How do I compute  $$\int_{|z|=r} x \, dz$$  by use of a parameter?
I have a complex integral $$\int_{|z|=r}x \, dz$$ for the positive portion of the circle.  I know that this integral seems easy enough but I am having trouble with it, and I'm fairly certain my answer is incorrect.  I wanted to use a parameter such that $x=r\cos\theta$, so we then have $dz=dx+i\,dy=dx$ since there is no reason to have have the $i\,dy$ since the imaginary part is equal to $0$.  So now I have the double integral  $$\int^r_0\int^\pi_0 r \cos\theta \, dx$$  which equals $0$ since integration the $\cos\theta$ gives $\sin\theta$ and evaluated from $0$ to $\pi$ makes the integral evaluate to $0$.  Am i misinterpreting part of the question?  Way off base?  The book suggests to also solve the problem by realizing $x=\frac{1}{2}\left(z+\frac{r^2}{z}\right)$ along the circle.  For this portion of the problem I assume that I would integrate the above by $z$ as if it were a normal calculus problem, but I have problems with what to put as upper and lower limits.  Thanks in advance!
I'd like to use a different + symbol in addition to + in math mode, preferably one which is just bolder than the standard one.  I'm currently using \pmb{+} from the amsbsy package, but it's a bit too lean for my taste, and it looks a bit pointy to the ends.   I am aware of \ding{58} from the pifont package, as well as \Plus from bbding, both of which seem to be non-math mode symbols, and look really bad next to math symbols.  Am I missing some other option to produce a bold +, apart from drawing the symbol myself? I just bought a new HP Pavilion dv7t Quad Edition laptop that has HDMI out. I already have a couple of monitors that are VGA and I'd like to hook up one of these to dual monitors.  I've read some sketchy reviews of products that try to convert directly from HDMI (digital) to  VGA (analog), so I was wondering if someone has a good recommendation on how to hook these two together. How to get a JavaScript object's class? I'm thinking about installing Ubuntu Desktop, but I don't know what flavor is the best for my system. What are the minimum and recommended hardware requirements? What kind of CPU? How much memory? Should I have Hardware Acceleration? What flavor should I use?  This is an attempt of a canonical answer. My answers have the &quot;official  requirements&quot;, the recommended are a mix of official sources and opinion based (along with the answer it's told the source). You can comment or edit if you feel that the information is obsolete or incomplete. It is a good rule of thumb that any system capable of running Windows Vista, 7, 8, x86 (Intel) OS X will almost always be a lot faster with any Ubuntu flavor even if they are lower-spec than described below. Suppose $G$ is a semigroup and it holds both left and right cancellation. Suppose $G$ is a semigroup and it holds both left and right cancellation. Also for each $a,b\in G$, $xa=b$ has solution in $G$. Prove G is a group.   I know this question looks very "old" style. First I think it's easy for me too since I already practice probelems like "semigroup, $xa=b$ and $ax=b$ have solution , then group" Or "finite semigroup, both left and right cancellation hold, then group ". But I still get stuck... My try:  First of course, for $xa=a$ , there exists $e_a$ such that $e_aa=a$. Now we need to proof $e_a$ is left identity for all elements. Then left inverse exists hence $G$ is a group. But the key is to proof $e_ab=b$ for all $b\in G$. First, this question is motivated by the imprecise question: Is there a sensible notion of parity (evenness and oddness) for real numbers?   Here are some properties a notion of parity for $\mathbb{R}$ should have:    It should be an equivalence relation on $\mathbb{R}$ with exactly two equivalence classes.  Each equivalence class should be dense in $\mathbb{R}$. There should be some kind of symmetry between the two equivalence classes. (This is intentionally imprecise.)   If we divide $\mathbb{R}$ into rationals and irrationals, this lacks symmetry, since the rationals have measure zero and are countable, while the rationals have infinite measure and are uncountable.   So we could require that each equivalence class be uncountable and/or have positive (or infinite) measure. Then take one equivalence class to be the rationals, union with countably many fat cantor sets with positive measure, and the other class its complement. Then both classes are now dense, uncountable, and have infinite measure. But again, symmetry is lacking. One of these classes is "uncountable everywhere" and has "positive measure everywhere" while the other does not.   I propose some definitions: A subset $A \subset \mathbb{R}$ is uncountable everywhere if for any open interval $(a,b)$, the intersection $A \cap (a,b)$ is uncountable. A subset $A \subset \mathbb{R}$ has positive measure everywhere if for every open interval $(a,b)$, the intersection $A \cap (a,b)$ has positive measure. We can see immediately that having positive measure everywhere implies being uncountable everywhere, since a positive measure set must be uncountable.   