Let $D = \{x \in \mathbb{C} : \text{Re}(z) &gt; 0\}$, and $g: D \to \mathbb{C}$ an analytic function. Does it follow that$$f(z) = \overline{g(\overline{z})}$$is also analytic?
I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean differentiable at each point. Here $f$ is a complex valued function.  What I do is write $f(z)=u(x,y)+iv(x,y)$, where $u$ and $v$ are real functions of two variables. Then $\overline{f(\bar{z})}=u(x,-y)-iv(x,-y)$.  These two forms look very similar, in the sense the one function being differentiable should immediately imply the other is differentiable, since the only thing really changing might be a $-$ sign popping out due to the chain rule.   How can I more rigorously express that $f(z)$ is analytic iff $\overline{f(\bar{z})}$ using this? Many thanks.
How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions If $n$ is an integer and $n&gt;2$, then $$\bar{z}=z^{n-1}$$ has $n+1$ solutions. Does this have something to do with the rational root theorem? Can crumbled feta cheese last almost a year? talks about feta cheese in a brine or milk bath lasting up to three months.  What about crumbled feta?  I was looking through the fridge, and found a 1/2 pound open container of crumbled feta.  I have only used about an ounce of it.  There is no sign of mold on it, and it smells like feta cheese.  A small sample tasted okay.  The amazing thing is that I bought it in October or November: 8-10 months ago!  I have a recipe that calls for almost a 1/2 pound of feta, and I will be the only one eating it (so I'm not trying to impress anybody!).  I dislike wasting food, but I dislike being sick even more!  Is it safe to use dry crumbled feta that is so old? How to read ext4 partitions on Windows? Is there a way to read ext4 partitions from Windows? If so, what can I do? This is a follow-on from my  about samples from a distribution.  Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that $P(X_n &gt; X_1 \wedge \ldots \wedge X_n &gt; X_{n-1}) = \frac{1}{n}$ (this is true intuitively and experimentally). To prove that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not uniformly converge to $f(x) = 0$ on $[0,1]$ My Approach. To prove the given statement it is sufficient to show that $$\exists \epsilon &gt; 0, \exists x \in [0,1], \forall N \in \mathbb{N}, \exists n &gt; N $$ $$ \Rightarrow | f_n(x) - f(x)| &gt; \epsilon$$    Let $\epsilon = \frac{1}{10}, x = \frac{1}{n}, n =N+1$. Hence $| f_n(x) - f(x)| = \frac{1}{2} &gt; \epsilon = \frac{1}{10}$. Hence, we have shown for every $N \in \mathbb{N}$, how to choose $n$ such that $| f_n(x) - f(x)| &gt; \epsilon$. Thus, $f_n(x)$ does not converge uniformly on $[0,1]$.    Is there any logical flaw or wrong steps in my proof. Any help would be appreciated. From :   (formal) showing great care and effort in your work synonym: diligent  (formal) working very hard and taking great care that everything is done as well as it can be synonym: diligent  (formal) showing care and effort in your work or duties  These definitions look alike. What they have in common is that these words mean to take care and to put effort into your work. All words are formal. None of these seems to be a proper term in a specific field, e.g. medical, law. I also looked at  to get more details. This adds the connotation of steady and earnest to diligent and unremitting attention and persistent application to assiduous. But in the end, these additional information are again describing an equal thing, i.e., continuously doing your work. The entry on sedulous doesn't add any more content; it refers to diligent and again describes the word with perseverance. From this point (checking dictionaries only), I don't see any difference at all. I continued with a corpora search. From : Diligent is most used in combination with work, effort, research, study but also student, people.  She's a diligent student. His diligent research identified the value of ...  Assiduous is also most used with work, effort, research but not with student, people. Merriam-Webster's Learner's Dictionary entry on  lists student as example though.  He's assiduous in his work. With assiduous attention to detail ...  Sedulous is most used, um, to little results.  has even less content. Merriam-Webster's example is, once again, a student. So, diligent is used most often, followed by assiduous and sedulous is quite rare. All words are formal. All words seem to be usable as an attribute to a person (e.g., student) but also specifying the intense care and effort on work, e.g., a research. I don't see any significant difference in these words, besides frequency. When is a student diligent, assiduous or sedulous? And when is a research diligent, assiduous or sedulous? Or simply: What are the subtle distinctions which aren't represented by dictionaries? Can Mage Hand be used in combat to grab something from an opponent?  For example, one could grab a bow out of their hands or the arrows from a quiver, or try to push them out of a tree, etc.  Mage Hand can carry up to 10 pounds, but what is its ability to interact in situations like these? I've created a Lightning component to view key details for the Account and Contact lookups from a Case, like so:    The problem is, I have been unable to get field content to render when  using the &lt;force:inputField/&gt; or &lt;force:outputField/&gt; components when I place them inside the markup to style them as cards. I've used &lt;ui:outputText/&gt; as a workaround, but that means I don't get any of the built-in behaviors I would get with the &lt;force:*Field/&gt; components -- like Lightning styling, which means I have to manually do it using SLDS.  