Let $G$ be a region in a complex plane. Suppose $f:G\to\mathbb C$ is analytic and one-one. Show that $f'(z)$ is not zero for any $z$ in $G$.    I was trying to do the contradiction method. But I can't conclude anything. Please help
I recently saw a lecturer prove the following theorem (assuming the result that every analytic function is locally 1-1 whenever its derivative is nonzero): Let $\Omega \subset \mathbb{C}$ be open, and let $f : \Omega \to \mathbb{C}$ be 1-1 and analytic on $\Omega$.  Then $f&#39;(z_0) \not = 0$ for every $z_0 \in \Omega$.  I got the basic idea behind the proof: we assume for contradiction that $f&#39;(z_0) = 0$, and, assuming without loss of generality that $z_0 = f(z_0) =0$, we have (from the power-series expansion) that $f(z) = z^kg(z)$ for some analytic $g$ in some disk at the origin (i.e., $z_0$) and some $k \ge 2$. Since $z^k$ is not 1-1 in any such disk (because there are multiple roots of unity), then $f$ isn't either.  However, the proof he gave was rather awkward and technical- it involved defining three different axillary functions, even though the idea was simple, and I've since forgotten how it exactly worked.  In any case, I'm convinced there's a better way.  The problem is that I'm having trouble turning the idea into a real proof- I know that it obviously follows if $g$ is 1-1, but I'm also pretty sure that that is too strong an assumption.  Am I missing something, or does the argument just have to be more complicated?
Option to contact moderators Due to some "recent issues", I think the following is necessary:  Users need a way to contact moderators through the StackOverflow website.  This is especially important for suspensions, where there is no way to appeal or discuss the suspension from within SO itself.  Basically, there would be some kind of a box on the user's page that allows them to mention something to the moderators and some way to read the replies. (Or they could just go to the user's email.) It looks as if the 10k-tools flagged post list is b0rken.  Next to the review link in the toolbar there is a flag counter showing there are 10 flags to be reviewed:    Clicking on that flag counter leads to an empty page:        The moment I visit that page (through clicking on the flags counter, or by going to the review queue first then clicking on Tools, selecting the flags tab if not already selected), the flags counter disappears from the toolbar:    but after a while (half an hour, an hour?) the flag count re-appears in the toolbar.  Either the counter is wrong (presumably taken from a Redis cache somewhere) or the 10k flag review page is falling over every time someone tries to load it.  This at first appeared to happen only on Meta, on Stack Overflow the flags page has plenty to review. However, there are now  on Stack Overflow. Why does work depend on distance? So the formula for work is$$ \left[\text{work}\right] ~=~ \left[\text{force}\right]  \, \times \, \left[\text{distance}\right] \,. $$  I'm trying to get an understanding of how this represents energy.  If I'm in a vacuum, and I push a block with a force of $1 \, \mathrm{N},$ it will move forwards infinitely. So as long as I wait long enough, the distance will keep increasing. This seems to imply that the longer I wait, the more work (energy) has been applied to the block.  I must be missing something, but I can't really pinpoint what it is.  It only really seems to make sense when I think of the opposite scenario: when slowing down a block that is (initially) going at a constant speed. How do I extract text that lies between parentheses (round brackets)? My very simple Greasemonkey script is not running? Given the pdf $f(y\mid\theta) = \theta y^{\theta - 1}1_{0<y<1}$, show $E\left(\frac{1}{2\theta\sum_{i = 1}^n W_i}\right) = \frac{1}{2(n-1)}$ $a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$  Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$  If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it. How to be sure API calls are using TLS 1.2 We are using .Net to make web service calls the Salesforce API. Is there a way to know for sure that the calls are using TLS 1.2? Are there any API logs or records of the protocol is use in Salesforce we can review to be confident the change is working? Short Question:  Why can we manipulate a read-only file in Vim using : + w + q + ! even without being an administrator?  