I don't need this statement to be true globally - even if it's locally true I'm fine with that. I'm on a noncompact Riemann surface but that doesn't matter because I only want this to hold locally anyway, so we can just look at little subsets of $\mathbb{C}$. Say $h$ is an arbitrary holomorphic function.  Say $f,g$ are holomorphic functions with no common zero. Then I want to show that there exist $\phi_1, \phi_2$ holomorphic such that $h = f\phi_1 + g\phi_2$ locally (so on some neighbourhood of some fixed point). I'm happy to just work with Taylor series of all these functions (which is what I was doing) because as I said, I'm fine with this holding just locally. But I can't quite figure out how it works. I'd appreciate some help, please!
Let $\Omega$ be a region in the complex plane and let $f_1$ and $f_2$ be holomorphic functions on $\Omega$ having no common zero. Show that there exist holomorphic functions $g_1$ and $g_2$ on $\Omega$ such that, $f_1g_1+f_2g_2$ is identically equal to one on $\Omega$.
How to have colors in Milky-way? Many photos of Milky-way show it in nice colors instead of just the black of sky and white of the stars. Is the key to having those colors in the capturing of the image or in the post processing, or both? Or am I just doing it in a wrong time of year when the visible part of Milky-way is simply not showing color?  Here is a sample photo of how Milky-way looks like in Finland in September:    APS-C sensor camera, 18 mm, 30 sec, f/3.5, ISO 3200 - captured in pitch-black darkness of Finnish countryside, far far away from any citylights, at time when moon was still below visible horizon.  What I'd like to achieve is something like the photo in Vivek's answer to   I've just bought a wireless shutter release for longer bulb exposures, and have plans to build a barn-door tracking mount. But is longer exposure the needed trick, or what is? Word for Party *Steward*? Is there a word that describes a person who is usually not too drunk and takes care of the rest of the folks at a party?  Typically, they're the ones who drag the sloshed ones back to beds, clean up the house and generally maintain a bit of sanity. Port forwarding with UFW I realize UFW is mainly just for enabling/disabling access on particular ports, but I'm trying to forward one port to another. I want any connection to port 1724 to be forwarded to 1723.   The file /etc/default/ufw already has DEFAULT_FORWARD_POLICY="ACCEPT". And I believe I need to uncomment net/ipv4/ip_forward=1 in /etc/ufw/sysctl.conf. Finally I need to edit /etc/ufw/before.rules to add an iptables rule.   And here is where I get lost because I can't find any documentation for this file. Where do I add the rule? And how exactly?  In particular:   On exactly which line do I add the rule? Do I put anything before or after it? root folder access via gui I'm new to linux and the terminal. As I learn how to use the terminal to navigate through directories, I like to use the gui file explorer to mirror my terminal navigation. The icons and file extension info in the gui file explorer help me figure out what everything is. However, I can't access the root folder via the gui, I can only access it via the terminal. Is there a way to enable access to the root folder outside of the terminal? Is hiding the ball allowed in football? Imagine all our football team except one player form a tight circle, with our hands on the shoulders of the people next to us.  Inside the circle there is the eleventh person with the ball. The whole "construction" is moving slowly towards the opponent's goal. It is inevitable that we will score.  Is this illegal? If yes, according to what rule? How to retrieve multiple rows from database in PHP What formats supporting animation are suitable for the web? What formats supporting animation are suitable for the web and what are the pros and cons of each. Listing only directories in UNIX I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on  and found this bewildering formula,   $$1 + 2 + 3 + \dots  = - \frac1{12}$$  The article also says that "Ramanujan summation of a divergent series is not a sum in the traditional sense". I am wondering then what this summation actually implies?  Is there any interpretation of this confounding result in the physical world that someone not so math-savvy can understand? Removing OS from Dual Boot computer, and recovering HD space. I have a computer that dual boots Linux Mint, and Ubuntu, I am wondering what the best way would be to remove Linux Mint, then recovering the hard drive space that is resides in so that Ubuntu will be able to access it. Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form I need to express $$1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb  + \frac{1}{n + 1}\binom{n}{n}$$ in a simplified form.  So I used the identity $$(1+x)^n=1 + \binom{n}{1}x + \binom{n}{2}x^2 + \dotsb   + \binom{n}{n}x^n$$ Now on integrating both sides and putting $x=1$.  I am getting $$\frac{2^{n+1}}{n+1}$$ is equal to the given expression.But the answer in my book is  $$\frac{2^{n+1}-1}{n+1}.$$ Where does that -1 term in the numerator come from? I am currently reading about the interactions between light and matter, but I keep coming across conflicting explanations.  My initial understanding (using classical electrodynamics) was that light (EM) waves do not actually travel through glass. Instead, they are absorbed by the atoms and/or electrons within the glass, which then emit new EM waves. The new EM waves emitted by the energised particles make up the forward-radiation, which is the light observed from the opposite side of the glass. So it appears as if light is travelling through the glass, but it's actually being absorbed and then re-emitted with (mostly) the same properties.  The article  by Chad Orzel, based on a related SE question, confirms the above:      To understand the propagation of a wave through a medium, you can   think of each component of the medium– atoms, in the case of a glass   block– as being set into motion by the incoming wave, and then acting   as a point source of its own waves.   But an answer to a different but related question,  conflicts with the above:     the electrons in glass are tightly bound to atoms so they are not free   to move like the electrons in a metal and therefore they do not absorb   the photons.   