When describing black body radiation in Physics, one comes across an integral of the form $$I(a)=\int_0^\infty\frac{x^3}{e^{ax}-1}dx \tag1$$ for $a&gt;0$. In our lecture, we were given that this integral evaluates to $$I(a)=\frac{\pi^4}{15a^4}$$ without a derivation of this result. While reviewing my course notes over the Christmas break, I came across this integral and have tried to evaluate it myself, but I have not been successful so far. My first step, of course, was to substitute $u:=ax$, which gives us $$I(a) = \frac{1}{a^4}\int_0^\infty\frac{u^3}{e^u-1}du \tag2$$ but beyond that, I'm stuck. I've tried differentiation under the integral sign on the original integral in $(1)$, which lead to the integral $$\int_0^\infty\frac{\log^4 u}{(u-1)^2}du$$ which I also no have no clue how to tackle. I've also thought about using contour integration, but I'm unsure as to what would be an appropriate contour to consider. I know that the integrand has singularities at $z=2\pi ik$ for $k\in\mathbb{Z},k\neq 0$ but I'm not even sure how one would go about calculating the residues at these singularities due to the peculiar form of the denominator, which is unlike any I have previously encountered.  How would one go about deriving that $$\int_0^\infty\frac{x^3}{e^{ax}-1}dx = \frac{\pi^4}{15a^4}$$ is true?
how to calculate $$\int_0^\infty \frac{x^{3}}{e^{x}-1} \; dx$$   Be $q:= e^{z}-1 , p:= z^{3}$ , then $e^{z} = 1 $ if $z= 2\pi n i $, so the residue at 0 is : $$\frac{p(z_{0})}{q&#39;(z_{0})} = 2\pi i n ^{3}$$  problem is that this is not symmetric, so how does one find the definite integral?
Short story with aliens who hear their own thoughts as god-like voices I would have read this about 20-25 years ago.  I believe the subject was an alien individual, but it could have been a non-standard human instead.  The hook of the story was that the subject lacked an organ like the  that allows the halves of the brain to communicate efficiently, allowing the illusion of a single consciousness.  The subject is conscious, and has thoughts, but is subject to hearing commands like "Wash your hands!" "Fetch water!" within its head.  I recall the subject (and others like the subject) believing that the voices are their gods talking to them.  To the reader it becomes clear that this is a separate part of the subject's own mind that is responsible for long-term planning and simply can't communicate with the rest of the mind in the way a "normal" human is accustomed to. When I try to install any software, I encounter this error, and therefore I cannot update.  Preconfiguring packages ... dpkg: error: parsing file '/var/lib/dpkg/status' near line 51245 package 'oracle-xe':  mixed non-coinstallable and coinstallable package instances present N: Ignoring file 'gcl' in directory '/etc/apt/sources.list.d/' as it has no filename extension N: Ignoring file 'gcl' in directory '/etc/apt/sources.list.d/' as it has no filename extension E: Sub-process /usr/bin/dpkg returned an error code (2) How can I use multiple overlay effects simultaneously in Photoshop? There are 3 overlay effects available in Photoshop as layer styles: Color, Gradient and Pattern. The problem: these effects come in a strictly fixed order!  Is it possible to make a translucent Gradient Overlay over a Color Overlay using only layer styles? Or a Multiply Pattern and Color Overlays? Alternative of `route -n` of Linux in Mac How can I see the list of routes and its corresponding gateway in a Mac ? The command in linux is route -n .  For a particular IP address, I am able to get the routes for an particular IP address via route get 10.0.0.0. how do i get my stolen phone& pictures back? On 7 july 15 someone broke in my house and stole my galaxy note 3 it did not have service it has been disconnect for about 6mo now i would like to get my phone back because it had picture of me and my dad who died the day after father day this past june &amp; other family pictures that can not be replaced.how do i go about getting those back? Including Interaction Terms in Random Forest Suppose we have a response Y and predictors X1,....,Xn. If we were to try to fit Y via a linear model of X1,....,Xn, and it just so happened that the true relationship between Y and X1,...,Xn wasn't linear, we might be able to fix the model by transforming the X's somehow and then fitting the model. Moreover, if it just so happened that the X1,...,XN didn't affect y independent of the other features, we also might be able to improve the model by including interaction terms, x1*x3 or x1*x4*x7 or something of the like. So in the linear case, interaction terms might bring value by fixing non-linearity or independence violations between the response and the features.  