I would like to solve:$$\int_{-\infty}^{+\infty}e^{-2i\pi xy-\pi x^2/r^2}dx$$  Where $i$ is the imaginary unit, and $y,r$ are constants. I've used Wolfram to get $re^{-\pi r^2 y ^2}$. Any way of solving it step-by-step?
Let $$G_\beta(w) = e^{\beta w^2}$$  Now I get the process of applying a fourier transform (or inverse) to get a new  gaussian:  $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$  but in doing the derivation the constant $G_\beta(0)$ never arose. I know how to calculate the constant, but where did it come from in the derivation?  edit: Fourier form used: $$\int_{-\infty}^{\infty} f(w) e^{-iwx} dw$$
At most finitely many (Hamel) coordinate functionals are continuous - different proof I'm sure this is a newbie question, but I can't figure this out. I have tried to follow the advice here:  and here:   But I can not seem to be able to generate any font change. I either get no change or error messages.  Here is the output of one failed attempt to change the font to Times using this command:  \rmfamily \renewcommand{\rmdefault}{ptm}     Apparently I either don't have a correct package installed or...  The output says " Metric (TFM) file not found." but I am not sure where to go from there. Was the "Eye of Sauron" Sauron's actual physical form? During the time frame of the Lord of the Rings, was the Eye of Sauron Sauron's actual physical form? If not, what form did he have? How to install Unity 8? I'm using Ubuntu 14.04 and found out that we can install Unity 8 on it. Will Unity 8 replace Unity 7 once I install it? And if it does can I switch back to Unity 7 if I want? How would one go about it? register_nav_menus() won't register menus Please help with menu registeration:  I have followed this guide (http://codex.wordpress.org/Navigation_Menus) step by step.  Added the following code to functions.php:  function register_my_menus() {   register_nav_menus(     array('header-menu' =&gt; __( 'Header Menu' ) )   ); }  add_action( 'init', 'register_my_menus' );   But the guide says that this will make the menu appear in the Appearance -> Menus admin screen in the "Theme Locations" box as "Header Menu".  When I go to Appearance -> Menus everything is greyed out and there is no option of "Theme Locations" I have a PostGIS database and have successfully consumed a table into GeoServer and published a WFS.  Are there REST endpoints available for these WFS' akin to ESRI's feature services?  How are parameters to be constructed? I'm adding a title clock in the foot line of a beamer slide, if I compile it with pdfLaTeX it works fine; while the clock doesn't align in the middle vertically if compiling with XeLaTeX. How can I fix this.  Here is my MWE:  \documentclass[10pt]{beamer} \usepackage[font=Times,timeinterval=1]{tdclock} \usetheme{Madrid} \title{Title} \author{Authors} \institute{Institute} \date[\initclock\tddate\ \ \tdtime]{\today} \begin{document} \frame{\titlepage} \end{document} How does searching a file (locate) or autocompletion work? I am a Windows user and coming from Windows with all the indexing and eternal wait times when searching for files, I find it surprising how fast does locate work or the autocompletion (that I know) work in linux.  Is there any indexing being done in the background or how is this achieved? I still have a pretty clean installation, so maybe this goes bad with time, but in Windows the moment you want to search on a folder which is not indexed you have to wait for some seconds. I'm trying to write a basic program in C using gedit or sublime text (which I prefer), but no matter what I do when I try to compile or run, or even while writing, I have no option to use 'stdio.h' or any similar package. Even when I start typing #include &lt;st the only word completion sublime suggests is 'struct'.  I've read dozens of threads, but all managed to fix this by installing gcc, which I've tried to reinstall multiple times, with no luck.  This is the output from gcc -v:  Using built-in specs. COLLECT_GCC=gcc COLLECT_LTO_WRAPPER=/usr/lib/gcc/x86_64-linux-gnu/7/lto-wrapper OFFLOAD_TARGET_NAMES=nvptx-none OFFLOAD_TARGET_DEFAULT=1 Target: x86_64-linux-gnu Configured with: ../src/configure -v --with-pkgversion='Ubuntu 7.2.0-8ubuntu3' --with-bugurl=file:///usr/share/doc/gcc-7/README.Bugs --enable-languages=c,ada,c++,go,brig,d,fortran,objc,obj-c++ --prefix=/usr --with-gcc-major-version-only --program-suffix=-7 --program-prefix=x86_64-linux-gnu- --enable-shared --enable-linker-build-id --libexecdir=/usr/lib --without-included-gettext --enable-threads=posix --libdir=/usr/lib --enable-nls --with-sysroot=/ --enable-clocale=gnu --enable-libstdcxx-debug --enable-libstdcxx-time=yes --with-default-libstdcxx-abi=new --enable-gnu-unique-object --disable-vtable-verify --enable-libmpx --enable-plugin --enable-default-pie --with-system-zlib --with-target-system-zlib --enable-objc-gc=auto --enable-multiarch --disable-werror --with-arch-32=i686 --with-abi=m64 --with-multilib-list=m32,m64,mx32 --enable-multilib --with-tune=generic --enable-offload-targets=nvptx-none --without-cuda-driver --enable-checking=release --build=x86_64-linux-gnu --host=x86_64-linux-gnu --target=x86_64-linux-gnu Thread model: posix gcc version 7.2.0 (Ubuntu 7.2.0-8ubuntu3) `   This is the code:   #include &lt;stdio.h&gt;  int main(void) { printf("hello, world\n"); }   When I compile using gcc, I get nothing, no error at least. but when I try to run using just ./hello.c this is what I get. Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$. Is this statement "Every element in the empty set is greater than itself." true or its negation is true? $p$ : Every element in the empty set is greater than itself.  $\neg p$ : Some element in the empty set is smaller than or equals to itself.      I do not have the answers to this exercise, but it came out in my exams. I simply do not understand. There is nothing in an empty set! How can nothing be greater/equals/smaller/ to itself? If I had to choose an answer, it has to be $\neg p$ since nothing is equals to nothing...   What is the answer? What is the use/meaning of 'swap area' while linux installation? When i installed ubuntu in my x86 machine, I had to configure some memory as 'swap area'.  What is the use of this memory and what is the importance of this memory in linux file systems?  How can I determine the exact size of 'swap area' in a machine for the safe working of linux inside in it? Nothing can ruin your day like a poorly-written bug report.  I've seen several sets of guidelines for bug reporting in different organizations. In your opinion, what guidelines/steps are most essential for good bug reporting?  Feel free to share complete guideline recommendations from prior/current projects/organizations. What is a good package for displaying algorithms? I found this page with some examples:   I'm wondering which one to use. Any suggestions? I'm using other packages too so whichever is less likely to interfere with them is better for me. I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new equation (adding or taking away from an expression) allows one to know what an unknown represents. What about equations lets us do this? How can I calculate sensor size (width and height) from focus length, vertical, and horizontal angle of view? Where can I find information about ubuntu compatible or certified hardware/PC models? Pay real money to set bounties (another method, not replacing rep method) It may seem like a stupid thing, but could we please be able to pay real money to set a bounty.  For example, if I had only 1 rep, I want to be able have a bounty.   Here late the suggested prices in USD. Convert money    $0.49 for each 50 reputation $0.99 for each week $1.99 for a month $9.99 for a year   Every month, the bounty fee would come in. This would mean you would have to enable your account with a bounty fee button on your profile. Extracting Windows 8 Start Screen Patterns Is there any way to extract the Windows 8 Start Screen patterns, in order to use them as standalone wallpapers on other systems?  For example see this screenshot:    I am interested in the dark blue background. I heard that this background is somehow adapted to your chosen color theme. So many different variations should exist.  Engadget has an article  briefly talking about these background patterns and the different color schemes. They call them "personalization tattoos".    Update:    I found the following page/guide:   This looks a very good way to achieve my goal. Unfortunately the guide seems to be dated.  I tried to find shsxs.dll on the RTM of Windows 8 without much luck. It looks like Microsoft changed the way they handle the background image. The file is nowhere to be found...  I also looked into uxtheme.dll, but it doesn't contain any interesting images.  Any new ideas? Any other *.dll with might be of interest here? Singular vs. Plural with Multiple Gerunds as Subject (IE: [Gerund] and [Gerund] are/is [something].) I'm trying to find out whether I should use a singular or plural verb when there are multiple gerunds as the subject of the sentence. For example:  Running the correct course and keeping a steady pace are/is necessary in order to win.  With either one of these by itself, &quot;is&quot; would be correct:  Running the correct course is necessary in order to win. Keeping a steady pace is necessary in  order to win.  With both gerunds combined, I can't seem to figure out whether the verb should stay singular since each phrase is singular, or if it should become plural since there are two connected by &quot;and&quot;. If we just treat the gerunds as regular nouns, then obviously it would become &quot;are&quot;, but I'm not sure if gerunds have the exact same rules as regular nouns. I know that if the sentence was:  Running the correct course and keeping a steady pace are both necessary.  That &quot;are&quot; would be correct, but without the &quot;both&quot; it sounds incorrect to me. Does anyone know the official rule here?