Suppose u is a harmonic function in disc $|z|&lt;1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary?  I know v can be represented by integral use the boundary value of u, but I don't think this can help.  Can anyone prove it or give a counterexample? Thank all of you.
Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we know that the constructed function has uniform limit which is the given continuous function as $r\to 1$, and we can complexify the solution by taking the Schwarz integral formula. Now (in this very particular case) does the imaginary part has a uniform limit? If it does, why don't we prove the uniqueness of the solution by a simple application of the Cauchy integral formula? Here is the proof that I reckon to be reasonable(which is what I expose here to be criticised)  We construct the Schwarz integral, or $\frac{1}{2\pi}\int _{0}^{2\pi}\frac{\exp(it)+z}{\exp{(it)}-z}u(\exp{(it)})\mathrm{d}t$. This function is an analytic function in the interior unit disk, and is continuous in the closed unit disk(its closure). Let this analytic function denoted by $f$, and another analytic function that has the same uniform limit function on the circle be $v$. Then $u-v$ is an complex analytic function in the interior of the unit disk, continuous on the closure, with boundary value identically zero. An application of Cauchy's integral formula gives rise to the conclusion that $u-v$ in identically zero on the entire disk.   Another question being if the boundary function has finite jump discontinuities, what is the behaviour of the convolution with the Poisson kernel near the discontinuous points?
Why are heatmaps on qgis 2.8.1 produced in the wrong CRS? All the info about TLP on the surface looks more like advertisement and marketing to me, I can't see any detailed comparison to default configurations. What are the precise differences compared to default (K/L)Ubuntu installs under the hood, are they really effective?  Update: And if they really are, why aren't they a part of a default config? Differences between ways to express future actions I asked this question  but I haven't gotten a useful answer. Could you tell me the difference in meaning between these sentences?        Do you think you will visit them next week?     Will you visit them next week?     Are you going to visit them next week?     Are you visiting them next week?     Are you going to be visiting them next week?     Are you planning to be visiting them next week?     Do you plan to visit them next week?      Will there be any difference in the answers?        I think I will visit them next week.     I will visit them next week.     I am going to visit them next week.     I am visiting them next week.     I am going to be visiting them next week.     I am planning to be visiting them next week.     I plan to visit them next week.      Can I use "planning to" instead of "going to"?  (E.g. "Are you planning to visit them next week? — Yes, I am planning to visit them next week.") Which of these sentences show that the subject of conversation has already been discussed? For example, I talked with a friend of mine about my plans to visit somebody. So my friend is aware of my plans.  And my last question. Which of these seven sentences express that the action will certainly take place? How to execute .sh file on Windows? When I am trying to execute a file(name.sh) in the command line by the command  ./name.sh  , I am getting the error that:     "." is not recognized as an internal   or external command, operable or batch   file   please help me execute the .sh file Is there a way that I can configure Visual Studio 2008 to understand CamelCase?  Specifically, I'd like to be able to get Ctrl + right or left cursor to take me to a subsection of a variable or type name. i.e., if my cursor was at the start of this line: LongNamedExampleClass longNamed = new LongNamedExampleClass();  and I hit Ctrl + right, I'd like to jump to &quot;Named&quot; in the class name, rather than the start of the variable name.  If I hit it again, I'd like to go to &quot;Example&quot;.  I hope that makes sense. I used this navigational feature in Eclipse quite a lot, and find it difficult to unlearn something so ingrained. $5$ tourists plan to attend Octoberfest. Each attends a location at random from the choices: Alpine, Bingeman, Concordia, Kitchener, Queens, Schwaben, Transylvania. Q What is the probability that two attend one location and three attend another same location?  I get that there are $\binom{5}{2}$ possibilities to create 2 groups. There are $\binom{7}{2}$ ways to choose 2 places so shouldn't it be $\frac{\binom{7}{2}\binom{5}{2}}{7^5}$? Word for "blissfully oblivious"? Is it possible to skip releases during an upgrade? Possible Duplicate:          Is it possible to upgrade a 10.04 lucid installation to 11.04 Natty, without upgrading to 10.10 first? What about other versions? I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me How does the Mac choose which connection to use when both Wifi and Ethernet are connected? When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? Can players enchant items? I know that the DM can give you enchanted items, but can players spend gold and downtime to enchant their own? Moving from HTTP to HTTPS in Google Search Console How can I import an obj sequence into blender? I have an obj sequence of a fluid simulation from Realflow. I've found a few tools that can import an obj sequence into blender, but most are outdated and don't work. Are there any scripts or plug-ins that can do that? What should my Objective-C singleton look like? My singleton accessor method is usually some variant of:  static MyClass *gInstance = NULL;  + (MyClass *)instance {     @synchronized(self)     {         if (gInstance == NULL)             gInstance = [[self alloc] init];     }      return(gInstance); }   What could I be doing to improve this? Sum with parameter convergence override parent theme configuration in child functions.php Handle .bib in subfiles I have a problem very similar to the question being asked in  . My .bib is references in main.tex. When compiling main, everything works like a charm. Working on the subfiles, I have two issues though (which are most likely related):   I can't access the bib with auto-completion When compiling the subfile, it doesn't insert the citations.   I tried including  \bibliographystyle{plainnat} \bibliography{mybib}   in the subfiles, which also worked smoothly. Obviously, the bibs are included in every chapter of the document when compiling main.tex, then. I tried setting a global boolean in main.tex:  \newboolean{printBibInSubfiles} \setboolean{printBibInSubfiles}{false}    and in the subfiles:  \ifthenelse{\boolean{printBibInSubfiles}}    {}    {\bibliographystyle{plainnat}     \bibliography{../betterbib.bib}}    For some reason, then setting the boolean to false, the "bibliography"-header was printed in the subfiles as well as for the main file, but there was no output of citations under any of the "bibliography"-headers. Furthermore, compiling only the subfiles didn't work correctly.  My document structure is the following:  main.tex:  \begin[document] ... \subfile{sub/dataprocessing} ... \bibliographystyle{plainnat} \bibliography{mybib} \end{document}   the subfiles:  \documentclass[../main.tex]{subfiles} \begin{document} ... \cite{Random2014} ... \end{document}   After all this explanation: the only thing I want is to access the .bib from the subfiles without having to manually uncommenting something when compiling the main file. Any takers?  PS: Obviously, I also tried the solution provided in "" but couldn't get it to work. I don't know whether that would solve my problem anyway, since I don't have the \printbibliography-issue of the thread opener. The game  has a very unique intro that I would like to show my friend again for the first time. I'm not sure how to reset all my progress on this game to achieve this though. Any thoughts? prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit How to analyze trend in non-periodic time series Suppose I have following non-periodic time series. Obviously the trend is decreasing and I would like to prove it by some test (with p-value). I am unable to use classic linear regression due to strong temporal (serial) auto-correlation among values.  library(forecast) my.ts &lt;- ts(c(10,11,11.5,10,10.1,9,11,10,8,9,9,                6,5,5,4,3,3,2,1,2,4,4,2,1,1,0.5,1),             start = 1, end = 27,frequency = 1) plot(my.ts, col = "black", type = "p",      pch = 20, cex = 1.2, ylim = c(0,13)) # line of moving averages  lines(ma(my.ts,3),col="red", lty = 2, lwd = 2)     What are my options?