Let $f$ be an entire function s.t. $\forall z \in \mathbb{C}, f(z)\text{ is real}\iff z\text{ is real}$. Prove that $\exists a,b \in \mathbb{R}$ s.t. $a\neq 0$ and $\forall z \in \mathbb{C}, f(z)=az+b$.  I think I have a proof for this using some advanced theorems, but I want to find one that can be done by elementary means (by a student in undergraduate complex analysis).
Let $f$ be a holomorphic function defined on $\Bbb C$ such that $f^{-1}(\Bbb R)=\Bbb R$. Prove that there exists $a,b \in \Bbb R, \; a \neq 0 \;$ such that $f(z)=az+b \;$ for every $z \in \Bbb C$.  I was given a hint: define the function $g(z)=\frac{f(z)-f(0)}{z}$. such $g$ is also holomorphic in $\Bbb C$ since the only singularity is in $0$, and it is a removable singularity since we can define $g(0)=f'(0)$. Next I thought about proving $g$ is bounded, and thus constant (using Liouville's theorem), but pretty much out of ideas. How can I finish the proof/solve in another way?
I want to have an index entry typeset as $a_\text{T}$, so I issue the command \index{at@$a_\text{T}$}. Each use of this index command creates a separate line in the index. How can I get \makeindex to recognize they all reference the same topic?  My index entries look like this: Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and . Every partial order can be extended to a linear ordering I'm having a problem with author display for a statute - the author is 'City and County of San Francisco' but the output (using the apalike.bst style file provided by Elsevier) gives the author as 'City and San Francisco', missing the 'County' in the authorname.  Any suggestions? What is the different between an Normal user account and the Admin account in Ubuntu?  You must always enter your password if you want do something. Can you give a Normal account  "limitations" to do not install or remove anything or to not open any program? Can you also "hide/deactive" the terminal and console for a normal user? When you energise a taut string, the following resonant modes of vibration occur:    Plotting on the frequency domain, you can see their corresponding frequencies:    But what is the underlying physical principle? Why does this happen?  Is there any way of explaining it that could be understood by a smart 15 year old?  EDIT: I'm going to give my best attempt so far. Here goes:   We can start with sympathetic resonance. Sounding a particular frequency, a pure sinewave.  And noticing the string resonates sympathetically at the frequency of each harmonic. Say this is explained and understood. Now imagine that plucking a string is equivalent to a burst of white noise, which contains frequencies all the way through the spectrum. This could be approached backwards, by starting with random frequencies and noticing that resultant wave produced looks like white noise.   If the above is scientifically correct, then it restricts the domain of the question.  I would really like to be able to understand it scientifically and also be able to explain it intuitively.  PS Images from I'm trying to connect to a sandbox using the sfdx cli and the JWT flow through force:auth:jwt:grant. According to , we can use the same connected app information from Production to connect to our Scratch Org.  Is it something that should work for sandbox too ? I'm getting a "Failed: Not approved" error.  Connected app in sandbox comes from the refresh but has a different consumer key. The Permission set is set correctly on my user.  Is there a way to connect to a recently refreshed sandbox using the JWT flow (without manually changing something like the consumer key) ? I got a flat/puncture on my bike tube, how can I fix it myself without going to the nearest bike shop? What kind of equipment would I need? Blocking comment spam without using captcha Azure DNS does not lookup SPF policies for certain domains Updated title of arxiv not recognised by Google Scholar? I submitted a paper to a journal and one of the referees suggested to slightly change the title of the paper. The paper was already posted on arxiv. I have updated the paper on arxiv after the revision, but this update is not recognised by Google Scholar. The main problem with this is that Google Scholar is not detecting the new citations to the arxiv paper since the new citations use the new title.  Any suggestions on how to solve this issue?  Update: I updated the arxiv version 2 months ago but Google Scholar does not detect it. It detects new papers that cite my manuscript, but not the updated arxiv.  Disclaimer. I am aware of the following questions, but they are related to different issues. Editing a question with only a JSFiddle link does not work I can't edit a question that has a jsfiddle link because it has a lack of code. Can we get privileged users able to bypass restriction? The question had some really poor formatting and linking and I couldn't fix it. :  Html is not taking the css code Some of my html tags are not reacting by the css which i gaved them for example is offerbox not reaction to the html part. all html parts down from offerbox are not taking the styles that i gaved the enter code here http://jsfiddle.net/8Bw7M/  pic from the html and css  I feel that as a privileged user, I should be able to make the changes needed. This is related, but I'm not adding any new links. Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b&gt;0$.   My Try:  So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\pi\}$.   On $\Gamma$, $z=a\cos t+i b \sin t$, so $|z|^2=z\bar{z}=a^2\cos^2 t+b^2 \sin^2 t$.   Now, $dz=-a\sin t+i b \cos t dt$.   Hence, the integral becomes $\displaystyle \int_\Gamma \frac{dz}{z(iab+(\sin t \cos t)(b^2-a^2))}$. I know that $\displaystyle \int_\Gamma \frac{dz}{z}=2\pi i$. Now, how do I get rid of $\sin t \cos t$ part? I am stuck here. Can somebody please explain how? Symmetric and exterior power of representation Does there exist some simple formulas for the characters $$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V},$$ where $V$ is a representation of some finite group?  Thanks. Find which date is missing in the results How can one format a cell to have word wrap in Google Sheets?  I've tried Shift+Enter, but that doesn't seem to work.  One thing I know works is to place  and then if you edit your cell you can have word wrap in it.  But is there a way to do this without switching modes? How do you uninstall wine 1.5? Open-source alternative to Adobe InDesign There is some open-source software, similar to Adobe InDesign? It would be awesome if it do exist...  I know all the others like Inkscape, Gimp, etc... but for InDesign i have none.  Could you give me one direction? Windows seems to have a length limit on file names when trying to delete, though it won't prevent those files from being created.  Our build process creates a number of temporary files (many build off of a WSDL) that run afoul of this limit.  Our ant script is somehow able to delete them when doing a clean, but sometimes I need to delete the workarea directory (where all the temp files go) without actually doing a full clean from ant.  This is the same errors , but the answers there don't really work for me as I'm dealing with a directory, not a file, and I don't always know what specific files or subdirectories are causing the problem.  And I'm trying to avoid any manual process (other than triggering a single command) to actually delete them.  If I try deleting the directory from Explorer I get the error   Cannot delete [file name]: The file name you specified is not valid or too long.   Specify a different file name   Trying Remove-Item in powershell gives the following error:  Remove-Item : The specified path, file name, or both are too long. The fully qualified file name must be less than 260 characters, and the directory name must be less than 248 characters. At line:1 char:12 + Remove-Item  &lt;&lt;&lt;&lt; -force -Recurse &lt;directory&gt;   Does anyone know of any tools or easy ways to get around this delete error without having to manually find the problem files and move/rename them? Share wireless Internet connection through ethernet