real analysis proof ? how to prove this problem
Real analysis - proof approach help
Multiple workspaces on Ubuntu 18.04.1 LTS and later with GNOME Shell Cauchy-Riemann equations in polar form. Show that in polar coordinates, the Cauchy-Riemann equations take the form   $\dfrac{\partial u}{\partial r} = \dfrac{1}r \dfrac{\partial v}{\partial \theta}$ and $\dfrac{1}r \dfrac{\partial u}{\partial \theta} = −\dfrac{\partial v}{\partial r}$.      Use these equations to show that the logarithm function defined by   $\log z = \log r + i\theta$ where $z=re^{i\theta}$ with $-\pi&lt;\theta&lt;\pi$ is holomorphic in the region $r &gt; 0$ and $-\pi&lt;\theta&lt;\pi$.   What I have so far:   Cauchy-Riemann Equations: Let $f(z)$ = $u(x, y)$ +$iv(x, y)$ be a function on an open domain with continuous partial derivatives in the underlying real variables. Then f is differentiable at $z = x+iy$ if and only if $\frac{∂u}{∂ x}(x, y)$ = $\frac{∂ v}{∂ y}(x, y)$ and $\frac{∂u}{∂ y}(x, y)$ = −$\frac{∂ v}{∂ x}(x, y)$. So we have $f'(z)= \frac{∂u}{∂ x}(z) +i \frac{∂ v}{∂ x}(z)$. Let $f(z)$ = $f(re^{iθ})$= $u(r,θ)$ +$iv(r,θ)$ be a function on an open domain that does not contain zero and with continuous partial derivatives in the underlying real variables. Then f is differentiable at $z$ = $re^{iθ}$ if and only if $r \frac{∂u}{∂r}=\frac{∂ v}{∂θ}$ and $\frac{∂u}{∂θ}$ = $−r \frac{∂v}{∂ r}$.  Sorry, if this is not very good. I just decided to start learning complex analysis today... Securely erase hard drive using the Disk Utility I am planning on selling my laptop. So I formatted my disk using the Disk Utility and chose to overwrite the disk with zeroes.  Two questions:  Is this the same as overwriting the disk using dd?  sudo dd if=/dev/zero of=/dev/sda   And is this method secure enough so that buyers can't easily recover the previous data? Or should I take additional measures (like encrypting the disk, destroying the headers, etc.)? If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces. "Install alongside" option missing. How do I install Ubuntu beside Windows using "Something Else"? Whenever I startup from the LinuxLive USB, and try to install Ubuntu 12.04, it only gives me two options instead of the usual 3 or 4. My options are "Replace Windows 7 with Ubuntu" and "Something Else". Whenever I go into "Something else", it looks extremely confusing. How do I install it side-by-side while using the "Something else" section? These are my partitions.   /dev/sda /dev/sda1 ntfs (104Mb, System Required Partition) /dev/sda2 ntfs (1000097Mb, Windows 7 Partition)   Or should I try reinstalling my installer on a CD rather than USB and maybe then it will have the "Install alongside" function? How to check type of files without extensions in python? What is the best way to randomize an array of strings with .NET? My array contains about 500 strings and I'd like to create a new Array with the same strings but in a random order.  Please include a C# example in your answer. How to force a table into page width? I have the following table:  \begin{table}[htb] \begin{tikzpicture} \node (table) [inner sep=0pt] { \begin{tabular}{ l | l }   {\bf Symptom} &amp; {\bf Metric} \\ \hline Class that has many accessor methods and accesses a lot of external data &amp; ATFD is more than a few\\ Class that is large and complex &amp; WMC is high\\ Class that has a lot of methods that only operate on a proper subset of the instance variable set &amp; TCC is low\\ \end{tabular} }; \draw [rounded corners=.5em] (table.north west) rectangle (table.south east); \end{tikzpicture} \caption{God class symptoms} \label{tbl:god_class} \end{table}   Now I want to force the width of the table to be the same as the \textwidth, either by linewrapping of table text or by scaling. How can I achieve that? How to show an alert when user navigates to another page without filling the form? I have a form created by a custom module. Now I want to show an alert like ("Please fil the complete form before navigating otherwise lost all the data") if the user navigate to another page without filling all required fields.  How can I do so? Is there any web browser with NPAPI plugins support? As per  article, Mozilla has dropped the support of NPAPI plugins from Mozilla Firefox version 52. Is there any web browser with NPAPI plugin support? I have to use a website frequently, which uses Oracle Java plugin. I'm using Ubuntu 16.04 64 bit with Java JDK 8u121. Helps are much appreciated. Thanks. Edit1: After 3 years of asking this question, Java plugin still works fine in Palemoon v28.9.2-1 64 bit on Ubuntu 20.04 LTS. Java version is JRE 8u251 64 bit Edit2: As of 20th October 2020, Oracle completely removed Java Plugin from 8u271 JRE and JDK packages mentioned in  article. But 8u261 or prior versions works just fine. Period character used for local computer name I'm wondering about the period character . used as a shortcut for the local computer name.  You can use it when logging into Windows 7, for example, to specify that the user account you're entering belongs to the local computer rather than to a domain:  .\MyLocalUserAccount   And you can use it within a Windows Management Instrumentation (WMI) query to indicate that the target is the local computer rather than a remote computer.  You can also use it when working with named pipes to for similar purposes.  However, you cannot use it as the UNC network name for the computer, like this:  \\.\MyShare   Does this character have an official name (e.g. "LocalHostToken") and when and where can it be used for this purpose? how many permutations of {1,2,...,9} How many permutations of {1,2,…,9} are there such that 1 does not immediately precede 2, 2 does not immediately precede 3, and so forth up to 8 not immediately preceding 9? One obvious example of such a permutation might be 987654321, but there are many others, such as 132465879 or 351724698. How to cite a book chapter written by an author who is not the editor or author of the book with BibTeX? I don't know how to cite a chapter written by an author who is not the editor or author of the book.   How can I get the following citation?     Grandstrand, O. (2004), 'Innovation and Intellectual Property Rights', in J. Fagerberg, D.C. Mowery and R.R. Nelson (Eds.), Oxford Handbook of Innovation. Oxford University Press: Oxford. Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$. Let $K$ be the rational function field $k(x)$ over a perfect field $k$ of   characteristic $p &gt; 0$. Let $F = k(u)$ for some $u\in K$, and write   $u = f(x)/g(x)$ with $f$ and $g$ relatively prime. Show that $K/F$ is   a separable extension if and only if $u\notin K^p$.   There is already a post on this but it has not yet been answered :   The truth is that I am very confused with this problem, I would like someone please help me to understand what happens with this exercise and explain to me how it could be done by giving me a help or something. I already have an approach for the left-to-right implication, but I do not know very well how good it is. Could someone help me with the other implication please? Thank you very much. I'm looking for all the ways to increase Friendship in Pokemon X and Y.  Note that Friendship is different from Pokemon Amie Affection; they both do different things. Notification that ‘someone might be watching what you browse’ Where can the viewport clipping parameters be changed? Do Cleric's domain spells add slots? I'm a little bit confused with the whole spell system and additions to them. If I pick for instance Tempest Domain Spell, are the spells additional to the other ones? (2 at level 1) So do that make it 3 spells? Does the domain add slots too?  And also, I've read about people choosing multiple domains, is that possible? Prove that any finite group $G$ of even order contains an element of order $2.$ I ran dolphin like this:  dolphin . &amp;   I see messages in the console like these:  QPixmap::scaled: Pixmap is a null pixmap   How can I keep the console clean or silent?