I have tried solving the equation but I keep getting zero. I also can't find a proof online, I think because it is a rather specific question.
How to show this function's discontinuity?  $ f(n) = \left\{     \begin{array}{l l}      \frac{xy}{x^2+y^2} &amp; \quad , \quad(x,y)\neq(0,0)\\      0 &amp; \quad , \quad(x,y)=(0,0)    \end{array} \right.$
How to make dot match newline characters using regular expressions General formula for evaluating integrals of the form $\displaystyle\int_0^\infty x^ae^{-bx}dx$ Find a general formula for evaluating integrals of the form $\displaystyle\int_0^\infty x^a e^{-bx}dx.$   I've been pondering about this question for some time. Obviously integration by parts is way too inefficient. I know there's recursion involved and maybe there's a way to find a general formula through intuition. How to flash a BIOS when only a Windows binary is provided? Very specific paragraph shape Output URL of open firefox tabs in terminal Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $ Express recurrence relation  of the integral  $$ I_n=\int\frac{dx}{(1+x^2)^n} $$  [My Answer]  $$ I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx $$  $$ I_n=I_{n-1}-\int x\cdot\frac{x}{(1+x^2)^n}dx $$  $$ I_n=I_{n-1}-\frac{x}{2(1-n)(x^2+1)^{n-1}}+\frac{1}{2(1-n)}I_{n-1} $$  $$ I_n=\frac{2n-3}{2(n-1)}I_{n-1}+\frac{x}{2(n-1)(x^2+1)^{n-1}} \ \ \ \ (n&gt;1) $$  $$ I_1=\arctan(x) $$  Is my answer correct? Let's have 2 numbers, N and K, where K divides N. The number of K-combinations from a given set S of N elements is a well known formula. Let's concatenate N/K groups (resulting in N elements) such as the resulting set is N. How many possibilities are there, i.e. what's the formula?  For instance:    N=4, K=2, C(4, 2) = 6 {1,2},{1,3},{1,4},{2,3},{2,4},{3,4}   The 3 possibilities are:     {1,2},{3,4}   {1,3},{2,4}   {1,4},{2,3}   I generated these combinations and I think the number goes like this:  (4, 2): 3 (6, 3): 10 (6, 2): 15 (10, 5): 126 (9, 3): 280 (10, 2): 945 (14, 7): 1716 (12, 4): 5775 (15, 5): 126126 (15, 3): 1401400   Apparently, the result always divides with (N-1). Probability of flipping coin What is the probability of obtaining THT before HHH when a toss is tossed sequentially? Could someone solve using state diagram and law of total probability? Duplicate notifications I receive every push notification from the iOS app twice on iPhone. The second comes immediately after the first. This happens every single time.   I also have the app installed on an iPad and signed in to the same account - could this be a possible cause?   App Version: 1.4.0 Device: iPhone7,2 OS Version: Version 9.1 (Build 13B143) I am building a java server that needs to scale. One of the servlets will be serving images stored in Amazon S3.  Recently under load, I ran out of memory in my VM and it was after I added the code to serve the images so I'm pretty sure that streaming larger servlet responses is causing my troubles.  My question is : is there any best practice in how to code a java servlet to stream a large (>200k) response back to a browser when read from a database or other cloud storage?  I've considered writing the file to a local temp drive and then spawning another thread to handle the streaming so that the tomcat servlet thread can be re-used. This seems like it would be io heavy.  Any thoughts would be appreciated. Thanks. Can a purely biological account of man show a superiority over other animals other than that of belonging to a later stage of evolution or of greater possibilities of action, a superiority “in his very being”? Suppose we have a set  $S=\{a,b\}.$ Obviously, the total number of binary operations on $S$ is the number of all the pairs such that $$*(a,b)=a \, \text{or}  \, b,$$ which gives  $ 2 \cdot 2 \cdot 2 \cdot 2=2^4$ possible ones. Is there a way to find which of these $16$ binary operations are associative without writing all the operation matrices and checking it by hand? Is the oil thrown on a creature consumed when it's dealt fire damage and lit? I had the idea of throwing oil flasks on enemies at level 1, then hitting them with a torch to deal a bunch of bonus fire damage.  PHB 152:     If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil.   PHB 152:     If you make a melee attack with a burning torch and hit, it deals 1 fire damage.   A flat 6 fire damage is pretty potent at level 1, but does the oil remain or does it burn out in a flash when hit (since it has no damage over time) when hit?  If not there is potential for a dual torch wielding PC to throw down 12 fire damage per round, or for multiple PCs to deal 6 damage per torch per PC per round. How to disconnect selected mesh geometry? I don't know how to disconnect a face from the rest of the mesh in edit mode.  This isn't like pressing P to separate the selection into a completely different object. I just want to select a face and disconnect/rip it away from the rest of the mesh but keep it as a part of the original object.  One way to do it would be to duplicate the face with ShiftD and then delete the original face. But I suspect there is a "native" command for disconnecting selected faces and I would like to learn it. Markdown does not allow for bold italic, then bold, then bold italic I am trying in one of my posts to do this:  ***how many*** **and** ***where***   Which for some reason comes out like this:    Why is that happening?  Shouldn't the middle word be bold?  Is this  for markdown or is it (as I assume) a ? Should flour be thrown out past its expiry date? Compiz Problems (Plugin Missing, Performance Issues, Development) Why does k-means clustering algorithm use only Euclidean distance metric? Is there a specific purpose in terms of efficiency or functionality why the k-means algorithm does not use for example cosine (dis)similarity as a distance metric, but can only use the Euclidean norm? In general, will K-means method comply and be correct when other distances than Euclidean are considered or used?  [Addition by @ttnphns. The question is two-fold. "(Non)Euclidean distance" may concern distance between two data points or distance between a data point and a cluster centre. Both ways have been attempted to address in the answers so far.] I'm going to guess that the answer is "no" based on the below error message (and ), but is there anyway to perform a cross-database query using PostgreSQL?  databaseA=# select * from databaseB.public.someTableName; ERROR:  cross-database references are not implemented:  "databaseB.public.someTableName"   I'm working with some data that is partitioned across two databases although data is really shared between the two (userid columns in one database come from the users table in the other database). I have no idea why these are two separate databases instead of schema, but c'est la vie... Is "until" inclusive or exclusive? I sent an email to someone and got an auto-reply saying:     I am out of the office until 09/15/2014.   Does this mean he will be available on the morning of the 15th?