An old complex analysis exam question:  Evaluate   $$\large I(a) = \int_{-\infty}^{\infty} e^{-\frac{1}{2}x^2+iax}dx$$  So far, I have completed the square in exponent, and now I have the integral   $$\large I(a) = \int_{-\infty}^{\infty} e^{(x+\frac{ia}{2})^2-\frac{3}{2}x^2+\frac{a^2}{4}}dx$$  Perhaps completing the square above gives me an integral where I now don't have to worry about whether a is positive or negative -- and so I don't need to break the integration out into two cases.  But I'm not sure what to do from here.  Most integration that I come across in old exam questions is an application of the Residue Theorem, but currently the integrand looks pole-free / entire, so there'd be no residues to compute.  One last thing I thought of so far is that the integral looks a bit like the  Gaussian integrals.  Would this be the better / correct path to follow?  If so, what should I start with?  The $iax$ term makes it tricky to know what sort of substitution I could go with to perhaps get something like $e^{-ax^2}$    Any hints or suggestions are welcome.  Thanks,
I would like to work out the Fourier transform of the Gaussian function   $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$  It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to work. Does anyone have any advice?  By the way, I am using   $$\mbox{FT}(f)(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,\mathrm d x$$  as my definition of a Fourier transform. Many thanks.
Determining audio level peaks with ffmpeg Is there a plugin I'm missing that allows me to determine the volume samples over the length of an audio file?  For example, a sample of every second to gain the dB vs timestamp? Using a question mark mid-sentence How to make child collide An invalid argument, the conclusion of which is a tautology Why it is not possible to have such argument?  I found some textbook Introduction to Formal Logic and this is from one of the first excercises. This is marked as impossible, but I do not understand why. I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! Rsync filter: copying one pattern only I am trying to create a directory that will house all and only my PDFs compiled from LaTeX. I like keeping each project in a separate folder, all housed in a big folder called LaTeX. So I tried running:  rsync -avn *.pdf ~/LaTeX/ ~/Output/   which should find all the pdfs in ~/LaTeX/ and transfer them to the output folder. This doesn't work. It tells me it's found no matches for "*.pdf". If I leave out this filter, the command lists all the files in all the project folders under LaTeX. So it's a problem with the *.pdf filter. I tried replacing ~/ with the full path to my home directory, but that didn't have an effect.  I'm, using zsh. I tried doing the same thing in bash and even with the filter that listed every single file in every subdirectory... What's going on here?  Why isn't rsync understanding my pdf only filter?    OK. So update: No I'm trying  rsync -avn --include="*/" --include="*.pdf" LaTeX/ Output/   And this gives me the whole file list. I guess because everything matches the first pattern... Facebook must have changed something again because I cannot find out how to view the list of subscribers.  In Googling for this, I came upon this question:    However, it doesn't take me to my list os subscribers.  Where do I view this list now?  Please include a screenshot and highlight the link I need to click on.  This way, when Facebook changes their layout again, your screenshot will at least clue future inquirers in on this fact so they know they're not going crazy.  Edit  Here's what I see when I log into Facebook: I'm hoping someone could check my work. I'm trying to simplify the equation for $x$. $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$\frac{a}{b} = \frac{174.1}{86}$$ Thus I began by substituting $a$ with: $$a = \frac{174.1}{86}b$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} = 1 - (\frac{y}{b})^{3.2}$$ $$(\frac{x}{b})^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}(b)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x^{\frac{16}{5}} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x = [\frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}]^{\frac{5}{16}}$$ Is my work correct, or am I completely off the board? Currently, Stack Exchange’s Markdown parser only allows four-space indents to represent code blocks:      // some code     // another line of code    and other Markdown implementations allow for an alternative syntax, that doesn’t require indenting each code line:  ``` // some code // another line of code ```   This is much more convenient to type out. It would be super useful if Stack Exchange could support this syntax.    By extension, this syntax also allow you to specify the source language right after the opening ```:  ```js // some code // another line of code ```   …which would then enable syntax highlighting for that specific language. Although it’s interesting metadata, I don’t think this feature is needed on Stack Overflow, as the syntax highlighting library works pretty well for various languages.  So, even if you would allow this syntax but ignore the ```language, this would greatly improve my productivity on Stack Exchange.  What do you think?  