I want to calculate $\displaystyle{ \int_{-\infty}^{\infty}\frac{\cos\left(a x\right)}{(x^{2} + 1)(x^{2} + 4)}\,{\rm d}x\, \quad}$. I started of by rewriting the integral cos(ax) into e^{iax}+e^{-iax} and then integrating along the line segment $\left[-R, R\right]$ and the semicircle $C = \left\{z: \left\vert z\right\vert = R, \; z\geq 0\right\}$. I am very new to complex integration and I got stuck there so I would appreciate any help on what to do next or guidance to solving this.
$$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx. $$  I am not sure how to solve this question. Can anyone help me to approach this problem. Thanks.
Adding a Method to an Existing Object Instance I've read that it is possible to add a method to an existing object (i.e., not in the class definition) in Python.   I understand that it's not always good to do so. But how might one do this? Does a percentage quantity take singular or plural verb agreement? Does a percentage require a singular or plural verb, for example, do we say ten percent "go" or "goes"? How does the Schengen 90/180 rule work? What's the best and quickest way to detect whether you're running a 32 or 64-bit version of Windows Server from the command line?  (Cygwin is installed) Find $\lim_{n \to \infty} \sqrt[n]{n!}$. I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test and I got stuck at something that I can't find on google. Right away while running the root test I encountered $\lim_{x \to +\infty}\left(\frac{1}{n!}\right)^\frac{1}{n}$. I have made the claim that $\left(n!\right)^\frac{1}{n}\geq 1 \hspace{3mm}\forall n\in \mathbb{N}.$ I have begun the proof and I think it is on the right track but some verification would be nice.  Proof: We can see that for n=1, this obviously holds, $\left(1!\right)^1\geq 1.$ Now suppose that this is the case for some $n=k$, then we have $\left((k+1)!\right)^{\frac{1}{k+1}}= \left((k+1)k!\right)^\frac{1}{k+1}$. It is at this point that I start to have a little trouble. Can somebody give me a push in the right direction? Thanks! How do I get the size of a java.sql.ResultSet? Shouldn't this be a pretty straightforward operation? However, I see there's neither a size() nor length() method. Why is my starting gravity low? In the fall of 2011 I made the leap from extract/partial kits to all-grain kits. I'm enjoying AG, but my SG is continuously below the expected. I have brewed 5 AG kits and 4 have had a low SG.   My assumption is my sugar extraction during mashing (or sparging) isn't optimal. I heat my mash water to 165 and mash the grains for 60 minutes. Mashing levels out at 152. For Lautering I heat the water to 175-180. After I drain the initial mash I add the lautering water and let that soak for 10 minutes. the temperature levels out around 170-172. Then I drain into the kettle and being the boil.  I aim for a gentle boil just enough to break the surface tension. I boil for 60 minutes, chill, rack, take a reading and pitch yeast.  Here are some of the numbers  type            expected     actual  kit st paul porter  1.052        1.039   http://www.northernbrewer.com/shop/st-paul-porter-all-grain-kit.html belgium strong  1.067        1.055   http://www.midwestsupplies.com/superior-strong-ale-all-grain-kit.html troegs hop back 1.063        1.049   http://www.byo.com/stories/recipeindex/article/recipes/90-american-amber-a-pale-ale/2196-troeegs-hopback-amber-ale-clone   The fuorth was the Phat Tyre kit from Northern Brewer. Can't remember the SG though. It was low. They taste great, but the numbers weren't "right". What can I adjust so the numbers are more inline with the recipe?   Update Thank you for all the feedback. There are a number of factors for me to review and adjust. Given my current setup I will follow @mdma advice and recirculate the wort once and extend lautering to 30 minutes.  It will be a week or two before I can attempt this, so it will be awhile before I report back again.  Update #2 I now have some awesome local contacts who are helping me out with technique and trouble shooting. Another (major) issue I could be experiencing is draining the wort way too fast. According to my friend. The drain should be low and slow after cycling a gallon or 2 of wort. sparging should take ~2 hours. for each gallon drained plato/brix should drop 3 points and go no lower than 3-5 points at the end.  the second recommendation is to heat up 3-5 gallons of water to 180 and poor it in prior to/while sparging.  my first attempt using this technique proved successful with the gravity being a lot closer to the anticipated OG. I had some temp issues with the batch, but I was so focused on sparging that I ignored the temp. My guess is I would have been right on if my temp didn't fall so low.  Update #3 I had the chance to brew with my mentor this weekend and he pointed out 2 glaring problems with my mashing technique.   Starting mash temp is too low. I was heating my water to 10-12 degrees above target temp. He recommends 17 degrees. between grain absorption &amp; the tun itself most of the heat dissipates within the first few minutes. This does go against what I originally said. That I was hitting my temp marks. Looking back I was hasty with my readings. I should not dump my sparge water (or lauter?) into the tun while draining the wort. The idea is to compact the liquid into the bottom of the tun thus compacting the sugars. I am stirring up the mash when I dump all sparge water in. We kept our sparge water 1-2 inches above the grain bed and our final results we excellent.   