Prove that if $f$ is analytic in a region symmetric with respect to the real axis and if $f$ is real for real $z$, then $f (z) = \overline{ f (\bar z)}.$ My attempt : If i take constant function $f(z)=1$ , then obviously it will satisfied $f (z) = \overline{ f (\bar z)} $ since $\bar1=1$ If $z=x \in \mathbb{R}$ then $f(x) =\sum_{n=0}^{\infty} a_nx_n $ Then $\overline{f(x)} =\overline {\sum_{n=0}^{\infty} a_nx_n}=\sum_{n=0}^{\infty}\bar a_n x^n$
Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$.  Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. Prove that $f(\overline{z})=\overline{f(z)}$ for any $z \in U$.    By definition, I know that $f$ analytic on $U$ means that for every $z_0 \in U$, there exists $r&gt;0$ and a sequence of complex numbers $\left(a_n\right)_{n=0}^\infty$ such that $f(z)=\sum_{n=0}^\infty a_n\left(z-z_0\right)^n$ on the disc $D(z_0,r)\subseteq U$. I see that $f$ takes points without imaginary components to other points without imaginary components. But I don't see how this implies a symmetry that $f(\overline{z})=\overline{f(z)}$ for $z \in \mathbb C \setminus \mathbb R$. It seems like a very strong conclusion and I'm not sure how I would prove it.  I have also shown in the preceding question that $U\cap\mathbb R$ contains an open interval, however I am not sure if that detail is meant to be helpful to this question.
After reading the great answer for  by @Ben, I am a bit confused by the part &quot; set the relative tail sizes as a control variable, and then you find the formula for the length of the confidence interval conditional on that variable&quot;. I understand this means you need to compute the length of the confidence interval as a function of the relative tail sizes and then minimize the function. However, what exactly is relative tail sizes? Is it the ratio between the areas of the two tails? Also, is there another way to find shortest CI for a given confidence level? For those who are interested, I know there are somewhat relevant results allowing us to compute sample size such that the length of a CI, say 95% CI, does not exceed a certain bound. Can I use my powers for good? Is it correct to say "What was your name?"? The reason I am asking this is, generally the name of the person will not change. One should say "What is your name?" Bonferroni adjustment in SPSS - what does it do? I'm probably being a bit simplistic here, but from what I understand the Bonferroni post hoc test is simply when we adjust the alpha to account for inflated error when several post hoc tests are conducted. You divide the alpha by number of tests. So what exactly does SPSS do when we click the button for Bonferroni?   Shouldn't the dividing of alpha be done by us in interpreting the result, e.g. if we do 10 post hoc tests our alpha criterion should be .005 so if the p value is .012 then it is not significant, but SPSS hasn't done anything there, we have just changed our interpretation. So what exactly is it adjusting? How to repair Ubuntu installation after interrupted dist-upgrade without losing any custom settings? During my attempt to upgrade my Xubuntu 14.04 installation with  sudo apt-get dist-upgrade   I made the mistake of interrupting the process by closing my laptop, which caused it to suspend to disk. After re-opening, the monitor remained black, so I was forced to reboot the system. I have since discovered many failures of drivers (wlan0 missing, USB memory stick cannot be mounted, USB mouse not recognised, and maybe more undiscovered problems with hardware).   My beginner's attempts to manually repair the single problems could have made it even worse. During the installation of the current distribution I chose to encrypt my whole hard disk. I installed Ubuntu 12.04 a month ago and am using it till now. I failed to notice that all this time there was no sound at all while running Ubuntu, even while playing a game in Wine. The weird thing is that only the startup sound comes when I log in (Indian/African drum tone), then comes the utter silence.  I tested both Digital Output (S/PDIF) and the speakers in the sound settings but can hear nothing.  Any help? Spurious space above align environment at top of page/minipage It seems that the align environment adds additional vertical space if it starts at the top of a page or minipage.  Below are two pages, and two minipages (on Page 1), where the first page/minipage has text followed by an align* environment, and the second page/minipage starts with an align* environment, and note that the equations are still aligned.  Attempted Solution: The closest related question I found was:    I attempted to use the  at the above by adding \setlength\abovedisplayskip{0pt} (after the preamble as per ).  Although, this did move the equations up vertically, they were still vertically aligned:  And since the problem occurs at top of a page as well, I don't think this is related to  Question:  How can I adjust the align environment to automagically not have additional space at the top of a page or minipage? Is there a reason for this behavior, and perhaps should not be changed?  Code: \documentclass{article} \usepackage[showframe,paperwidth=3.5in,paperheight=2.2in,margin=0.5in]{geometry} \usepackage{amsmath}  \begin{document}%\setlength\abovedisplayskip{0pt} Top of Page 1 \begin{align*}% Text above at top of page  e &amp;= m c^2 \end{align*}  \hrule \begin{minipage}[t]{0.45\linewidth}     Top of minipage 1     \begin{align*}% Text above this minipage      e &amp;= m c^2     \end{align*} \end{minipage} \begin{minipage}[t]{0.45\linewidth}     \begin{align*}% No text above in this minipage      e &amp;= m c^2     \end{align*} \end{minipage}  \newpage \begin{align*}% No text above at top of page  e &amp;= m c^2 \end{align*} \end{document} Why is "letter" not plural in "two letter words"? Is there any program to provide a consistent interface across multiple archive types? At the moment, if I download a compressed file, it could be any of a .tar.gz archive, a tar.bz2 arhive, a .zip archive or a .gz archive. And each time I do so, I have to remember what the command line options for that program are.  