Does any of you know how to solve  $$\int_0^\infty \frac{u^\alpha}{1+u} du$$  analytically for $\alpha&lt;-1$? E.g., for $\alpha=-\pi$, the solution can be expressed by the Digamma Function according to Wolfram Alpha.  Prior solutions on   only cover $0 &lt; \alpha &lt; 1$.
I've been looking at  $$\int\limits_0^\infty  {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$  It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:  $$\displaystyle \int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^3}}}dx = } \frac{\pi }{3}\frac{1}{{\sin \frac{\pi }{3}}} = \frac{{2\pi }}{{3\sqrt 3 }}$$  $$\int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^4}}}dx = } \frac{\pi }{4}$$  $$\int\limits_0^\infty  {\frac{{{x^2}}}{{1 + {x^5}}}dx = } \frac{\pi }{5}\frac{1}{{\sin \frac{{2\pi }}{5}}}$$  So I guess there must be a closed form - the use of $\Gamma(x)\Gamma(1-x)$ first comess to my mind because of the $\dfrac{{\pi x}}{{\sin \pi x}}$ appearing. Note that the arguments are always the ratio of the exponents, like $\dfrac{1}{4}$, $\dfrac{1}{3}$ and $\dfrac{2}{5}$. Is there any way of finding it? I'll work on it and update with any ideas.    UPDATE:  The integral reduces to finding   $$\int\limits_{ - \infty }^\infty  {\frac{{{e^{a t}}}}{{{e^t} + 1}}dt} $$  With $a =\dfrac{n+1}{m}$ which converges only if   $$0 &lt; a &lt; 1$$  Using series I find the solution is     $$\sum\limits_{k =  - \infty }^\infty  {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}}} $$
Switching from one area of graduate study to another? Is it possible to switch from one area of graduate study to another in US universities? For example, suppose someone has enrolled in a computer science phd program. Can he switch over to math(or physics) phd program in the same school later?(or say from applied mathematics to pure mathematics?)  What are the steps for doing for doing that? Why can't a domain's root be a CNAME? I've just read  which essentially says that when I set up DNS for example.com, the root record can't be a CNAME, it has to be an A record.  My question is, why?   I'm sure the clever people who designed DNS didn't make arbitary restrictions for no reason, but I don't see what we gain by requring root domains to be A records. I would love to just point my example.com domain to someserver.somewebhost.example and forget about it, but I can't.  Please enlighten me, billpg. Iridescence In Cycles? Is it possible to create an  shader in Cycles? Specifically for something like , , , etc.. James Paterson in his novel writes, “she was shrink by trade, a forensic profiler, and Jack Morgan’s number two at Private “.   What does whole sentence mean? Amongst North Indian Students, the phrase "give an exam" is very popular. These students use the phrase to describe the act of writing the answers to examination questions.  The reason being, in Hindi, it's known as "exam dhena" which translates literally to "to give an exam".  In South India, however, a majority of people use the phrase "to take up an exam".  Which of these usages is more common/appropriate/grammatical? (As a prof I have developed a strange aversion toward the "give an exam" phrase; I find it wrong and ugly.) Can my spouse receive an emergency UK visa to visit a funeral? I'm a British national. My grandmother just passed. My wife is Chinese. In general it's a major pain to get my wife a visa to visit the UK, we have never been before.  Is there any way to get an emergency visa to attend the funeral?  What sort of documents would we need? Can I purge configuration files after I've removed the package? I was doing a package removal with apt-get remove but then realized I should have done a --purge along with it to remove the configuration files.  Can I remove the packages configuration files easily or do I need to reinstall the package and then remove with a --purge? How can I pass a command line argument into a shell script? I know that shell scripts just run commands as if they were executed in at the command prompt.  I'd like to be able to run shell scripts as if they were functions... That is, taking an input value or string into the script.  How do I approach doing this? I'm reading through  for the , which contains this figure on page 16:    This is of course an instrumentation amplifier that apparently uses an inverting amplifier's output instead of a ground in the lower-right corner of the above figure, but what really perplexes me is the 100 k&Omega; resistors attached directly to the noninverting inputs of three of the four amps. I don't recall seeing an instrumentation amplifier circuit in books or application notes that have them, and all of the instrumentation amplifiers I've built using the three op-amp scheme works fine without them.  The datasheets specifies an input resistance of 1012 &Omega;, which is 10,000,000 times greater than 100 k&Omega;, so it doesn't seem to add anything to the already high-impedance JFET inputs. I thought perhaps it has something to do with input bias currents, but that's just me making a wild stab in the dark.  Curiously, figure 26 in the same datasheet (page 18) shows a two-op-amp version of an instrumentation amplifier without the 100 k&Omega; resistors at the noninverting op-amp inputs!  