I know the above is referring to the quantum electrodynamics understanding, but I think it still conflicts as it claims that EM waves are not absorbed by the electrons in glass because they are too tightly bound to their atoms.  Also, the  provides a small explanation of the mechanics behind refraction within glass, which kind of conflicts with both of the above:     In the case of dielectrics such as glass, the electric field of the   light acts on the electrons in the material, and the moving electrons   generate fields and become new radiators. The refracted light in the   glass is the combination of the forward radiation of the electrons and   the incident light. The reflected light is the combination of the   backward radiation of all of the electrons.   From my understanding, the above claims that some of the light is absorbed and re-emitted by the electrons (which conflicts with the linked answer), and that the rest of the incident light travels through the glass (makes up the rest of the refracted light) (which I think conflicts with the linked article, claiming the light is absorbed (although the article doesn't technically state whether all of the light is absorbed and re-emitted, so maybe that's where I'm going wrong(?))  I'm sure this is my misunderstanding, so I would be very grateful if you could clear it up for me:  When a light wave is incident on a sheet of glass, do the particles that make up the glass absorb the waves and re-emit them in the forward direction? Or do the light waves manage to travel through the glass, without being absorbed and then exit the glass? In the last 30 years, human technology has developed massively.  Every year a new smartphone is released with a faster processor, more memory, better features.  Why hasn't the same level of progress happened in the Star Wars universe?  In The Phantom Menace, we see a number of droids - R2 units, C-3PO, Pit Droids, etc.  In Return of the Jedi - the droids we see pretty much have the same capabilities.  I would have thought that technology would still be progressing in the Star Wars universe - but it appears to have stopped.  During wars on Earth, our technology capabilities grow - so it can't just be the effect of a massive war which is causing the stagnation of development.  So, what gives? Sum of the series:$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\cdots$ I have a question where i couldn't find any clue. The question is  $$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+\cdots$$  I could get the general term as $t_n=\frac{1\cdot3\cdot5\cdot7\cdots(2n-1)}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdots2n}$  I have also tried it to form the sequence in the telescopic form.But couldn't get. Any hint will be appreciated. Running a .desktop file in the terminal From what I can gather, .desktop files are shortcuts that allow application's settings to be customized. For instance, I have lots of them in my /usr/share/applications/ folder.   If I open that folder in nautilus, I can run these applications just by double clicking its associated file, e.g. double-clicking firefox.desktop runs Firefox. However, I can't find a way to do the same thing via terminal.   If I do gnome-open foo.desktop it simply opens foo.desktop as a text file. If I make it executable and then run it in bash it simply fails (which is expected, it's clearly not bash script). EDIT: Doing exec /fullpath/foo.desktop gives me a Permission denied message, even if I change ownership to myself. If I make executable and do the same command, the terminal tab I'm using simply closes (I'm guessing it crashes). Finally, if I do sudo exec /fullpath/foo.desktop, I get an error reporting sudo: exec: command not found.  That's my question, how can I run a foo.desktop file from the terminal? What is the difference between display: inline and display: inline-block? This question has an inverse:     What would be the most efficient way to translate a combination of $k$-tuple into its positions within the $\left(\!\!\binom{n}{k}\!\!\right)$ combinations? I need this to be fast for combinations of $\left(\!\!\binom{70}{7}\!\!\right)$ order of magnitude - very large, but not exceeding 2 billion (fits into int32 maximum value).  Below is an example of $\left(\!\!\binom{6}{3}\!\!\right)$, where the aim is to quickly translate (a, d, f) tuple to value 9, while in the real problem $k$ ranges between 5 and 8.  $$\begin{array} {cccccc|c} a&amp;b&amp;c&amp;d&amp;e&amp;f&amp;^{combination}/_{sort\_order}&amp; \\\hline x&amp;x&amp;x&amp; &amp; &amp; &amp;1\\ x&amp;x&amp; &amp;x&amp; &amp; &amp;2\\ x&amp;x&amp; &amp; &amp;x&amp; &amp;3\\ x&amp;x&amp; &amp; &amp; &amp;x&amp;4\\ x&amp; &amp;x&amp;x&amp; &amp; &amp;5\\ x&amp; &amp;x&amp; &amp;x&amp; &amp;6\\ x&amp; &amp;x&amp; &amp; &amp;x&amp;7\\ x&amp; &amp; &amp;x&amp;x&amp; &amp;8\\ x&amp; &amp; &amp;x&amp; &amp;x&amp;9\\ x&amp; &amp; &amp; &amp;x&amp;x&amp;10\\ .&amp;.&amp;.&amp;.&amp;.&amp;.&amp;.\\  &amp; &amp; &amp;x&amp;x&amp;x&amp;20\\ \end{array}$$  I know that I could pre-calculate all the combinations and reverse the lookup dictionary. However, such dictionary would not be efficient in terms of memory usage. Therefore I am looking for either calculation-based approach, or a more efficient data structure to perform this mapping. I have this run.sh file, and I've create a symbolic link into my desktop.   When I double-click on it the file the following dialog appearsr    And I have to click on "run" each time. How can I create a link that runs by default? Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $ I'm having trouble calculating the integral  $$\int_{- \infty}^\infty \frac{\sin x}{x+i}\,dx $$  using residue calculus. I've previously encountered expressions of the form   $$\int_{- \infty}^\infty f(x) \sin x \,dx $$  where you would consider $f(z)e^{iz}$ on an appropriate contour (half circle), do away with the part of the contour that wasn't on the real axis by letting the radius go to infinity, then recover the imaginary part of the answer to get back the sine. However here, I can't replace the sine with $e^{iz}$ in my complex function because $$\operatorname{Im} \frac{e^{ix}}{x+i} \neq \frac{\sin x}{x+i}\cdots$$  How to remedy this? I'm not sure if substituting $\sin x = \frac{1}{2i}(e^{ix}-e^{-ix})$ and solving two integrals is how this problem is meant to be solved, although I'm 99% sure it would work I have 16GB RAM. Do I need 32GB swap?