Finally, my questions:    Is there a partition of $\mathbb{R}$ into two sets that are both uncountable everywhere?  Is there a partition of $\mathbb{R}$ into two sets that both have positive measure everywhere?  Is there a partition of $\mathbb{R}$ into two sets that split each interval $(a,b)$ into two parts of equal measure?   Potential generalizations for extra credit: What about partitions of $\mathbb{R}$ with $n$ equivalence classes, where $n \in \mathbb{N}$, or even with countably many, or even uncountably many equivalence classes?  Note: If you've seen these definitions of uncountable everywhere or positive measure everywhere somewhere under a different name, please let me know. I've never found anything where other people were thinking about these notions.  EDIT: By "measure," I mean Lebesgue outer measure, so we don't have to worry about anything being measurable. Conditional Expectation of Bernoulli R.V. Let $X_1, X_2,\ldots, X_n$ be iid bernoulli r.v. with parameter $p$. Let $S=X_1+\cdots+X_n$ and $Y=X_1X_2$. Compute $\mathbb{E}(Y\mid S)$.  I know that $\mathbb{E}(X_1\mid S) = S/n$. So If I could split $\mathbb{E}(Y\mid S)$ into two then I would be done. But I don't think that is allowed. How to proceed? Cause of Alpha or Black Artefacts in Render Edit Mode Transparent Objects Can all fundamental forces be fictitious forces? After reading many questions, like   and , I wonder:   Is it possible to consider the other fundamental forces too, the electroweak interaction and the strong interaction or ultimately the unification of these, to be fictitious forces like gravity in the framework of general relativity?   If we want a final unification of all fundamental forces, hasn't this feature of gravity to become a feature of the other forces as well? How do I check if a given string is a legal/valid file name under Windows? I want to include a batch file rename functionality in my application. A user can type a destination filename pattern and (after replacing some wildcards in the pattern) I need to check if it's going to be a legal filename under Windows. I've tried to use regular expression like [a-zA-Z0-9_]+ but it doesn't include many national-specific characters from various languages (e.g. umlauts and so on). What is the best way to do such a check? Zoom lenses and cameras with superzooms are often sold with a times ("×") zoom number — like a 12x zoom or 30x zoom. The question  explains how this relates to focal length numbers, like 70-200mm or 18-55mm. But how does ×-zoom relate to magnification, when used for macro or just to make something far away look bigger?  How can two 70-200mm lenses have different magnification? I've just installed Mir and I'm wondering if I'm actually running it or if it fell back to X. How can I tell if Mir is running? Automatically login and lock Vito and Luca question remove isp-provided dns My isp provides a dns service; but unfortunately they do block certain sites. I would like to use opendns or google; but it seems like it doesn't work.    I have set up a wired connection; given a manual ip and entered opendns-ip's in the nameserver field (all through the gui). Still url's are being resolved through my ISP. I have tried to configure the same on my girlfriends laptop (win 7) and I also can't seem to do the setup. My previous computer also was a win 7 and had no problem. (so I believe the problem is 'me') If it were only one site, i could edit the hosts file; but I prefere to circumvent their whole dns server.  Is there something an isp can do to block the use of other dns-servers than theirs? Is it possible to re-route the ISP dns to the opendns? Where in the registry does Windows store with which program to open certain file types? After installing Adobe PDF reader, it opens files with the .PDF file extension. Ones' default web browser opens .html files and notepad opens .txt files, etc.  Where exactly in the registry are these filetype - program associations stored in the registry?   In other words, how does Windows know with which software/program it must open a certain file(type)? I'm positive that vs I strongly believe that In general, which one is conveys a stronger conviction/opinion? I did some quick google search but didn't find any interesting results.      I'm positive that   or     I strongly believe that Drawing cross section in QGIS? Does anyone have a simple work-flow for drawing cross sections of earth features in QGIS?   My thing is that QGIS is being used to present the plan view, but we have a little interest in some elevations as well. All the data is one set and is simply an elevation value in the attribute table, I don't want to have to move all that to CAD.  One logical workaround is to present the z value as the y.  I am a dunce in paper space in CAD, and would be more comfortable with the QGIS Composer. In the movies, the pack of Uruk-Hai capture Merry and Pippin, and are bringing them back to Saruman as quickly as possible.  Why didn't they use Wargs to carry the Hobbits instead of packing them on their backs?    Or, better yet, why didn't the Nazgul fly down and take them on their Fell Beasts?