What's the purpose of the 100 k&Omega; resistors at the noninverting inputs in the above circuit? Am I missing something completely obvious? Can PS4 Minecraft join with PC Minecraft? LCM is associative: $\ \text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$ Definition: The least common multiple of $a.b\in\Bbb{Z}$ is the smallest  $k\in\Bbb{N}$ that is divisible by both a and b.         Proposition: $\text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))\tag{2}$    Proof: We want to show $\text{lcm}(\text{lcm}(a,b),c)\mid\text{lcm}(a,\text{lcm}(b,c))$ and $\text{lcm}(a,\text{lcm}(b,c))\mid \text{lcm}(\text{lcm}(a,b),c)$ such that $\text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$.  Let $k=\text{lcm}(\text{lcm}(a,b),c)$, then $c\mid k$ and $\text{lcm}(a,b)\mid k$, but if we let $l=\text{lcm}(a,b)$, then $a\mid l\,\,\text{and}\,\, b\mid l$. Therefore $c\mid k\,\,,a\mid k\,\,\text{and}\,\, b\mid k$, but then $\text{lcm}(b,c)\mid k$. Thus $\text{lcm}(a,\text{lcm}(b,c))\mid k$      Question: Does this makes sense, ifso can I do the same for $\text{lcm}(\text{lcm}(a,b),c)\mid \text{lcm}(a,\text{lcm}(b,c))$? How do I set my wireless network to be private instead of public? Trying to maintain good security settings on my laptop, I want my home network to be interpreted as private and the university network at school to be interpreted as public. However, my home network is currently showing as public.  Here is what my Network and Sharing Center looks like:    How do I change my home network from public to private on Windows 8 Consumer Preview? Where should I start in tracking down Firefox malware Sign of the vector (especially $g$) while calculating I'm a high school student and I'm studying Newton's second Law. While my teacher is calculating a net force of an object he always treat all vectors as positive numbers. I think what he meant is the magnitude of the vector which is always positive, but the problem is that he didn't use the magnitude symbol, for example like $||Fg||$. He said the acceleration due to gravity , g, is positive, but shouldn't it be negative since it is pointing down. This also cause a problem: every time when I calculate the net force I can't add all forces together, instead I need to choose to use minus sign when the vector is pointing down. So in general which way is correct? How to create vector graphics In Photoshop? I know people prefer using CorelDraw or Adobe Illustrator, but I want to know if there is a technique I can follow to use Adobe Photoshop to make vector Images. Why does Ubuntu force users to create a password upon installation? Java: Detect duplicates in ArrayList? Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$. Is there an (easy) way to create or personalize .bst files? When you prepare an article for a journal (in my case, Quantum Information and Computation, Rinton Press), often they ask you to use some format for your references; in my case, the following:     [1] R. Calderbank and P. Shor (1996), Good quantum error          correcting codes exist,  Phys. Rev. A, 54, pp. 1098-1106.      [2] M.A. Nielsen and J. Kempe (2001), Separable states   are  more disordered globally than locally, quant-ph/0105090.      [3] A.W. Marshall and I. Olkin (1979), Inequalities:   theory of majorization and its applications, Academic Press (New   York).   The issue is, sometimes (and this is the case) they do not provide a .bst style file for bibtex users. I (and I guess many other people) prefer not to copy all the references of a bibtex file one by one into a tex document. So here is the question I am trying to answer:     What is the easiest way to create a .bst file? (for Rinton or any other publisher)   At first sight, I thought this would be very easy to do: after all, a .bst file is just a mere script to give format to a list. Strangely to me, after a long search I have been unable to find a simple clean solution to this problem. These are the the options I have been looking at:   I have tried to adapt a previous .bst file: concretely, apsrev4-1.bst, which has a similar format. Other people have tried this solution before , . However, it is an ugly option. First, it requires going through the script of the .bst and understanding it. Second, the script format used in different .bst files differs (quite annoyingly), so it is not easy to modify them unless you do understand what's written. The above solution is quite hideous and you really lose time on it, so I was wondering: isn't there any software tool (hopefully with a GUI) that does this job? It should, after all, pretty easy to program for a .bst script expert some sort of program to generate .bst files. There are some related questions in this site asking for something similar , ; there I have seen a couple of proposed apps, but going through their manuals they seem not so easy to use. Isn't there some software tool available to generate a .bst in less than 1 hour of work?   Remarks: ideally I would like to use/personalize/create .bst styles that support eprint fields (e.g. aps4-1.bst). On Irix's  4Dwm, there was an ability to minimise windows into a box (contrary to the task bar used by modern window managers). I have seen this also on an old HPUX.  See the "console" square in the linked image:    Is it possible to accomplish on Ubuntu, either with a plugin or maybe some  window manager other than Unity?