So I had this question as extra credit on my exam and no one was able to solve it.  I tried to do integration by parts but that just keep on expanding with no solution in sight.  I have also tried to take the derivative of the integral but that didn't seem to help either.  How would I integrate $\int_{0}^{\infty}e^{{-u}^{2}}\cos(ux)du$?
Prove that :  $$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$    the only thing I can think of is differentiating the RHS and trying to get :  $$ -2 f'(a) =a f(a) $$ But I couldn't do it.  Can anyone show me how to do this ?
Importing Nodes to D7 with all custom fields I've got a Drupal site with an obscene about of content that has been given to me in CSV format. I'm toying with the idea of building a module to let me do this but if there's one already out there that I can use to just get this site out of the way that would be awesome.  My question is: Is there anything out there that allows you to import a CSV file into D7 that will populate the node table and all the custom fields tables as well?  Thanks. If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication) Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But  writes that there is a converse: "If $L(V,W)$ is Banach, and if $V$ is non-trivial, then $W$ is Banach". This is pretty interesting since I never seen a converse before. I was wondering if anyone has a nice proof.  I tried reversing the proof for the usual direction, but the inequalities can't be reversed. I also tried to start with a Cauchy sequence in $W$, and construct linear operators (using Hahn-Banach to control the operator norms), but alas I cannot say much about the distance between these operators (much less say that it is Cauchy) Can I have multiple primary keys in a single table? My exterior door deadbolt has gotten stuck - it'll open with the lever but not the key. By spraying WD-40 inside the lock, I've been able to get it to work but it's very stiff. I'd like to remove the lock and inspect the inside, either to clean/lube it or replace it. But I can't figure out how to remove the lock; there's no obvious way to do this. (I'm used to seeing two screws on the inside of the deadbolt, usually under the lever arms when it's horizontal.)    How do I remove this? I can loosen the cover plate on the inside but it doesn't do anything other than rotate a few degrees. Ping is a a program owned by root with the user id bit set.  $ ls -l `which ping` -rwsr-xr-x 1 root root 35752 Nov  4  2011 /bin/ping   As I understand it, if a user runs the ping process, then the effective user id will change from the real user id (i.e. the user id of the person who launched the process) to the user id root. However when I try this and look at the output of ps to see if the ping process is running as the root user, I still get the real user id showing.  ps -e -o user,ruser,euser,cmd,args | grep ping sashan   sashan   sashan   ping -i 10 -c 1000 www.goog ping -i 10 -c 1000 www.google.com Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$. In Burton's Elementary Number Theory, when introducing absolute pseudoprimes (also called Carmichael numbers), the author states that they are composite numbers $n$ such that $a^{n-1} \equiv 1 \pmod n$ for all integers $a$ with $\gcd(a,n) = 1$.   When showing that 561 is an absolute pseudoprime, he then concludes that $a^{561} \equiv a \pmod{561}$ for all $a$.   I searched a bit on the internet, and it seems that these two are actually equivalent definitions of a Carmichael number. However I have not been able to find a proof; the fact that the implication in the title of the post is stated so plainly in my textbook makes me think it is quite simple to see, but I can't seem to understand the argument. Possible Duplicate:          If I am using the word model in the context of financial models and the UK, then which of these words should I use? Also, are there any key difference in the meanings of the two? Can I get the "Diamonds to you!" achievement in singleplayer? How can one get the "Diamonds to you!" achievement in singleplayer? The achievement list cannot be completed without it. Perhaps it can be possible by passing a diamond to a zombie? Generate four random words from a list for XKCD-like passwords For start, we know how can we output random lines from a txt file:   perl -MList::Util -e 'print List::Util::shuffle &lt;&gt;' words.txt   But we need a general solution (perl is usually installed on every unix-like OS) to output 4 random words from a txt file, how can we do it?    $ cat words.txt ... alpha beta gamma delta epsilon ... $   The "..." represents many other words.   Example output (without newline between them, but one unix newline is needed at the end, spaces should separate them):  $ perl SOMEMAGIC words.txt gamma alpha delta beta $   Using only perl, the shortest possible solution (oneliner).   It will be used as human memorable password generator:    Since afaik perl is enough good to be named "random generator".   Example common english wordlist with 32768 unique lines: Safely moving and creating new tempdb files Two things I'd like to know:   how do you safely move tempdb with minimal downtime? how many tempdb files do you need?   Is it 1 file per core? So quad-core = 4 tempdb files, creating three new ones? Formula for calculating $\sum_{n=0}^{m}nr^n$ Why does Type I error always occur in a NHST? Or, is NHST too weak to tell the truth? Recently I have noticed that Type I error of a test is too much persistent. Even if all the values are almost equal (variance is extremely low) and tested to that value (true mean), still there is nearly 5% Type I error of the test. I am frightened and curious to rethink, are the people who have banned (or skeptic about) NHST (null hypothesis significance testing) truly correct? Let's have a look at the following r code:       n=10000 t1err=0 set.seed(1235) for (i in 1:n){    x=rnorm(100, 10, 20)    if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)  } cat("Type I error rate in percentage is", (t1err/n)*100,"%")   It will produce roughly 5% as Type I error. Seems ok. But let's see this:      n=10000   t1err=0   set.seed(1235)   for (i in 1:n){      x=rnorm(100, 10, 0.000001)      if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)    }   cat("Type I error rate in percentage is", (t1err/n)*100,"%")   Each values are extremely close to 10. Still it produces about 5% Type I error. What conclusion we may draw from it? Or, am I missing something? 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. How to install Android Studio on Ubuntu? Please tell me the full process of installing Android Studio and its necessary dependencies. Add anchors to the how-to-ask and how-to-answer pages Why Windows sets new created file's "created time" property to old time? I would like to label an equation with an arbitrary symbol, rather than a number, or letter.  Is this possible? Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics Marker shadows in Google Maps v3 How to boot WUBI installed Ubuntu within Windows 8? I have Windows 8 Release Customer Preview on my laptop. I tried to install Ubuntu on it with Ubuntu Windows installer (I have always used this to install, although that was on Windows 7), it installs and asks to reboot, then immediately Windows 8 boots up.  I do not have the option to choose what OS to boot. How can I boot to Ubuntu? VM is not exactly what I would like to use, but it is an option, however a last resort. Any help?