This displays the contact name as expected:  &lt;aura:component implements="flexipage:availableForRecordHome,force:hasRecordId" access="global" controller="LightningCaseTestController"&gt;   &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;   &lt;aura:attribute name="recordId" type="Id" /&gt;   &lt;aura:attribute name="contact" type="Contact" default="{sobjectType: 'Contact'}"/&gt;    &lt;force:outputField value="{!v.contact.Name}"/&gt;   &lt;force:inputField value="{!v.contact.Name}"/&gt;  &lt;/aura:component&gt;   This does not:  &lt;aura:component implements="flexipage:availableForRecordHome,force:hasRecordId" access="global" controller="LightningCaseTestController"&gt;   &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;   &lt;aura:attribute name="recordId" type="Id" /&gt;   &lt;aura:attribute name="contact" type="Contact" default="{sobjectType: 'Contact'}"/&gt;    &lt;article&gt;     &lt;force:outputField value="{!v.contact.Name}"/&gt;     &lt;force:inputField value="{!v.contact.Name}"/&gt;   &lt;/article&gt;  &lt;/aura:component&gt;   Basically, it appears that as soon as any markup is placed around the &lt;force:*Field/&gt; tags -- even just a &lt;div&gt; -- they no longer render. Examples in Trailhead and other places all seem to use &lt;ui:outputText/&gt; or &lt;ui:input*/&gt;.  This seems like a lot of extra manual work. Am I missing something?  Update: it only fails when the component is included in a Lightning page -- such as the record detail page -- from Lightning App Builder. If I add it to a simple Lightning app, it works:  &lt;aura:application &gt;     &lt;c:LightningCaseTest recordId="SOMECASEID" /&gt; &lt;/aura:application&gt;   So it appears to be a bug? Algorithms for Time Series Anomaly Detection I'm currently using Twitter's AnomalyDetection in R: .  This algorithm provides time series anomaly detection for data with seasonality.  Question: are there any other algorithms similar to this (controlling for seasonality doesn't matter)?  I'm trying to score as many time series algorithms as possible on my data so that I can pick the best one / ensemble. Media keys behaving like function keys When a user on our site loses his password and heads off to the Lost Password page we need to give him a new temporary password. I don't really mind how random this is, or if it matches all the "needed" strong password rules, all I want to do is give them a password that they can change later.  The application is a Web application written in C#. so I was thinking of being mean and going for the easy route of using part of a Guid. i.e.  Guid.NewGuid().ToString("d").Substring(1,8)   Suggesstions? thoughts? $dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is. Command to move a file to Trash via Terminal Do "twice as high xxxxx as" , "twice as high" and "twice higher" mean the same thing? I have a strange issue with Windows 10  When I open .nef files they load fine. However after a second or two the contrast changes noticeably. I tried Windows 10 Photos and Picasa Image Viewer.   What am I trying to accomplish? I just want to open files quickly to check for focus and composition before importing into Lightroom  Lightroom does not have this issue   Here is the example opening with Windows 10 Photos    And here is after a second when it 'fully' loads    Picasa Photo Viewer after it loads a bit. It initially opens fine but then it goes all purple How to pass arguments to addEventListener listener function? The situation is somewhat like-  var someVar = some_other_function(); someObj.addEventListener("click", function(){     some_function(someVar); }, false);   The problem is that the value of someVar is not visible inside the listener function of the addEventListener, where it is probably being treated as a new variable. hello everyone I have a little problem, how do I bring up the menu in the circle? when I select another object the menu becomes non-existent How to Configure Launcher Placement in Multi-monitor Setup I use a large monitor as primary and laptop screen as secondary. Is it possible to move the launcher to my big monitor or do I have to turn off my laptop monitor?   "Monitor Preferences" (Launcher search -> "Monitors") does not have a way to set the default/primary monitor. nvidia-settings which I just installed says "You do not appear to be using the NVDIA X driver. Please edit your X configuration file (just run nvidia-xconfig as root), and restart the X server.  sudo nvidia-xconfig says sudo: nvidia-xconfig: command not found.  I guess I don't have an AMD driver because I don't have "AMD Catylist Control Center" in the launcher search. Accidently deleted /usr/lib, how do I restore it? Is it possible to have two background images?  For instance, I'd like to have one image repeat across the top (repeat-x), and another repeat across the entire page (repeat), where the one across the entire page is behind the one which repeats across the top.  I've found that I can achieve the desired effect for two background images by setting the background of html and body:  html {     background: url(images/bg.png); }  body {     background: url(images/bgtop.png) repeat-x; }   Is this "good" CSS?  Is there a better method?  And what if I wanted three or more background images?