How to perform the integral $\int_{-\infty}^\infty\frac{e^{-x^2}\sin(x) }{x }$? Does anybody know how  to perform the integral $$ \int_{-\infty}^\infty\frac{e^{-x^2}\sin(x) }{x } $$ Thanks.
Integral of sinc function multiplied by Gaussian I am wondering whether the following integral $$\int_{-\infty}^{\infty}  \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If analytical form does not exist, I really appreciate any alternative means for evaluating the integral. One method would be numerical quadrature including Gaussian quadrature, but it may be inefficient when the parameters $a$ and $b$ are very different in scale.  EDIT : In view of  discussion, we have decided to add OP's self-answer to the end of the question, for it does not qualify as an answer yet contains vital details. The copying is unabridged. Thanks very much for your comments, and the following result was obtained including the case for $x_0 \ne 0$: $$ \int_{-\infty}^{\infty} dx \exp[-a(x-x_0)^2] \frac{ \sin(bx) }{ x } = \pi \exp(-a x_0^2) \mathrm{Re}\left(\mathrm{erf}\left[\frac{b+2iax_0}{2\sqrt{a}}\right] - \mathrm{erf}\left[\frac{2iax_0}{2\sqrt{a}}\right]\right) $$ where $a\gt0, b, x_0$ are assumed to be all real. (note: coefficients etc may be still wrong...) This integral appears in a type of electronic structure calculation based on a grid representation (sinc-function basis). I believe the above result should be definitely useful. Thanks much!! --jaian
A word for a person who always needs there to be a problem? I have a WinXP installation that is used to only connect to another PC with RDP.   A dummy user account gets logged in automatically, then the user must click a link to start the RDP connection with predefined parameters.  Is there a way to run the RDP client without Windows Explorer being accessible? Getting Session ID in Lightning If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$ Change equation font in Pages During a Pages update which was pushed out a few months ago, Pages obtained the ability to write equations using LaTeX or MathML commands. According to an Apple Support article, the size and color of the equation can be changed ():  The equation appears at the insertion point in your document (or before the selected text). The equation is an inline object, the same size and color as the surrounding text. If you change the size or color of the surrounding text, the size and color of the equation’s font also change.  This does not mention anything about the font used for the equation itself, which seems to be something similar to a Times New Roman font. Is there any way to change the font used, preferably to something like Latin Modern Math? I was considering the possibility of replacing the font file Pages uses with a downloaded Latin Modern Math font file, but I'm not sure what to replace. Could I replace Times New Roman in /System/Library/Fonts/ or something like that? Batch renaming files with pattern I have some files named like these:  63018933.mp4?token=1325697436_b0c3e70c6e339380b4a484c576a8c287  63808488.mp4?token=1325697401_4ae5f7a68d93873c8881b303e7655e14  How do I rename all them to, for example 63018933.mp4(remove characters after mp4)? Number of reflexive relations defined on a set A with n elements I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function. It is a well known fact that Alex is more soft-spoken than (she/her).    Why would "her" be wrong? Why must the sentence end with "she"? Difference in meaning of "frustrated at" and "frustrated with" What is the difference in meaning of "frustrated at" and "frustrated with" used in sentences like     He is frustrated with me   or      He is frustrated at me   When is each of these sentences above are used? Is there any way, shape, or form that ultraviolet light is recorded by a digital sensor?  Ideally, this question is related strictly to the sensor, without even a glass lens in between which could cut or block UV light.  But of course I don't take pictures with a bare sensor, I also have a lens, so I'd also like to know how much UV gets to the sensor to begin with. how to show only changed rows if want querying the following table :  EventDateTime (DateTime), Host (guid), Flag ( bit) 2013-04-01 05:00, {id-x}, 0 2013-04-01 05:01, {id-x}, 0 2013-04-01 05:02, {id-x}, 0 2013-04-01 05:03, {id-x}, 0 2013-04-01 05:04, {id-x}, 1 2013-04-01 05:05, {id-x}, 1 2013-04-01 05:06, {id-x}, 1 2013-04-01 05:07, {id-x}, 0 2013-04-01 05:08, {id-x}, 0 2013-04-01 05:09, {id-x}, 0 2013-04-01 05:10, {id-x}, 0 2013-04-01 05:11, {id-x}, 1 2013-04-01 05:12, {id-x}, 1 2013-04-01 05:13, {id-x}, 1 2013-04-01 05:14, {id-x}, 1 2013-04-01 05:15, {id-x}, 0   and the result should be:  2013-04-01 05:04, {id-x}, 1 2013-04-01 05:07, {id-x}, 0 2013-04-01 05:11, {id-x}, 1 2013-04-01 05:15, {id-x}, 0   i have try it with LAG,LEAD and OVER Clause. but no luck :( Since the  would it make sense to create a Stack Exchange site dedicated to homework help? It wouldn't have to be limited strictly to homework either, possibly including discussions of puzzles and brain teasers and possible approaches to solving them.  I have a programming background so I would be interested in helping with CS or math related homework, discussions of programming contest problems, etc. Topics don't necessarily have to be related to programming either. That's just where my interests are.  What are your thoughts? Exclude folder using `find` command I'm using the find command on a Mac to search for a folder called test1. Now test1 folder could be present in the .Trash folder also. How do I exclude the .Trash folder from getting reported in search results or basically any folder I wish to exclude?  $ find $HOME  -name test1 -type d /Users/theuser/.Trash/test1 /Users/theuser/test1  /Users/theuser/Downloads/test1    I want the result to be just  /Users/theuser/test1 /Users/theuser/Downloads/test1    I used grep:  find $HOME  -name test1 -type d | grep -v '.Trash'    to filter out the .Trash result, but I'm interested in knowing if using find alone achieves the same results. Google Chrome Address Bar Font has gone too large, how can we reduce it Running Unity on my desktop (2*20" LCD displays), I initiated a standard daily update on my Ubuntu 14.04 LTS today and Google Chrome updated itself to Version 35.0.1916.114.  The URL address bar input now shows IMO a too larger font size which then proceeds to auto show in green. I prefer the smaller font size and have no need for 'gooey slime green' auto complete font colours. Any advice on how to revert back to smaller, generic Address font styling.  I use both FF and Chrome for numerous tasks, Chrome being my primary browser and I've checked my system Unity 'default' theme (Ambiance! LOL.. yes I'm tired of that also).  Thanks for sharing. I have installed the developer preview for Yosemite.  Huge mistake. Everything in my workspace is broken.  I don't have any sort of backup image. I was wondering if I could re-download Mavericks and run the installer - would that effectively downgrade me back to Mavericks? Why does the $\sum_{n>1}(\zeta(n)-1)=1?$ While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \left(\zeta(n)-1\right) = 1 $$ Furthermore, if you use only the even numbers for the zeta function, the sum will converge to $\frac{3}{4}$, or $$\sum_{n=1}^{\infty} \left(\zeta(2n)-1\right) = \frac{3}{4}$$  Leaving $$\sum_{n=2}^{\infty} \left(\zeta(2n-1)-1 \right)= \frac{1}{4}$$  This is probably common knowledge among mathematicians, but I couldn't find much about it on the internet. Is there a proof of this or perhaps even a simple explanation why this is so? Recovering a password for a Word 2007 document When and why should I use % !TEX TS-program and % !TEX encoding? How to create a sub array from another array in Java?