The moderators are currently collecting feedback regarding this feature, so please post a comment with your thoughts. Good idea? Bad idea? Don’t really care? What do you think are the benefits/drawbacks? Experiences? Let them know! Thanks! How can one cite theorems of references? I would like to cite the item (iii) of proposition 3 of a reference. How can I do it? Which ways do you suggest? This is a question not only about LaTeX, but also about style. One option I have seen to cite pages of a reference is \cite[p.~333]{Audet}. Is there something similar to cite theorems, propositions and so on? I'm actually at the cemetery at White Orchard. There is a white question mark and some little red arrows.  What are this little red arrows?  Do they point to something dangerous? If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant. How do I prove that if $f: \Bbb{R}\rightarrow\Bbb{R}$  is continuous at $0$ and $$f(x)=f(2x)$$ for each $x\in\Bbb{R}$, then $f$ is constant? Reading/Writing a MS Word file in PHP Is it possible to read and write Word (2003 and 2007) files in PHP without using a COM object? I know that I can:  $file = fopen('c:\file.doc', 'w+'); fwrite($file, $text); fclose();   but Word will read it as an HTML file not a native .doc file. Two envelope problem revisited I was thinking of this problem.  I believe the solution and I think I understand it, but if I take the following approach I'm completely confused. Problem 1:  I will offer you the following game. You pay me \$10 and I will flip a fair coin. Heads I give you \$5 and Tails I give you \$20. The expectation is \$12.5 so you will always play the game.  Problem 2:  I will give you an envelope with \$10, the envelope is open and you can check. I then show you another envelope, closed this time and tell you: This envelope either has \$5 or $20 in it with equal probability. Do you want to swap? I feel this is exactly the same as problem 1, you forgo \$10 for a \$5 or a \$20, so again you will always switch.  Problem 3:  I do the same as above but close the envelopes. So you don't know there are $10 but some amount X. I tell you the other envelope has double or half. Now if you follow the same logic you want  to switch. This is the envelope paradox.  What changed when I closed the envelope?? EDIT: Some have argued that problem 3 is not the envelope problem and I'm going to try and provide below why I think it is by analysing how each views the game. Also, it gives a better set up for the game. Providing some clarification for Problem 3: From the perspective of the person organising the game:  I hold 2 envelopes. In one I put \$10 close it and give it to the player. I then tell him, I have one more envelope that has either double or half the amount of the envelope I just gave you. Do you want to switch? I then proceed to flip a fair coin and Heads I put \$5 in and Tails I put \$20. And hand him the envelope. I then ask him. The envelope you just gave me has twice or half the amount of the envelope you are holding. Do you want to switch?  From the perspective of the player:  I am given an envelope and told there is another envelope that has double or half the amount with equal probability. Do I want to switch. I think sure I have $X$, hence $\frac{1}{2}(\frac{1}{2}X + 2X) &gt; X$ so I want to switch. I get the envelope and all of a sudden I am facing the exact same situation. I want to switch again as the other envelope has either double or half the amount. Why does the flash freeze a picture? Why does a flash freeze a picture even if I use a shutter speed like 1/10 sec?  I don't understand it, because the light will hit the sensor the whole time, not just when the flash is fired. Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ I am trying to solve the following problem at the level of a senior undergrad analysis level. So, the problem is as follows: We are given a function $f$ which is continuous on the interval $\left [ 0,1 \right ]$, and the question is to find the limit: $$\lim_{n\rightarrow \infty}\int_{0}^{1}x^{n}f(x)dx\;.$$ The second part of the problem is to deduce the following limit: $$\lim_{n\rightarrow \infty}n\int_{0}^{1}x^{n}f(x)dx\;.$$  For the first part: I just did the following: For every $0\leq x&lt; 1$: $x\leq M$, where $0&lt; M&lt; 1$. Then: $$\int_{0}^{1}x^{n}f(x)dx\leq M^{n}\int_{0}^{1}f(x)dx\;.$$ Then: $$\lim_{n\rightarrow \infty }\int_{0}^{1}x^{n}f(x)dx\leq \lim_{n\rightarrow \infty }M^{n}\int_{0}^{1}f(x)dx= 0.\int_{0}^{1}f(x)dx=0\;,$$ so $$\lim_{n\rightarrow \infty }\int_{0}^{1}x^{n}f(x)dx=\lim_{n\rightarrow \infty }f(1)\int_{0}^{1}1dx=f(1)\;.$$ Does that make sense? If not, please show me the correct one.  As for the second part, I have no idea what to do. Any help? Area 51 login no longer redirects nor logs in JavaScript object: access variable property by name as string Goal: I want to be able to securely use the internet via my home PC while my notebook is connected to an open hotspot/access point.  I do know that I can use a SSH tunnel/SOCKS proxy, but I don't want to fiddle around with applications (making them use it, if even possible). I guess what I need is an OpenVPN setup, so I'm looking for a detailed guide on how to:   Install and setup the OpenVPN server Setup the OpenVPN client (NetworkManager)   Ubuntu versions this should work on are 10.10 and 11.04.