As stated in the title, I want to evaluate the integral $$\int_0^\infty \frac{\tan^{-1}(t)}{e^{2\pi t}-1}\,\mathrm{d}t$$ Because of the $e^{2\pi t}$, it seems that complex analysis techniques are required for this one, which I am not so familiar with. I have tried some substitutions and integration by parts. Besides, I tried using the property $$\int_0^\infty f(t)\,g(t) \,\mathrm{d}t =\int_0^\infty \mathscr{L}[f](s)\cdot \mathscr{L}^{-1}[g](s)\,\mathrm{d}s$$ But no luck here. I would like a solution without complex analysis techniques, if that's possible. Thank you!
I want to calculate $$ \int_{0}^{\infty} \frac{\arctan{2t}}{e^{2\pi t}+1} \, {\rm d}t $$ and am wondering if there at all is an analytical expression for the result. I haven't tried much so far, but expressing the denominator as the geometric series and then substituting $z=2\pi t (n+1)$. Equivalently I get $$ \sum_{n=0}^{\infty} \frac{{\rm si}\left(\pi(n+1)\right)}{2\pi (n+1)} $$ where $$ {\rm si}(x) = -\int_{x}^{\infty} \frac{\sin(t)}{t} \, {\rm d}t $$ is the shifted ${\rm Si}(x)$ function. The value is approximately $0.03618247146$, in fact I know from comparison with something else that it is exactly $\frac{\ln \pi -1}{4}$, but there should be some (direct) derivation, or?  Thanks for your input if you have any.  Edit: I tried to follow your instructions, but am now stuck. So I wrote $$ {\cal L}^{-1} \left(\frac{\psi(t)-\psi(1)}{t-1}\right)(s) =\sum_{n=0}^{\infty} \frac{e^{-sn}}{n+1} = e^s \int_s^\infty \frac{{\rm d}x}{e^{x}-1} $$ and taking the Laplace-transform \begin{align} \psi(t) &amp;= \psi(1) + (t-1)\int_0^\infty {\rm d}s \, e^{-s(t-1)} \int_s^\infty \frac{{\rm d}x}{e^{x}-1} \\ &amp;= \psi(1) + (t-1) \int_0^\infty {\rm d}x \int_{0}^{x} {\rm d}s \, \frac{e^{-s(t-1)}}{e^{x}-1} \\ &amp;= \psi(1) + \int_{0}^{\infty} {\rm d}x \, \frac{1-e^{-x(t-1)}}{e^x-1} \tag{1} \\ &amp;= -\frac{1}{t} + \psi(1) + \int_0^\infty {\rm d}x \, \frac{1-e^{-tx}}{e^x-1} \tag{2} \\ &amp;= \dots \\ &amp;=-\sum_{k=0}^{n} \frac{1}{t+k} + \psi(1) + \int_0^\infty {\rm d}x \, \frac{1-e^{-(t+n)x}}{e^x-1} \\ &amp;\stackrel{(\rm{2})}{=} -\frac{1}{2t} + \ln(t) - \int_0^\infty {\rm d}x \, \left\{ \frac{1}{2} - \frac{1}{x} + \frac{1}{e^x-1}\right\} e^{-tx} \tag{Binet 1} \end{align} where $\psi(1)=-\gamma$.  Now we integrate over $t$ from $1$ to $y$ yielding $$ \ln \Gamma(y) \stackrel{({\rm 1})}{=} \psi(1)\,(y-1) + \int_0^\infty {\rm d}x \, \frac{y-1+\frac{e^{-x(y-1)}-1}{x}}{e^x-1} $$ but what now? Substitute?
