Problem. Suppose that $f$ has an isolated singularity at the point $a$, and $f'/f$ has a first-order pole at $a$. Prove that $f$ has either a pole or a zero $a$  This is a problem from a past qualifying exam in the institution I'm attending now which might appear in the final exam on tomorrow.(so if asking this is a problematic, then let me know, I'll delete it.)Because $f$ should have pole or removable or essential singularity at $a$, I checked that if $f$ has removable singularity at $a$ then $f(a)$ has to be $0$ to satisfy the condition in the problem, and also checked that if $f$ has pole at $a$ then $f$ satisfies the condition in the problem. So, I was trying to show that $f$ can't have essential singularity at $a$, and tried to use Casorati-Weiestrass theorem to derive some contradiction and it didn't work. I also tried to just plug in Laurent series expanded at $a$ in $f'/f$ and do calculation to eliminate some coefficients with negative indices but calculation would result infinite system of linear equations so I gave up doing it. Therefore, I'm stuck and please give me some hint.
The following is an old qualifying exam that I cannot solve:  Suppose $f$ is holomorphic on the punctured unit disk and $ \frac{f '}{f} $ has a simple pole at $0$.  Show that $f$ does not have an essential singularity at $0$.
Can't access https despite port 443 is opened String concatenation: concat() vs "+" operator Assuming String a and b:  a += b a = a.concat(b)   Under the hood, are they the same thing?  Here is concat decompiled as reference. I'd like to be able to decompile the + operator as well to see what that does.  public String concat(String s) {      int i = s.length();     if (i == 0) {         return this;     }     else {         char ac[] = new char[count + i];         getChars(0, count, ac, 0);         s.getChars(0, i, ac, count);         return new String(0, count + i, ac);     } } Gauge symmetries and elementary particles The  (disclaimer: I don't know this wikipedia entry) is usually mentioned as the reason why gravitons may not be composite particles. I do understand the proof of the theorem, but not the previous conclusion.   The theorem states that in an interacting and Poincare invariant quantum field theory, there are not massless, spin-2 particles unless there exists a gauge symmetry — which makes the energy-momentum tensor non-covariant (actually covariant up to a gauge transformation) under Lorentz transformations in the Fock space. So the immediate conclusion of the theorem is that the existence of a massless, spin-2 particle (like a graviton) requires linearized diffeomorphisms.  My question is: why do linearized diffeomorphisms imply that gravitons are elementary particles? Or, more in general, why  the particle corresponding to a gauge field must be elementary (I know that a gauge symmetry must be exact, but why this implies that the corresponding particle must be elementary?). This question is pretty simple: I'd like to have the possibility to get the exact size of a picture in LaTeX with, for example, the macro \the.  In fact, I've founded the way \settoheight{} and so on, but it didn't work at all. What I've got doesn't match with what I can see in the image properties (I suppose that's the name in English). The result: 8pt versus 1300 pixels. It's may be a mis-use...  Edit: In the end, my aim was to find a way to fit the picture in LaTeX. If width and height were both less than the dimensions of the body, then it should just be centred. It look like this :     Require:   \usepackage{graphicx} \usepackage{xifthen} \usepackage{calc}   Code (pretty ugly, I agree):   % Centrering+fitting for image, with captation and label (not auto) \newlength\imageheight \newlength\imagewidth \newcommand{\pic}[3]{% \settoheight\imageheight{\includegraphics{pictures/#1.png}}% \settowidth\imagewidth{\includegraphics{pictures/#1.png}}% \ifthenelse{\lengthtest{\imagewidth &gt; \textwidth}}{\ifthenelse{\lengthtest{\imageheight &gt; \imagewidth}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[height=0.85\textheight]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[width=0.95\textwidth]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}}{\ifthenelse{\lengthtest{\imageheight &gt; \textheight}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[height=0.85\textheight]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}}} %   To use it : put your png files in a "pictures" directory, then, just simply call the picture like this : \pic{name_whitout_extension}{Captation_if_needed}{Label_if_needed}.    Edit 2 :   Thanks to Martin Scharrer's easyfig package, I've got a more or less 1,6 times faster solution to put my pictures, centred, with the perfect size.  CTAN:   VC:   Version v1.2 – 2012/05/15    This could be useful for some others, I hope. I have a material assigned to a few test objects with an image sequence as a texture. If I view the scene in render preview, and scrub the timeline, (or render a sequence,) the texture behaves as expected.  However, with the viewport set to 'texture' or 'material' I only see the first frame of the sequence, wherever I am on the timeline. 'Auto Refresh' is set in the cycles texture node. Is this the expected behaviour? How can I download this flash game for offline play? is one of the best flash games ever.    Is there some way to download it, so that I can play it offline? Which graphics cards are supported by the new AMDGPU driver in Ubuntu 16.04? Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?   PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely? find exclude directory When can we interchange “who/which" and "that”? So far I thought the words who and that are usually interchangeable:   There is the man who saw me.  There is the man that saw me.    Both sentences seem fine, who can be replaced with that. But in the following examples:   This is the man to whom I spoke.  This is the man to that I spoke.    The #2 sounds odd, which is why I don’t think it might be correct. The same observation for the words which and that, while   This is the ball which we used today.  This is the ball that we used today.    Seem to be in order and which can be replaced with that. But in the following examples the #2 sounds odd again.   This is the ball with which I scored.  This is the ball with that I scored.    I wonder if there’s any rule regarding the interchangeability of the above mentioned words. I would like to set the wallpaper in the login screen (Unity Greeter), how do I do that? Proving Geometric Hahn-Banach Implies Analytic Hahn-Banach I've come across a proof online of the H.B theorem using the goemetric version.   There is a step in the proof which im not sure why is true and it is as follows:   Let $X$ be a linear space and $M \subset X \times \Bbb R$ be a maximal subspace.  Then $M = G(F)$ for $F :X \to \Bbb R$ linear.   Someone can show me why this is true?   Maybe I misunderstood the proof, its the last paragraph of Matrin's answer here -.   Thanks a lot for helping! Smooth reload of uitableview data when rows are deleted What are the salary and perks of Captain America? Brooklyn is Captain America's home, but times have changed since he was kid 8-9 decades ago. He can't afford a house there, now. This thing really troubled me when he said it in Avengers: Age of Ultron. I mean, I really became sympathetic, despite it being intended for fun.  What's the salary Captain America collects from (fallen) S.H.I.E.L.D. and/or Tony's Avengers organization? Can't they gift him a house in Brooklyn? I mean, he is one of Earth's mightiest heroes who help saved the world lots of times. SemanticImport vs Import differ on same CSV file? Moderncv adding date of birth to personal information Honey changing into unusable, hardened clump I buy raw, unfiltered honey from the grocery store in jars sold/packed directly by the farmer. When I had used the last jar to about half, I noticed the honey beginning to dry(crystalize?) to the point where I couldn't use the honey around these spots.  Not too long after this, the entire jar had crystallized into a block of unusable hard stuff.   I didn't do anything differently with this jar than the last jar, but only now am I having the issue. My question(s):   Why is the honey crystallizing, and why so rapidly?   1) Is the honey unusable at this point?  2) Should I store the honey differently after purchasing it(move from jar to other container)?  3) I've never seen this happen to the filtered, strained(fake) honey from the store, so what causes this to happen in the raw and unfiltered form? Setting custom UITableViewCells height The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$ Does the Planck scale imply that spacetime is discrete?