I want to show given that $f:\mathbb{C} \to \mathbb{C}$ is holomorphic then  $g(z)=\overline{f(\overline{z})}$ is also holomorphic. I used the definition of the derivative to write $$g'(z_0)=\lim_{z\to z_0} \frac{\overline{f(\overline{z})}-\overline{f(\overline{z}_0)}}{z-z_0}=\lim_{z\to z_0}\frac{f(\overline{z})-f(\overline{z}_0)}{\overline{z}-\overline{z}_0}.$$ Now I argue that conjugation is continuous so if $z\to z_0$ then $\overline{z}\to \overline{z}_0$, does this mean $g'(z_0)=f'(\overline{z}_0)$?   Since $f$ is differentiable on an open set $U$ then $g$ would be differentiable for $\overline{z}_0\in U$ and also holomorphic.
Prove that the functions $f(z)$ and $\overline{f(\overline{z})}$ are simultaneously holomorphic.   I take this to mean that $f(z)$ is holomorphic if and only if $\overline{f(\overline{z})}$ is holomorphic.  Let $g(z)=\overline{f(\overline{z})}$. Note that $\overline{g(\overline{z})}=f(z)$. So it suffices to prove that if $f(z)$ is holomorphic, then $g(z)$ is holomorphic.  Write $f(z)=u(z)+iv(z)$. Since $f(z)$ is holomorphic, the real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(z)}}{\partial{x}} = \frac{\partial{v(z)}}{\partial{y}}, \frac{\partial{u(z)}}{\partial{y}} = -\frac{\partial{v(z)}}{\partial{x}}.$$  We have $g(z) = u(\overline{z})+i(-v(\overline{z}))$. To prove that $g(z)$ is holomorphic, we must prove that its real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(\overline{z})}}{\partial{x}} = \frac{\partial{(-v(\overline{z}))}}{\partial{y}}, \frac{\partial{u(\overline{z})}}{\partial{y}} = -\frac{\partial{(-v(\overline{z}))}}{\partial{x}}.$$  How can we obtain this from the above relations?
How to launch a command on network connection/disconnection? Compilers and argument order of evaluation in C++ 0:root@SERVER:/root # echo "something" | sed -e 's/^/\t/g' tsomething 0:root@SERVER:/root #    AIX/ksh .. why doesn't it works? I just want to put a "tab" before every line! How to check if a string starts with a specified string? Recovering a password for a Word 2007 document Vertical alignment in table: m-column, row size - problem in last column For example, on this page:   I see the following:   What does this 499 mean? I can't seem to correlate it with upvotes, posts, edits or anything else. I have multiple Ubuntu machines at home and a pretty slow internet connection, and sometimes multiple machines need to be updated at once (especially during new Ubuntu releases.)  Is there a way where only one of my machines needs to download the packages, and the other machines can use the first machine to get the debs?  Does it involve setting up my own local mirror?  Or a proxy server?  Or can it be made simpler? Can't install Ubuntu 18.10 on XPS 15 - EFI\BOOT\mmx64.efi not found I tried to install Ubuntu 18.10 on my XPS 15 9570 earlier. Everything was working fine until I got to the partition selection part of the installation. That's when the installer crashed and I had to shut down my machine. I think it's because I had my SATA configuration set to RAID ON instead of AHCI, which is now fixed.  Now when I try to run the installer from my bootable USB I get the following error  Failed to open \EFI\BOOT\mmx64.efi - Not Found Failed to load image \EFI\BOOT\mmx64.efi: Not Found Failed to start MokManager: Not Fond Something has gone seriously wrong: import_mok_state() failed   Hoping someone might have some idea as to what is going on I am running logistic regression on a small dataset which looks like this:    After implementing gradient descent and the cost function, I am getting a 100% accuracy in the prediction stage, However I want to be sure that everything is in order so I am trying to plot the decision boundary line which separates the two datasets.  Below I present plots showing the cost function and theta parameters. As can be seen, currently I am printing the decision boundary line incorrectly.    