A 3rd recommendation he made was to lower the pH of the water. tap water is ~7 and the ideal mash pH is 5.2. He added a solvent, which I cannot remember the name of, which did this for us. I'll be picking up some of that this weekend.  Thanks to all who helped me out with this problem. I need to get the size of directory in terminal for signing purposes. I'm using following command:  du -s /path/to/dir   I'm multiplying the result by traditional UNIX block size (512 bytes) and get actual directory size in bytes. However, the Finder's "Get Info" dialog window shows the size slightly smaller than the one calculated with the terminal command. And it seems that it's reproducible on any folder/bundle. What am I missing? I am using Eclipse 3.3 ("Europa"). Periodically, Eclipse takes an inordinately long time (perhaps forever) to start up. The only thing I can see in the Eclipse log is:       !ENTRY org.eclipse.core.resources 2 10035 2008-10-16 09:47:34.801     !MESSAGE The workspace exited with unsaved changes in the previous session; refreshing workspace to recover changes.   Googling reveals  that I remove the folder:  workspace\.metadata\.plugins\org.eclipse.core.resources\.root\.indexes   This does not appear to have helped.  Short of starting with a new workspace (something which I am not keen to do, as it takes me hours to set up all my projects again properly), is there a way to make Eclipse start up properly? How should I cite a website URL? What's the difference between iterating over a file with foreach or while in Perl? Plotting solutions to NDSolve How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$? Assume $A \in \textbf{S}^n_{++}$, an ellipsoid centered at the origin given by $$\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$ Then we have $\mathcal{E}_A \subseteq \mathcal{E}_B $ if and only if $B-A \succeq 0$. This is the proposition in the Boyd &amp; Vandenberghe's  (pages 45-46). The authors gave only the result, without proof. How to prove it? dropping trailing '.0' from floats I'm looking for a  way to convert numbers to string format, dropping any redundant '.0'   The input data is a mix of floats and strings. Desired output:  0   --> '0'  0.0 --> '0'  0.1 --> '0.1'  1.0 --> '1'  I've come up with the following generator expression, but I wonder if there's a faster way:  (str(i).rstrip('.0') if i else '0' for i in lst)   The truth check is there to prevent 0 from becoming an empty string.  EDIT: The more or less acceptable solution I have for now is this:  ('%d'%i if i == int(i) else '%s'%i for i in lst)   It just seems strange that there is no elegant way to handle this (fairly straightforward) case in python. Comment hint not disappearing when I start to type comment If I click "add comment", the comment hint (placeholder; "Request clarification or add information.") is not a hint, but editable text. If I delete the text and click outside of the comment then the hint appears as it should and when I click back into the comment box the hint disappears as it should.   I am on Internet Explorer 10.0.7.  Am I just oblivious and this is how it has always been (having to delete the comment hint), or is this a bug? Once a lead is converted it automatically creates an account, contact and one opportunity. But in my case i want to display the result without creating an Account. Please provide me with a sample code.  Any help will be appreciated... Changing underline color of LanguageTool in TeXstudio I've recently installed LanguageTool to check grammar and spelling mistakes in TeXstudio. However, I often miss the errors because the gray-bluish underlining makes them hard to spot. Is there a way to change the underlining to a distinctive color, such as red for example?   I checked . It says I have to do some changes to the .languagetool.cfg file, but for some reason this file doesn't appear in File Explorer. “Unable to Connect to World” Minecraft PE 1.16.201 So my brother and I have been playing Minecraft together on our iPads for years now, but in September I got a new iPad Pro 3rd Generation and I’ve been having issues with Minecraft ever since. Every time we have tried to join each other’s worlds we get the message “Unable to connect to world”. Recently after the update on December 16th, it worked for about a week without having any issues, but now it has gone back to not letting us connect to the worlds. Both of us are able to connect to other people’s worlds, but not each other’s, we’ve tried many things to fix this but so far nothing has worked. If you have any ideas for what we should do please let me know, thank you so much! Technical term for associating a group with the actions of a few When polynomials f(x) and f'(x) are relatively prime, f(x) has no repeated roots. Why? The problem is to show that a polynomial $f(x) \in F[x]$ (F is a field) has no repeated roots if and only if f(x) and f'(x) (the derivative of f(x)) are relatively prime. I've managed to prove one direction of this equivalence (if there are no repeated roots, f(x) and f'(x) are relatively prime), but the other one gives me headaches... can anyone help out?