Long Question:  I have a text file (myFile.txt) which is read-only for everyone:  navid@navid-ThinkPad-T530:~/ubuntuTest$ ls -l myFile.txt  -r--r--r-- 1 navid navid 26 Aug 22 21:21 myFile.txt   I can open it with Vim without having admin privileges:  navid@navid-ThinkPad-T530:~/ubuntuTest$ vi myFile.txt    I modify it and press: Esc + : + w + q + Enter and I see this error message:  E45: 'readonly' option is set (add ! to override)   So far, everything makes sense.  But when I press: Esc + : + w + q + ! + Enter, Vim saves the changes.  I'm using Ubuntu 16.04 and VIM 7.4. Conditional formatting of an entire row based on multiple cells I'm trying to make a conditional formatting rule in Google Spreadsheet that would highlight a row green based on text in multiple cells.   If cells L2, M2, and N2 = APPROVED; and cell O2 = X; then row 2 = highlighted green. load() method deprecated? $\sum_{n=1}^\infty \log \frac{n+1}{n}$  $\sum_{n=1}^\infty \log \frac{n+1}{n}$ =  $\displaystyle\lim_{n \to{+}\infty}(\log2 - \log1)+(\log3-\log2)+...+(\log(n+1)-\log n)$=$\displaystyle\lim_{n \to{+}\infty}\log(n+1)\to \infty$. So, the series diverges.  Is my procedure correct? How can I get the list of available databases on a SQL Server instance? I'm planning to make a list of them in a combo box in VB.NET. We currently have around 20 apex classes and 5 triggers which are not used in our production org. We want to delete them in our production instance. We dont have ant scripts ready yet to be used for our deployments and so we cannot use destructivechanges file. Currently our production deployments are done by our support team and they need documentation from us to do these deletes. Changesets do not allow to delete apex classes and triggers. So what is the best way to delete these apex classes and triggers from production? ASP.NET MVC 3 localization with DisplayAttribute and custom resource provider What exactly is voltage? Bit of a strange question, but what is it? My physics teacher said it was kind of like a "push" that pushes electrons around the circuit. Can I have a more complex explanation? Any help is much appreciated. How to Set a Short User Password in macOS Mojave and Later (10.14+) I just installed macOS Mojave and put in my usual short desktop-only password I've been using for years (yes, you read that right). Now though, the operating system rejects setting a password less than 4 characters. It also does not allow having no password at all.  I am completely comfortable using the shell if need be.  How can I change the password policy to allow quite short passwords? Is my way of solving and my answer correct?  Let $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$   And $g(x)=f^{-1}(x)$  Find $g'''(0)$  My attempt:  We know that $g'(x)=\frac{1}{f'(g(x))}$  $f'(x)=1+x+x^2+x^3+x^4=\frac{x^5-1}{x-1}$  $f'(g(x))=\frac{(g(x))^5-1}{(g(x))-1}$  $\Rightarrow g'(x)=\frac{(g(x))-1}{(g(x))^5-1}$  $\Rightarrow g'(x)[(g(x))^5-1]=g(x)-1$  Similarly differentiating again,  $\Rightarrow g'(x)[5(g(x))^4]+[(g(x))^5-1]g''(x)=g'(x)$  Similarly differentiating again,  $\Rightarrow g'(x)[20(g(x))^3g'(x)]+[5(g(x))^4]g''(x)+[(g(x))^5-1]g'''(x)+g''(x)[5(g(x))^4]=g''(x)$  Putting $x=0$,  $g(0)=f^{-1}(0)=0$  $g'(0)=\frac{1}{f'(g(0))}=\frac{1}{f'(0)}=1$  similarly,$g''(0)=-1$  $\Rightarrow g'''(0)=1$ On my Windows 7 system I installed via  from Fonts.com:   The font does not appear in C:\Windows\Fonts. However, it shows up in certain desktop applications, for example  as: Sabon Next Com Display  My intention was to use the font in a LaTeX document test.tex:  \documentclass{article} \usepackage{polyglossia} \setromanfont{Sabon Next Com Display}  \begin{document} Hello World! \end{document}   Unfortunately, xelatex test.tex fails:  […] Couldn't find `Sabon Next Com Displ.cfg' miktex-maketfm: No creation rule for font "Sabon Next Com Display".   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! fontspec error: "font-not-found" !  ! The font "Sabon Next Com Display" cannot be found. !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................    l.4 \setromanfont{Sabon Next Com Display}                                            % Sabon Next Com Display, SabonNex...  ?    Can XeLaTeX even work with fonts installed via SkyFonts? If not, what could be a workaround? What are the visa (or any other crucial) requirements of Bhutan for tourists? I heard that access to Bhutan is quite restricted for all people of outside South-Asian countries?  Can I have the details on that?