Let $f(z); z = x + iy,$ be an analytic function and $u$ be its real part. If   $u$ is a polynomial in the variables $x$ and $y,$ then show that $f(z)$ is a   polynomial in $z.$   I need to confirm my attempt: $u,v$ must satifies the C-R equation. So, $u_x=v_y\implies v=\int u_x dy,$ a polynomial in $x,y.$ Consequently $f=u+iv$ is a polynomial in $x+iy.$
given that $f$ is an analytic function with real part $u$ is a polynomial in the variable of $x,y$, $z=x+iy$, we need to show $f$ is a polynomial in $z$, I am kind of puzzled to see the problem, first I would like to say:any hint please
set session - custom variable to store user id I want to store user id in custom session variable and use it (read) in trigger procedures to authorize user actions. I found something like this:  set session "myapp.user" = '12345'; ... SELECT current_setting('myapp.user');   and it seems to work - I thought that "myapp.user" should be declared in .conf file but it seems that I can create session variables on the fly (I didn't change .conf file at all).  Is there any disadvantage of doing like this? WiFi not working Kubuntu 20.04 - Intel 8265 Network Interface Please add a system to allow file uploads attached to questions and answers How do I enable Remote Desktop connection on Xubuntu 11.10? Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks How can I get the output from Image Trace to be Strokes instead of a Fill? I need to trace picture below to vector.  After tracing it with black and white mode I got result below that I don't like. I would like to have all lines to be in the same constant width.  What tracing strategy I should use to get needed result? How to make dot match newline characters using regular expressions Is $\sqrt{64}$ considered $8$? or is it $8,-8$? Last year in Pre-Algebra we learned about square roots. I was taught then that $\sqrt{64}=8$ and $\sqrt{100}=10$, which I understood and accepted. I was also taught that $\pm\sqrt{64} = 8,-8$ because both of those numbers squared is 64, which I also get. But this year, with a new school and teacher in a different state, our teacher is telling us that: $\sqrt{64}=8,-8$ and $\pm\sqrt{64}$ also is $8,-8$. The way to get the positive root of something is: $+\sqrt{64}=8$ And these seem to contradict each other. I was always taught that a regular square root returned a positive number and only a positive number, but now my teacher is saying a regular square root gives two numbers, and considering the square root of a number $n$ is defined as $y^2=n$ I see where he is coming from. Upon researching this Wikipedia says:  For example, $4$ and $−4$ are square roots of $16$ because $4^{2} = (−4)^{2} = 16$  And Wolfram MathWorld says:  Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of $9$ are $-3$ and $+3$  But on the other side, Wolfram Alpha, when given &quot;The square root of 9&quot; gives only 3.  So, which is right? Is $\sqrt{64}$ considered $8$? or is it $8,-8$? How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$ $A$ is a $n\times n$ matrix such that $ A^m = 0 $ for some positive integer $m$.   Show that $A^n = 0$.   My attempt: For $n &gt; m$, it's obvious since matrix multiplication is associative.  For $n &lt; m$, $A^n\times A^{m-n} = 0$; not sure what to do next. Also I know that $\det A = 0$. Mcrypt extension is missing in 14.04 server for mysql Intuitive explanation for dividing by $n-1$ when calculating standard deviation? I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?! Let $R$ be a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal. Prove: Let $R$ a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal.     I got stuck on this. I understand that every ideal is generated by finitely many elements. Here is my approach:  It's enough to prove the following implication: $$ R/I \ \text{is a domain} \quad \Rightarrow \quad R/I \ \text{is a field}$$ Then only finding an inverse is left. So let $x \in R$ in arbitrary. We have to find an inverse element of $x+I \in R/I$, that as an element y so that $x \cdot y \in I+1$. We denote $I = (x_1, x_2, \cdots, x_n)$.  Now I have to find y so that $xy -1 = r_1 x_1+ r_2 x_2 + \cdots + r_n x_n$. This is where I no more knew what to do.   Your advise will be appreciated. How can I securely erase a hard drive? I'm planning on selling a USB external hard drive that currently contains an old Ubuntu installation with stored passwords and banking information.  How can I securely erase the drive before selling it? I wrote a script to create a big number of heat maps for a given scenario. I would like to save the result of every heat map as an PNG image. I have figured it out doing it with simple Map Rendering but the quality of the result is too low. Furthermore I would like to export to OpenLayers using the same code, which does not seem to be possible with simple rendering.   Now I am trying to do it with QgsComposition instead. Manually in the Map Composer you can just set the paper color to transparent and disable the background of the map canvas.   This is the code that I have so far. I just cannot figure out why it is not working. The result png-file always has a white opaque background.   My OS is Windows 7 and I'm using QGIS 2.6 Brighton.  I hope someone can help me out here.  # PARAMETERS mainPath = 'path/to/folder/' filename = 'filename' imageType = "png" imageWidth_mm = 400 imageHeight_mm = 160 dpi = 300  mapRenderer = iface.mapCanvas().mapRenderer()  c = QgsComposition(mapRenderer) c.setPlotStyle(QgsComposition.Print)  c.setPaperSize(400, 160)  x, y = 0, 0 w, h = c.paperWidth(), c.paperHeight() composerMap = QgsComposerMap(c, x ,y, w, h) c.addItem(composerMap)  # get all items and disable the backgrounds itemList = c.items() c.removeItem(itemList[2]) c.removeItem(itemList[2]) itemList[2].setBackgroundEnabled(False) itemList[3].setBackgroundEnabled(False) # add them to the composition c.addItem(itemList[2]) c.addItem(itemList[3])  c.refreshItems() c.refreshDataDefinedProperty(QgsComposerObject.AllProperties)  c.setPrintResolution(dpi) dpmm = dpi / 25.4 width = int(dpmm * c.paperWidth()) height = int(dpmm * c.paperHeight())  # create output image and initialize it # image = QImage(QSize(width, height), QImage.Format_ARGB32) image = QImage(QSize(width, height), QImage.Format_ARGB32_Premultiplied) image.setDotsPerMeterX(dpmm * 1000) image.setDotsPerMeterY(dpmm * 1000) #image.fill(0)  imagePainter = QPainter(image)  # This part does not seem to have any effect #brush = QBrush(Qt.BrushStyle(0)) # transparent brush # #imagePainter.setBackgroundMode(0) #c.setBackgroundBrush(brush)  c.renderPage( imagePainter, 0 ) imagePainter.end()   imageFilename =  mainPath + filename + '.' + imageType image.save(imageFilename, imageType) print 'image saved' print 'done' Hypernym for "audiobook", "podcast", "speech", etc.? I have a set of polygons representing administrative districts.  The districts were digitized by hand, and there are small spaces (slivers) between polygons where the polygons should be touching. How can I have the polygons snap together, removing the spaces? How do I copyright my website? What are the summary rules for when to use image stabilization and when not to? When I got my DSLR, I took a lot of pics and all of them came excellent, even though I had VR at the "Off" position.   Since those came out so well, I wonder when should I turn VR on, and when I should just leave it off.   I found  which answer this question in detail, but that's too much for me to understand easily. Can someone explain the strict rules in short?  Should I use VR with tripod? When should I use it when I'm not using a tripod, and when should I leave it off? How to install Redis on Ubuntu 16.04? I am working with Laravel but i have installed all on my own (php, mysql, composer, nginx) and now i need to install Redis and configure it so i could use it for queue driver in Laravel.  How to install it, because it cannot be installed through apt-get install redis  i get this error:  ubuntu@ubuntu:~$ sudo apt-get install redis [sudo] password for ubuntu:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package redis What's better here - a single column or multicolumn primary key? Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste?