Suppose complex valued $h(z)$ is harmonic on star-shaped domain $D$. I want to prove that there exists complex valued analytic functions $f, g$ such that $h=f+\bar g$ This is my approach. Since $h$ is harmonic, then by writing $u+iv:=h$, $u, v$ are also harmonic, so we can find harmonic conjugate $u_1, v_1$ of $u, v$, respectively. By defining $f=(u+iu_1)+i(v+iv_1), g:=v_1+iu_1$, we can check $f$ is obviously analytic, and $h=f+\bar g$. So I only need to check $g$ is analytic. But I could not go ahead. Could I get some help?
How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\overline{\psi}$ is an anti-holomorphic function (conjugate of a holomorphic function)?
Sending form POST request to PHP How could Argus Filch clean up the whole castle by himself without magic? Plotting Complex Quantity Functions Using a question mark mid-sentence Image capture opens after iPhone is connected Everytime I connect my iPhone to my laptop, Image Capture opens. How to I stop it? I just need to use iTunes, not Image Capture. Flash/SWF Editor For Ubuntu 12.10 Multiply cited reference in footnotes How can I separate a single object file into multiple objects based on closed geometry? Let's say you have a single obj with a number of separate closed 3D geometries in it. For example's sake, imagine you have an obj called neighborhoodfirehydrants.obj that includes the model for all of the fire hydrants on a street, which themselves are closed, but as I said above, are all within the same obj file.  Now let's say you wanted to be able to process each of those fire hydrants one at a time, separately. To the user, it's very obvious to see visually that these faces can be subgrouped into separate "objects" but in the Blender world, it sees them all as one.  I'm looking for a way (script would be ideal) to iterate through a blender obj neighborhoodfirehydrants.obj and find each of the "objects" (fire hydrants) that make it up. I've been looking into using the looping tools, but that's not making sense in this context. R linear regression categorical variable "hidden" value See code:   var file1 = "50.xsl"; var file2 = "30.doc"; getFileExtension(file1); //returns xsl getFileExtension(file2); //returns doc  function getFileExtension(filename) {     /*TODO*/ } How do you properly use namespaces in C++? I come from a Java background, where packages are used, not namespaces. I'm used to putting classes that work together to form a complete object into packages, and then reusing them later from that package. But now I'm working in C++.  How do you use namespaces in C++? Do you create a single namespace for the entire application, or do you create namespaces for the major components? If so, how do you create objects from classes in other namespaces? Is it okay to use my US visa after a name change? Manga where the protagonist need to solve girl's quests Can someone tell me the manga in which the protagonist is teleported to an alternative world where he needs to make the girls fall in love with him, then he gains some skills like an invisible sword. Then, when he finishes these &quot;quests&quot;, he will be able to bring one of them to the real world. He has the ability to save and load like checkpoints of games. MWE:  \documentclass{article} \begin{document} $\wedge{\small\wedge}{\tiny\wedge}$ \end{document}   All the \wedges have the same size... Is it possible that using jQuery, I cancel/abort an Ajax request that I have not yet received the response from? Maximum weight an average bicycle can carry? Geodesics equations via variational principle I would like to recover the (timelike) geodesics equations via the variational principle of the following action:  $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$  Using an arbitrary auxiliary parameter $\lambda$, then one is able to rewrite the action and obtains: $$ d\tau = \sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$ $$ -m \int d\tau = -m \int (d\tau / d\lambda) d\lambda =  -m\int\sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$  Now we can vary the paths $x^{\mu}(\tau) \rightarrow x^{\mu}(\tau) + \delta x^{\mu}(\tau)$.  I need the following equation to be true, so i get the geodesics equations, but I don't know how one can say the following: $$ \int d\tau \,\delta g_{\mu\nu}\frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = \int d\tau\, g_{\mu\nu, \rho} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} \delta x^{\rho} $$ I know there is a simpler action, which leads to the same equation of motion, but there one has to do the same manipulation, which I sadly don't understand.  Thanks for helping me! How can I improve Ubuntu overall system performance? What are your tips for improving overall system performance on ubuntu? Inspired by this  I realized that some default settings may be rather conservative on Ubuntu and that it's possible to tweak it with little or no risk if you wish to make it faster.  This is not meant to be application specific (e.g. make firefox load pages faster), but system wide.  Preferably 1 tip per answer, with enough detail for people to implement it.  A couple of mine would be:   Install  (via Software Center or sudo apt-get install preload); Change  value - "which controls the degree to which the kernel prefers to swap when it tries to free memory";   What are yours?  PS: Since this is not intended to have a unique answer but rather, several useful tips, I'm making this community wiki out-of-the-box. Ubuntu 14.04, problem when install gstreamer1.0-libav, unmet dependencies Suddenly my Ubuntu stopped playing video. When I open a video, the player suggests that I install  gstreamer1.0-libav. However, it fails to install.  I tried installing it manually, here is the error message:  $ sudo apt-get install gstreamer1.0-libav Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  gstreamer1.0-libav : Depends: libavcodec54 (&gt;= 6:9.1-1) but it is not going to be installed or                                libavcodec-extra-54 (&gt;= 6:9.13) but it is not going to be installed                       Depends: libavformat54 (&gt;= 6:9.1-1) but it is not going to be installed E: Unable to correct problems, you have held broken packages. Textbook clarification: $\log = \ln$ Textbook reads:     All logarithms are natural logarithms: $\log = \ln$.   Does this mean $n\log(n) = n\ln(n)$?