta) + i \sin (-\theta) \right )$  However, my trig is pretty rusty and I'm not quite sure where to go from here.
I'm trying to use the \citep command in Latex and my document class is elsarticle. The problem is that it displays numbers not authoryear citations in the text which is what I'm looking for. Here are related parts of my code:      \documentclass[preprint,11pt]{elsarticle}      \usepackage{natbib}   ...      \bibliographystyle{plainnat}      \bibliography{ML}   And I'm obviously using the \citep command like this:      I once read \citep{edelman1993magnetic}   And my .bib file looks like this:  @article{wolf2009regional, title={Regional brain activation changes and abnormal functional connectivity of the         ventrolateral prefrontal cortex during working memory processing in adults with attention-deficit/hyperactivity disorder}, author={Wolf, Robert C and Plichta, Michael M and Sambataro, Fabio and Fallgatter,     Andreas J and Jacob, Christian and Lesch, Klaus-Peter and Herrmann, Martin J and     Sch{\"o}nfeldt-Lecuona, Carlos and Connemann, Bernhard J and Gr{\"o}n, Georg and others}, journal={Human brain mapping}, volume={30}, number={7}, pages={2252--2266}, year={2009}, publisher={Wiley Online Library} }   The result looks like:  I once read[1]   I'm looking for a result like this:  I once read(wolf., 2009)   Elsevier is actually looking for:  Text: All citations in the text should refer to:    Single author: the author's name (without initials, unless there is ambiguity) and the year of publication;  Two authors: both authors' names and the year of publication;  Three or more authors: first author's name followed by 'et al.' and the year of publication.    Thanks in advance Rack Mount Servers - Build them or Buy them? This is the  for "Should I build coomputing hardware myself?" questions.   I have put together countless PCs, but never a large server. The geek in me says build it, but the realist in me says let the manufacturer handle it when there is a problem. Ignoring the time penalty involved with the initial assembly time of a built one, which is a better solution? Have you ever run into a problem with a home build server that would have been solved easier/quicker/cheaper by going with a manufacturer? Are there any features that manufacturers give that aren't easily attainable with a home built server? About two weeks ago I had a problem with my PC that my brother wrote a question about at .  I still haven't solved the problem, so my PC is unusable now.  Now I think that it would be simpler to re-install both Win7 &amp; Ubuntu than repair them, but there're some data on their partitions. I want to recover this data.  These partitions are not mountable for some reason. I don't know why.  sudo fdisk -l about the problematic partitions:     Device    Boot      Start         End      Blocks   Id  System /dev/sda1            2048     147912703    73955328   83 Linux /dev/sdb2            206848   174079999    86936576    7  HPFS/NTFS/exFAT   And in GParted:      And the error of the mount command:  Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   And the exact same message for /dev/sda1  How can I recover my data from those HDDs?  EDIT     I have tried to specify the type of the drive:  $ sudo mount -t ext4 /dev/sda1 /olddisk mount: wrong fs type, bad option, bad superblock on /dev/sda1,        missing codepage or helper program, or other error        In some cases useful info is found in syslog - try        dmesg | tail  or so $ sudo mount -t ntfs /dev/sdb2 /olddisk NTFS signature is missing. Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   EDIT     sudo ntfsfix /dev/sdb2 returns:  Mounting volume... NTFS signature is missing. FAILED Attempting to correct errors... NTFS signature is missing. FAILED Failed to startup volume: Invalid argument NTFS signature is missing. Trying the alternate boot sector Unrecoverable error Volume is corrupt. You should run chkdsk. How can I add a new user as sudoer using the command line? How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ How does one show that $$\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$$ for each nonnegative integer $n$?  I tried using the  but I guess I am applying it incorrectly. With the snake oil technique we have $$F(x)= \sum_{n=0}^{\infty}\left\{\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}\right\}x^{n}.$$ I think I have to interchage the summation and do something. But I am not quite comfortable in interchanging the summation. Like after interchaging the summation will $$F(x)=\sum_{k=0}^{n}\sum_{n=0}^{\infty}\binom{n+k}{k}\frac{1}{2^k}x^{n}?$$ Even if I continue with this I am unable to get the correct answer.  How does one prove this using the Snake oil technique?  A combinatorial proof is also welcome, as are other kinds of proofs. Motivation for Napier's Logarithms In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this . Please help me understand:       By repeated subtractions Napier   calculated $10^7(1 − 10^{−7})^L$ for L   ranging from 1 to 100. The result for   $L=100$ is approximately $0.99999 = 1  − 10^{-5}$. Napier then calculated the   products of these numbers with $10^7(1  − 10^{−5})^L$ for L from 1 to 50, and   did similarly with $0.9995 ≈ (1 −  10^{−5})^{20}$ and $0.99 ≈ 0.995^{20}$.   These computations, which occupied 20   years, allowed him to give, for any   number ''N'' from 5 to 10 million, the   number ''L'' that solves the equation      $$N=10^7 {(1-10^{-7})}^L$$      In modern notation, the relation to   natural logarithms is:      $$L = \log_{(1-10^{-7})} \!\left(  \frac{N}{10^7} \right) \approx 10^7  \log_{ \frac{1}{e}} \!\left(  \frac{N}{10^7} \right) = -10^7 \log_e  \!\left( \frac{N}{10^7} \right)$$      where the very close approximation   corresponds to the observation that      $${(1-10^{-7})}^{10^7} \approx  \frac{1}{e}. $$   I understand the computations, but I don't know what's going on. Why does $\sum_{k=1}^{\infty}\frac{{\sin(k)}}{k}={\frac{\pi-1}{2}}$? Inspired by  (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$.   Presumably, this can be derived from the similarity of the Leibniz expansion of $\pi$ $$4\sum_{k=1}^{\infty}\dfrac{(-1)^{k+1}}{2k-1}$$to the expansion of $\sin(x)$ as $$\sum_{k=0}^{\infty}\dfrac{(-1)^k}{(2k+1)!}x^{2k+1},$$ but I can't see how...  