Is there any CLI program where I can just go:  undocompression somefile.??  and let it figure out what format the archive is in? (overly long name used to avoid conflicting with any real program) Time taken to hit a pattern of heads and tails in a series of coin-tosses Inspired by Peter Donnelly's talk at , in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two patterns 'hth' and 'htt', it calculates how long it takes (i.e. how many coin tosses) on average before you hit one of these patterns.  coin &lt;- c('h','t')  hit &lt;- function(seq) {     miss &lt;- TRUE     fail &lt;- 3     trp  &lt;- sample(coin,3,replace=T)     while (miss) {         if (all(seq == trp)) {             miss &lt;- FALSE         }         else {             trp &lt;- c(trp[2],trp[3],sample(coin,1,T))             fail &lt;- fail + 1         }     }     return(fail) }  n &lt;- 5000 trials &lt;- data.frame("hth"=rep(NA,n),"htt"=rep(NA,n))  hth &lt;- c('h','t','h') htt &lt;- c('h','t','t')  set.seed(4321) for (i in 1:n) {     trials[i,] &lt;- c(hit(hth),hit(htt))     } summary(trials)   The summary statistics are as follows,        hth             htt          Min.   : 3.00   Min.   : 3.000    1st Qu.: 4.00   1st Qu.: 5.000    Median : 8.00   Median : 7.000    Mean   :10.08   Mean   : 8.014    3rd Qu.:13.00   3rd Qu.:10.000    Max.   :70.00   Max.   :42.000    In the talk it is explained that the average number of coin tosses would be different for the two patterns; as can be seen from my simulation. Despite watching the talk a few times I'm still not quite getting why this would be the case. I understand that 'hth' overlaps itself and intuitively I would think that you would hit 'hth' sooner than 'htt', but this is not the case. I would really appreciate it if someone could explain this to me. How come such different methods result in the same number, $e$? Execute shell script or command on idle time I'm searching for the possibility to execute a shell script or command based on idle time in the bash shell.  As an example, if I'm in a SSH session to a server of mine and after X minutes of inactivity in the terminal, I want a command to start a desired program like cmatrix, and when I come back and press a key I will get back where I left. How can one prove the Bohr-Sommerfeld quantization condition $$ \oint p~dq ~=~2\pi n \hbar $$ from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation? With $S$ the action of the particle defined by Hamilton-Jacobi equation $$ \frac{\partial S}{\partial t}+ \frac{(\nabla S)^{2} }{2m}+V(x)~=~0 .$$ Reapplying for a UK Standard Visitor visa after a refusal What does "official Ubuntu flavour" mean? What kind of support do they receive from Canonical or/and Ubuntu community? Who decides a distro should become official flavour? What privileges do "official flavours" have over unofficial derivatives of Ubuntu like Linux Mint, Elementary OS, Deepin OS,etc.? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive.  The laptop has windows 7 installed with ubuntu 11.04 installed through wubi.  What I want to do is remove windows completely, and make ubuntu the only OS installed on the system.  Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?)  or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in). How to use host names rather than ip addresses on home network? I've a small home network up and running and managed to set up ssh. Can anyone tell me what I need to do to refer to one computer on the network from another by host name rather than IP address (which is what I'm doing now)? How does a zener diode and a resistor regulate voltage? limsup of intersection of events as a subset of intersection of limsups Let $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ be two sequences of events in some probability triple $(\Omega, \mathcal{F}, \mathbf{P})$. Now, it is true that $\left(\limsup_n A_n\right) \cap \left(\limsup_n B_n\right) \supseteq \limsup_n (A_n \cap B_n)$, and I can think of many different cases where this relation holds true for equality (e.g., let $A_n = B_n$, or $A_n = B_n^c$, or $B_n = \emptyset$, or $B_n = \Omega$). However, I cannot think of any case where $\left(\limsup_n A_n\right) \cap \left(\limsup_n B_n\right) \supset \limsup_n (A_n \cap B_n)$ holds for strict inclusion.  My proof for the relation is simple. Since $A_n \cap B_n \subseteq A_n$ and $A_n \cap B_n \subseteq B_n$, then we can perform some set algebra as follows:  $$\bigcup_{k=n}^\infty (A_k \cap B_k) \subseteq \bigcup_{k=n}^\infty A_k$$ $$\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty (A_k \cap B_k) \subseteq \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k$$  Then we can use the same logic for $B_n$, and come to the conclusion as stated in the beginning. Javascript: negative lookbehind equivalent? Is there a way to achieve the equivalent of a  in javascript regular expressions?  I need to match a string that does not start with a specific set of characters.  It seems I am unable to find a regex that does this without failing if the matched part is found at the beginning of the string.  Negative lookbehinds seem to be the only answer, but javascript doesn't have one.  EDIT: This is the regex that I would like to work, but it doesn't:  (?&lt;!([abcdefg]))m  So it would match the 'm' in 'jim' or 'm', but not 'jam'