Let $f(z)$ be an entire function whose modulus is constant on some circle. Show that $f(z)=f(z_0) + c(z-z_0)^n$ for some $n \geqslant 0$ and some constant $c$, where $z_0$ is the center of the circle.   I only know a sketch of the proof, using Schwarz reflection Principle. I would seriously appreciate giving details to it.
Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you !
I can buy Dell Latitude E7270 through my employer. It will come with Windows, but I would like to install Ubuntu (most likely 14.04). This model does not appear on the list of certified hardware, however. Does anyone know whether I should anticipate any problems with his laptop? Thanks! When I set up a site in multiple languages, how should I set up my URLs for search engines and usability?  Let's say my site is www.example.com, and I'm translating into French and Spanish. What is best for usability and SEO?  Directory option:  http://www.example.com/sample.html http://www.example.com/fr/sample.html http://www.example.com/es/sample.html   Subdomain option:  http://www.example.com/sample.html http://fr.example.com/sample.html http://es.example.com/sample.html   Filename option:  http://www.example.com/sample.html http://www.example.com/sample.fr.html http://www.example.com/sample.es.html   Accept-Language header:  Or should I simply parse the  header and generate content server-side to suit that header?  Is there another way to do this? If the different language versions don't have different urls, what do I do about the search engines?    UPDATE 2011-12-06  Google has new recommendations for meta tags for explicitly pointing to other language content: .  UPDATE 2012-05-25  Related but not precisely:   UPDATE 2013-06-12  includes discussion of several URL schemes directly relevant to the question. If $2a_{n+2} \le a_{n+1}+a_n$, then $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$ This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does. What are the differences between Holmes, Moldvay, and Mentzer D&D? How do I boot from ISO file stored on hard disk? Square root of Positive Definite Matrix Flagging a deleted answer results in a message saying I am flagging too quickly Derivation of a generalization of Mertens' Third Theorem. One of Mertens' Theorems states $$\prod_{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma}}{\ln(x)}.$$ I have seen a generalized version that states  $$\prod_{m&lt;p\le x}(1-\frac{m}{p})\sim \frac{c(m)}{(\ln(x))^m},$$ where $m$ is a positive integer and $c(m)$ is a real number that depends on $m$. How can I derive this result? Is there a way to find what the values of $c(m)$ are? Thank you very much. How do I install .run files? My PC has both Ubuntu and Windows 7 installed. I have installed all my drivers in Windows like Nvidia graphics, motherboard etc. Shall I install them in Ubuntu too?  I looked on the web for my Nvidia drivers and I have found a .run file. How could I install it on my Ubuntu? Choosing between meaningful and meaningless hostnames Is it possible to find a smooth function $p$ such that $$p(x)=0 \text{ if } x\le -1$$ $$p(x)=0 \text{ if } x&gt; 0$$ $$0\le p(x) \le 1 \text{ if } -1&lt;x&lt;1$$ $$p(x) = 1 \text{ if } -\frac{1}{2}\le x \le \frac{1}{2}$$ Hint: The following function has continuous derivatives of all orders $$f(x) = 0 \text{ if } x\le 0$$ $$f(x) = e^{-\frac{1}{x^2}} \text{ if } x&gt;0$$  My thought: Considering the left-half of the function $p$, I tried something like $e^{-\frac{1}{\tan^2(x\pi)}}$. The shape looks good but I am not sure about the smoothness at $-1$ and $-\frac{1}{2}$. For $x=-1$, since $\tan$ is infinitely differentiable so we have smoothness there. But for $x=-\frac{1}{2}$, I am not sure about the smoothness after &quot;gluing&quot; with $y=1$. GUI to CLI command translator Is there any application (or sequence of commands) for this function: seeing the result of the actions, while using a GUI interface, being automatically translated into a Terminal (or similar CLI environment) on the same machine.  I saw this in action long years ago in VBA for Applications (for example the Macros in Excel). I wonder if there is same function in Linux. What is NDSolve`FEM`*? Use the new helpcenter for the question/answer ban message Visual Studio Editor does not underline errors anymore My Visual Studio (2008) Editor has stopped to underline Errors (this nifty wavy red lines). I can't really tell when, but it can be related to the installation of .Net Framework 3.5 SP 1 or the MVC Beta (which I guess is unlikely). Furthermore have I installed and uninstalled both CodeRush and Resharper for evaluation purposes (decided not to keep either one of them).  Does anyone know the problem and how to restore this functionality again? What is the CVE-2014-6271 bash vulnerability (Shellshock) and how do I fix it? How do I recover lost/inaccessible data from my storage device? Error occurred in deployment step 'Add Solution': Operation is not valid due to the current state of the object.? What is the Big-O time complexity of the following nested loops: for (int i = 0; i &lt; N; i++) {     for (int j = i + 1; j &lt; N; j++) {         System.out.println(&quot;i = &quot; + i + &quot; j = &quot; + j);     } }  Would it be O(N^2) still? What are the best ways to improve a site's position in Google? What are the best ways to improve a site's position in the Google search results?