Could someone please explain how $\dfrac{\pi-1}{2}$ is arrived at? Disable the guest user completely in El Capitan without having to disable 'Find My Mac' I know it should be straight forward but unfortunately, it  isn't.  I did disable the guest user in the System Preferences by unchecking the checkbox for Allow guests to login in to this computer, but it still shows up on the login screen.  I have heard that disabling 'Find My Mac' would fix it for OS X Lion, but I want to keep it.  Is there a way to do this? How to answer a negative question in English? Q: "Don't you know?"   If I really don't know, how should I respond to this question? Yes or No?  There's a similar question on EL&amp;U, but the answer isn't clear how it works:      Also here on ELU too:      These answers are confusing. I want to know what one word answer I can use. In other words I want to know:   What will native speakers understand if I reply No to this question? What will native speakers understand if I reply Yes to this question? What should I do when my Minecraft Launcher 1.3.11 wouldn't log me in? The console output is:                                              Sorry, but we couldn't connect to our servers. Please make sure that you are online and that Minecraft is not blocked. (CertificateNotYetValidException:NotBefore:Tue Apr 08 20:47:05 CST 2014) Add a "voted to close" activity event somewhere on the user profile page Often when I vote to close, I want to go back and check the comments on the question, to see if it's been improved, to comment further, or to answer the question.  It would be helpful to add a bit of a trail so I can find those questions again. I'm suggesting adding an event to the Activity page on the user profile, because this is the logical spot for the given action.  I realize the Activity page is public, but I don't see why this content couldn't be added only for the currently logged-in user.  An alternative would be to add a new tab on the user profile page that only lists current-user content, and is not shown publicly. I'm sure there are other events, data, or statistics that could be shown here as well, so I'm certain a new tab like this wouldn't be underutilized.    One thing that could be done instead of this is to always add a comment stating why I voted to close. This works well enough, but only some of the time -- for example, if there is already a vote to close as a duplicate and a comment with a link to that duplicate, I'm not going to add a comment myself if I have nothing to add aside from a close vote.  Another way to do it would be to favourite the question. This only works well to a point since questions that get voted to close don't necessarily end up being downvoted (i.e., for my "closed favourites," I couldn't just look at the last page of my favourites).    Related: Jon Skeet  that votes to close on your own questions be shown in the Recent Activity area, which is not what I'm requesting here. change environment of a running process How might it be possible to alter some variable in the env of an already running process, for example through /proc/PID/environ? That "file" is read-only.  Need to change or unset the DISPLAY variable of a long-running batch job without killing it. Add 1Password extensions support to the SE iOS app Please implement the URL scheme to call 1Password for password lookups on the iOS app.   a URL scheme for switching to 1Password to search item titles for a custom term, usually a service name like “twitter” or “evernote”. This speeds up the process for the user so they can copy something like their password and get back to your app  That will ease sign up speed using long, complicated or unmemorable passwords for many users of both apps. In the two years since the above blog was published there are now updated instructions for implementation and code on github: How do I remove the proprietary ATI drivers? I've installed 11.10 and the proprietary ATI drivers using "additional drivers" The performance of my system is absolutely awful and it shouldn't be. I tried to remove the proprietary drivers using the Additional Drivers tool and it appears to remove them. However after I reboot I cant get back into my desktop properly (the panel and launcher go missing). This doesn't seem to be an isolated problem in 11.XX.  covers how to restore the desktop (panel and launcher), but the guide doesn't fix my problem though.   Whenever I do sudo unity --reset it runs through its normal processes until it hangs at setting update "run_key" and never gets past that. I must reinstall the proprietary drivers using jockey-text or jockey-gtk in order to get back to my proper desktop.   Interestingly enough the system performance seems improved while it is in its "broken" state (missing panel and launcher).  I think restoring the default drivers may solve my problems but I cant figure out how to do it. How do I turn on workspaces? (Why do I only have one workspace?) It appears that Unity on Raring only comes with a single virtual desktop (aka workspaces) by default, and CCSM's Wall plugin has no way to add more. Is this configurable at all? Are there any valid reasons to not rollover a former 401(k) when changing jobs?  I have several little ones that I never quite got around to rolling together - with an upcoming job change, should they all be combined into my new employer's retirement plan?  If not, what else might be considered? Equicontinuity on a compact metric space turns pointwise to uniform convergence I know that   If $\{f_n\}$ is an equicontinuous sequence,  defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly.  I'm having trouble proving this.  I see the same problem  but am having trouble following the proof, particularly with part (3).  Can someone guide me through a proof of this result? I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. After the latest Win8.1x64 Updates, my system got stuck restarting or shutting down.. When I installed the last Windows update, all seemed fine except it got stuck doing the restart.  I waited about 5 minutes then used the power button.  Startup was normal.    But then when I told it to shut down it closed the programs then went to the "Shutting Down" screen and just stays there.    Disk activity lights aren't showing any activity.  Here is a screen grab of the most recent updates:    Before I do anything I wanted some input.  Question:  Any ideas of how I can figure out how to fix it? Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show that $f(z)=0$.