Extracting data  clear all; close all; clc;  alpha = 0.01; num_iters = 1000;  %% Plotting data x1 = linspace(0,3,50); mqtrue = 5; cqtrue = 30; dat1 = mqtrue*x1+5*randn(1,50);  x2 = linspace(7,10,50); dat2 = mqtrue*x2 + (cqtrue + 5*randn(1,50));  x = [x1 x2]'; % X  subplot(2,2,1); dat = [dat1 dat2]'; % Y  scatter(x1, dat1); hold on; scatter(x2, dat2, '*'); hold on; classdata = (dat&gt;40);   Computing Cost, Gradient and plotting  %  Setup the data matrix appropriately, and add ones for the intercept term [m, n] = size(x);  % Add intercept term to x and X_test x = [ones(m, 1) x];  % Initialize fitting parameters theta = zeros(n + 1, 1); %initial_theta = [0.2; 0.2];  J_history = zeros(num_iters, 1);   plot_x = [min(x(:,2))-2,  max(x(:,2))+2]  for iter = 1:num_iters  % Compute and display initial cost and gradient     [cost, grad] = logistic_costFunction(theta, x, classdata);     theta = theta - alpha * grad;     J_history(iter) = cost;      fprintf('Iteration #%d - Cost = %d... \r\n',iter, cost);       subplot(2,2,2);     hold on; grid on;     plot(iter, J_history(iter), '.r');  title(sprintf('Plot of cost against number of iterations. Cost is %g',J_history(iter)));     xlabel('Iterations')     ylabel('MSE')     drawnow      subplot(2,2,3);     grid on;     plot3(theta(1), theta(2), J_history(iter),'o')     title(sprintf('Tita0 = %g, Tita1=%g', theta(1), theta(2)))     xlabel('Tita0')     ylabel('Tita1')     zlabel('Cost')     hold on;     drawnow      subplot(2,2,1);     grid on;         % Calculate the decision boundary line     plot_y = theta(2).*plot_x + theta(1);  % &lt;--- Boundary line      % Plot, and adjust axes for better viewing     plot(plot_x, plot_y)     hold on;     drawnow  end  fprintf('Cost at initial theta (zeros): %f\n', cost); fprintf('Gradient at initial theta (zeros): \n'); fprintf(' %f \n', grad);   The above code is implementing gradient descent correctly (I think) but I am still unable to show the boundary line plot. Any suggestions would be appreciated.    logistic_costFunction.m  function [J, grad] = logistic_costFunction(theta, X, y)      % Initialize some useful values     m = length(y); % number of training examples      grad = zeros(size(theta));      h = sigmoid(X * theta);     J = -(1 / m) * sum( (y .* log(h)) + ((1 - y) .* log(1 - h)) );      for i = 1 : size(theta, 1)         grad(i) = (1 / m) * sum( (h - y) .* X(:, i) );     end  end   EDIT:  As per the below answer by @Esmailian, now I have something like this:  [m, n] = size(x);  x1_class = [ones(m, 1) x1' dat1']; x2_class = [ones(m, 1) x2' dat2'];  x = [x1_class ; x2_class] Files disappearing on linux server I've got 4 specific files that seem to keep disappearing from a user's home directory. As far as we know, there are no cronjobs or other automated tasks that would be removing them.  I've setup auditd on them but the logs aren't really showing anything of interest.  I can see our backup utility accessing them every night until the point they aren't there anymore, but nothing else.  Is there anything that would be causing those files to be removed that would get around auditd?  The files in question are these:  /home/username/.bashrc /home/username/.bash_profile   as well as a couple of files in that user's .ssh directory.  Copies of these files placed into a subfolder called "keepers" get deleted at the same time as well.  Changing the permissions on them to 000 and having them owned by root hasn't helped.   I've currently got inotifywait setup to log create,delete,move on that subfolder, so hopefully that will turn up something, although it doesn't log much aside from when it happened, not what caused it. Change Text Color in Blender Game Engine? I want to change the color of some text in my game. I've tried adding materials, but whenever I press P to play the game, the text stays the same white color. Any ideas? Installing VMware on 18.04: Failed to build vmmon Dropdownlist control with s for asp.net (webforms)? Can anyone recommend a dropdownlist control for asp.net (3.5) that can render option groups? Thanks Converge or diverge? : $\sum_{n=1}^{\infty}\frac{\tan{n}}{2^{n}}$ Picture rotation Is there a way to quickly rotate a picture in the SE editor without saving it on disk and rotating it in some image tool and reuploading it?  