However, Random Forests don't really make these assumptions. Is including interaction terms important when fitting a Random Forest? Or will just including the individual terms and choosing appropriate parameters allow Random Forests to capture these relationships? How can I set up password-less SSH login? I want to be able to login to a remote via ssh without having to enter the password all the time.   How do I set it up? Is a different command required to execute a password-less session? A new search engine for Stack Exchange Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$?  I once was given an assignment where I was asked to deduce and write down some physical quantity. It turned out that this quantity was $0$ in one unit or another, so I decided to drop the physical units because I figured that it didn't matter anyhow.  But I got to thinking:  If $0 \, \mathrm m = 0$, then we could add units of different dimensions (an operation not normally defined) like so: $5 + 0 \,\mathrm m = 5$.   Does $[ \cdot ]$ map $0 \, \mathrm m$ to both $0$ and $L$? Is there an established regime of thinking about physical quantities as an abstract algebra? Perhaps as a vector space where formal sums such as $5 \, \mathrm m + 10 \, \mathrm s$ are allowed? Would it be improper to drop the units of measurement of $0 \,\mathrm m$ like this in an academic paper? How to disable notifications for a single mailbox in Mail? Is it possible to make an ASP.NET MVC route based on a subdomain? Is it possible to have an ASP.NET MVC route that uses subdomain information to determine its route?  For example:   user1.domain.com goes to one place user2.domain.com goes to another?   Or, can I make it so both of these go to the same controller/action with a username parameter? Going into a directory linked by a link When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? How to "pretty" format JSON output in Ruby on Rails I would like my JSON output in Ruby on Rails to be "pretty" or nicely formatted.   Right now, I call to_json and my JSON is all on one line.  At times this can be difficult to see if there is a problem in the JSON output stream.  Is there way to configure to make my JSON "pretty" or nicely formatted in Rails? How can I make schematic of this board? I want to make a schematic of this board. Can anyone tell me how? To prove the convergence of the p-series   $$\sum_{n=1}^{\infty} \frac1{n^p}$$  for $p &gt; 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.  I am wondering if there is a self-contained proof that this series converges which does not rely on either test.    I suspect that any proof would have to use the ideas behind one of these two tests. Remove duplicate rows in MySQL Why does pluralization of compound words typically occur in the middle as opposed to the end of the word? As I understand, correct plural versions of passerby and attorney general become passersby and attorneys general.  But why?  With passerby, the the preposition "by" has been combined with the noun "passer" to become a one word noun "passerby".  So why shouldn't this be passerbys?  If I am an attorney general, and you are an attorney general, why are we not attorney generals?  As an American I have always heard it as attorneys general, however  suggests British English does use "attorney-generals" (as well as an interesting 2nd paragraph that further probes my question).  Are there hard rules to this topic?  Or merely generally accepted practices based on locale? I currently have a 75% credit utilization, which I know is not favorable on about 35,000 revolving credit.  I am getting my year end bonus and plan on using that to completely pay off all of my credit cards leaving me with a 0% credit utilization.  Currently my credit score is 668, mostly due to my utilization, I pay everything on time, but just have too much debt, which is why i'm paying it all off.  My question is how much will paying off all my revolving credit and having a 0% utilization on 35,000 credit availability positively effect my credit score? What's in "standard system utilities" in the server install? My googlefu has failed me. What packages are part of the "standard system utilities" that is preselected during install?  TIA! (-: \mathbb{Z} yields undefined control sequence error