Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z\in\mathbb{C}:|z|&gt;1\}$, and let $$\lim_{z\to\infty} f(z)=0.$$  Prove that  $$f(z)=\dfrac{1}{2\pi i}\int_{\Gamma}\dfrac{f(\zeta)}{z-\zeta}d\zeta$$,  for all $z\in\mathbb{C}$ with $|z|&gt;2$.  I am quite bad at complex analysis and I would appreciate any hints/directions here. Also, if there are generic methods to attack such problems, please share them as well.  Thanks in advance!
Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z \in\mathbb{C} : |z|&gt;1 \}$, and let $$\lim_{z\to \infty}f(z)=0.$$  Prove that $$ f(z)=\frac{1}{2\pi i} \int_{\Gamma} \frac{f(\xi)}{z-\xi}d\xi,  $$ for all $z\in\mathbb{C}$ with $|z|&gt;2$.  I have no idea how to even start. Any hint will be appreciated.
An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$. I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.   I have a really bad attempt:  Define $a_{0}=a_{1}=0$, and $a_{k}:=\log(\log(k))$ for $k\geq 2$, then $a_{k}\longrightarrow\infty$. But $$\dfrac{1}{n}\sum_{k=0}^{n-1}a_{k}=\dfrac{\log\Big(\prod_{k=2}^{n-1}\log(k)\Big)}{n}$$ and Wolframalpha told me that this series converges to $0$ as $n\longrightarrow\infty$.  However, I have no idea about how to show this convergence to $0$. Also, I wish to have a sequence as simple as possible, since at some stage I need too how $f(k):=a_{k}$ is positive semi-definite.   Is there any simpler example?   Thank you! Why commutator subgroup is normal to G? If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help. I have a SQL Server instance that has a linked server to an Oracle server. There is a table on the Oracle server called PersonOptions which contains the following data:  ╔══════════╦══════════╗ ║ PersonID ║ OptionID ║ ╠══════════╬══════════╣ ║        1 ║ A        ║ ║        1 ║ B        ║ ║        2 ║ C        ║ ║        3 ║ B        ║ ║        4 ║ A        ║ ║        4 ║ C        ║ ╚══════════╩══════════╝   I need to pivot that data so the results are:  ╔══════════╦═════════╦══════════╦══════════╗ ║ PersonID ║ OptionA ║ Option B ║ Option C ║ ╠══════════╬═════════╬══════════╬══════════╣ ║        1 ║       1 ║        1 ║          ║ ║        2 ║         ║          ║        1 ║ ║        3 ║         ║        1 ║          ║ ║        4 ║       1 ║          ║        1 ║ ╚══════════╩═════════╩══════════╩══════════╝   Any suggestions? Famous puzzle: Girl/Boy proportion problem (Sum of infinite series) Puzzle  In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?  My solution (not finished)  If we assume that the probability of having a girl is 50%, the set of possible cases are:  Boy (50%)  Girl, Boy (25%)  Girl, Girl, Boy (12.5%)  ...  So, if we call G the number of girls that a family had and B the number of boys that a family had, we have:  $B = 1$  $P(G = x) = (1/2)^{x+1}*x$  So  $G = \Sigma (1/2)^{x+1}*x$  I feel like the sum of this infinite serie is 1 and that the proportion of girls/boys in this country will be 50%, but I don't know how to prove it!  Thanks! How do I manage double-faced cards in Innistrad (MTG) tournament play? How to include graphics with spaces in their path? Can two particles remain entangled even if one is past the event horizon of a black hole? Windows 10 K&N Edition Anniversary update - Media player update not working I own a Windows 10 K or N Edition which doesn't include the media player package. My problem is: I want to have preview thumbnails in the explorer for my MP3s and videos.  I had the issue before when I updated to Win10 but there  saved me. Since the Anniversary update they are gone again.   Now I can install the old update but I still don't get the media player nor the preview images.  I tried most of the things Microsoft offers me but they are either not installable or don't work.  Any ideas what might be the correct update package?  Thx!  Ron How to search text throughout entire file system? For example, is it possible to drink a flying potion and cast greater invisibility spell on myself (both spells would require concentration), so I can fly around invisible? Dual boot Win 8 / Ubuntu loads only Win My computer is a Sony Vaio Notebook. I used Ubuntu and Win 8 with dualboot but windows somehow managed to remove the menu to choose the OS at startup. I think the BIOS is UEFI. Secure boot was disabled already and Boot Repair somehow fails.  How can I fix this? Git for beginners: The definitive practical guide SEO for websites with very less content and a lot of images Let's say we have an image exhibition website that has got a lot of images but next to nothing content. The focus keyword for this website is "Artistic wallpapers". What would be the best approach w.r.t SEO for such a website.  Here are a few things I can imagine:   Submitting sitemaps ( image and web ) Having a good pagespeed score. Nice navigation Breadcrumbs sharing the website content on Social media. Having appropriate titles.   Having worked on all these things, what can be done if Search engines are picking wrong keywords ( For example views, votes, ratings) Can anything be done so that Search engines don't track these specific items? Also, is it a good practice to have all the pages submitted to Search engines at once? I want to stop having to use sudo everytime I work in /var/www. How can I do that? I simply want to put all of my sites into this directory and work with them without too much pain. How to install updates via command line? Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward. When I reinstall Windows XP by mistake I deleted all five partitions. I did not format them yet though.  How can I recover the partitions? When I attach the hard disk to another PC it just says that the partition is unformated and asks "Would you like to format it?". Command line command to auto-kill a command after a certain amount of time I'd like to automatically kill a command after a certain amount of time.  I have in mind an interface like this:  % constrain 300 ./foo args   Which would run "./foo" with "args" but automatically kill it if it's still running after 5 minutes.  It might be useful to generalize the idea to other constraints, such as autokilling a process if it uses too much memory.  Are there any existing tools that do that, or has anyone written such a thing?  ADDED: Jonathan's solution is precisely what I had in mind and it works like a charm on linux, but I can't get it to work on Mac OSX.  I got rid of the SIGRTMIN which lets it compile fine, but the signal just doesn't get sent to the child process.  Anyone know how to make this work on Mac?  [Added: Note that an update is available from Jonathan that works on Mac and elsewhere.] What is this story about 20 y.o. inexperienced "kids" fighting against a stronger foe in space? When editing a question on , I observed that a link which was not rendered in the post was hyperlinked in the edit preview (see pictures below). I tried reproducing it in the  to see if this also happens when the post is posted for the first time and observed the same. It seems to be a bug after the shift to CommonMark, leading to the different rendering of links on client and server-side. A link following a special character (without any space) is not hyperlinked in the post but is hyperlinked in the preview. A link separated from the special character with a white space appears to have no such problem. (see pictures below and my ). Link not hyperlinked in post  Same link hyperlinked in preview