E.g. consider the picture below, assume I would like to quickly rotate it 90 degrees and wanted to do it expressly in the editor without the drudgery of saving it on disk, is there a shortcut in the stack exchange &quot;markup language&quot;?  If this feature is not available, I would like to nominate it for implementation. gnome maps program never runs I have this problem while running GNOME Maps ~$ gnome-maps  (org.gnome.Maps:10414): Gjs-WARNING **: 19:43:34.553: Some code called array.toString() on a Uint8Array instance. Previously this would have interpreted the bytes of the array as a string, but that is nonstandard. In the future this will return the bytes as comma-separated digits. For the time being, the old behavior has been preserved, but please fix your code anyway to explicitly call ByteArray.toString(array). (Note that array.toString() may have been called implicitly.) 0 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmTypes.js&quot;:32] 1 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEditDialog.js&quot;:35] 2 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEdit.js&quot;:25] 3 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/contextMenu.js&quot;:33] 4 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/mainWindow.js&quot;:33] 5 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/application.js&quot;:35] 6 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/main.js&quot;:43] 7 start() [&quot;resource:///org/gnome/gjs/modules/package.js&quot;:209] 8 &lt;TOP LEVEL&gt; [&quot;/usr/bin/gnome-maps&quot;:2]   (org.gnome.Maps:10414): Gdk-WARNING **: 19:43:36.525: Native Windows taller than 65535 pixels are not supported Gdk-Message: 19:43:36.574: Error 71 (Protocol error) dispatching to Wayland display.  19:43:36.574 (This message has changed many times) What's this? What is the difference between Θ(n) and O(n)? Piping STDERR vs. STDOUT According to "" (pg. 44), you can pipe only STDERR using the |&amp; redirection symbols.  I've written a pretty simple script to test this:  #!/bin/bash echo "Normal Text." echo "Error Text." &gt;&amp;2   I run this script like this:  ./script.sh |&amp; sed 's:^:\t:'   Presumably, only the lines printed to STDERR will be indented. However, it doesn't actually work like this, as I see:      Normal Text.     Error Text.    What am I doing wrong here? Can I run out of disk space by creating a very large number of empty files? It is well-known that empty text files have zero bytes:    However, each of them contains , which according to my research, is stored in , and do use space.   Given this, it seems logical to me that it is possible to fill a disk by purely creating empty text files. Is this correct? If so, how many empty text files would I need to fill in a disk of, say, 1GB?    To do some checks, I run df -i but this apparently shows the % of inodes being used(?) rather than how much they weigh.  Filesystem             Inodes  IUsed    IFree IUse% Mounted on udev                   947470    556   946914    1% /dev tmpfs                  952593    805   951788    1% /run /dev/sda2            28786688 667980 28118708    3% / tmpfs                  952593     25   952568    1% /dev/shm tmpfs                  952593      5   952588    1% /run/lock tmpfs                  952593     16   952577    1% /sys/fs/cgroup /dev/sda1                   0      0        0     - /boot/efi tmpfs                  952593     25   952568    1% /run/user/1000 /home/lucho/.Private 28786688 667980 28118708    3% /home/lucho