I am trying to connect to a Linode (running Ubuntu 12.04 LTS) from my local machine (also running Ubuntu 12.04 LTS)  I have created a private and public key on my local machine and copied my public key to my Linode's authorized_keys file.  However, whenever I try to ssh to my Linode I get the error message Permission denied (publickey).    It's not a problem with how ssh is set up on my Linode because I can ssh to it from my Windows machine using key authentication.    In my .ssh directory on my local Ubuntu machine, I have my id_rsa and id_rsa.pub files.  Do I need to create an authorized_keys file on my local machine?    EDIT:  This is what I get when I run ssh -vvv -i id_rsa [youruser]@[yourLinode]:  debug3: authmethod_lookup publickey debug3: remaining preferred: keyboard-interactive,password debug3: authmethod_is_enabled publickey debug1: Next authentication method: publickey debug1: Offering RSA public key: id_rsa debug3: send_pubkey_test debug2: we sent a publickey packet, wait for reply debug1: Authentications that can continue: publickey debug2: we did not send a packet, disable method debug1: No more authentication methods to try. Permission denied (publickey). Abstract class with all concrete methods What does a "Legacy USB Mouse" option in a BIOS mean?  Yes legacy usually means older revision, or obsolete. So legacy usb mouse means support for "old style usb mouses".     That's not really answering the question. As Mr. Tamm, my high school electricity teacher, liked to say, "Yes, and oranges taste orangey."   What does legacy USB mouse support mean?   What is a legacy USB mouse?  There's an "old style" USB mouse?  What changed between what version of usb and what version of usb that made mice incompatible? Why was the change made?  When was the change made? Who made the change? What was the virtue of the "new USB mouses" over the "old USB mouses"?   Put it another way:     What is the BIOS doing when "Legacy USB Mouse" option is enabled?   What is the BIOS doing when "Legacy USB Mouse" option is disabled?   See also Equation numbers are shown in reverse parentheses I am using   \begin{equation} a = b \end{equation}   to write the equations in my paper. However, the problem is that the number related to each equation is not shown in the correct format like the picture   How can I fix this?  This is what I have at the first of my code:  \usepackage{graphicx} \usepackage[justification=centering]{caption} \usepackage{float} \usepackage{url}% typeset urls \usepackage{times} \usepackage{url} \usepackage{latexsym} \usepackage{fontspec} \usepackage{multirow} \usepackage{listings} \usepackage{setspace} \usepackage{listings} \usepackage{fontspec} \usepackage{amsmath} \usepackage{polyglossia} \usepackage{color} \usepackage{bidi} \newfontfamily\faFont[Script=Arabic]{Yas}  \newenvironment{Fa}{\begin{RTL}\faFont}{\end{RTL}} \newcommand{\fa}[1]{{\faFont\RL{#1}}} \newcommand\Mark[1]{\textsuperscript#1} \definecolor{backcolour}{rgb}{0.95,0.95,0.92} \definecolor{darkblue}{rgb}{0.95,0.95,0.92} \setmainlanguage{english} \setotherlanguage{farsi} \newfontfamily\farsifont[Script=Arabic]{Yas} How do I bypass restrictions on the length of the PATH variable System Properties      This environment variable is too large. This dialog allows setting values up to 2047 characters long.   I use/have used a lot of software, and my PATH has grown as a result. When I try to add new paths, I get the above message. As a workaround, I'm editing my user path, but prefer not to. Find the asymptotic tight bound for $T(n) = 4T(n/2) + n^{2}\log n$ Find the asymptotic tight bound in  $$ T(n) = 4T\left(\frac{n}{2}\right) + n^{2}\log n. $$  where $ \log n= \log _{2}n $ and $T(1) = 1$.  I should solve this using all three common methods: iteration, master theorem and substitution.  It is highly likely that this kind of recurrence equation will be in my test in two days. Thank you very much in advance! Why can a discontinuous function not be differentiable? Rings, ideals and units-proof Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice. How to format a duration in java? (e.g format H:MM:SS) What does "proverbial" mean? Using Regex to generate Strings rather than match them Move all files of type to a new folder in windows 7 Why does the order in which libraries are linked sometimes cause errors in GCC? I have build a website for a finance company.  The Cognito Form is on the website as an application form.  I would like the information sent automatically to the finance company's email address (as soon as the form is submitted) without having to log into Cognito Forms. The finance company needs to know the exact moment the application for finance comes in. The finance company can not possibly log in to Cognito Forms 24/7 to check if there are any finance applications. I know a lot of female programmers, and I know there are a good number of them out there.  But I don't recall ever having one of my questions answered by, nor have I ever answered a question by a female programmer here at Stack&nbsp;Overflow.  Now I mostly deal in the Delphi tag, and that may make a difference. But I've got the feeling there is something about the programming social network that Stack&nbsp;Overflow is, along with its form of mini-competition (via reputation) that turns most ladies off.  When I go down the first few pages of top users by rep, there is nary a lady to be found.  Go over to Quantcast and . Go down to the Demographics, and you'll see that a full 20% of the visits are by females. So they are here, lurking and using the wonderful info that Stack&nbsp;Overflow provides.  Then why aren't there more of them participating, both with questions and answers?  What can Stack&nbsp;Overflow do to persuade female programmers to participate more?    So after a dozen downvotes offset by a dozen upvotes, and accusations by many that I am a bigot, the bottom line seems to be that Stack&nbsp;Overflow appeals to males more than females.  This is my observation and I believe a proper analysis would support the hypotheses that a much higher percentage of male programmers use Stack&nbsp;Overflow than the percentage of female programmers.    Assuming that is the case, then Stack&nbsp;Overflow is geared more towards the male psyche than the female. If so, I again ask, what can they do about that?      This question was asked in StackOverflow Meta on November 20, 2009. It was made Community Wiki. It was asked in a constructive manner and should be considered an appropriate question for Meta.   Yet it was closed over two years after it was asked as "not constructive" after more than 40 upvotes and 10 favorites and 16 answers.   Now it resides here on Meta StackExchange, still closed. I just shake my head. How do I invoke a private static method using reflection (Java)? How can I add a new user as sudoer using the command line? After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? What part of a multi-type damage roll is reduced by a non-type-specific effect? Effects like  (PHB, p. 54) and Spirit Shield (XGtE, p.  10) reduce damage by a roll, irrespective of damage  (PHB 196). However, single damage rolls can have multiple damage types, as confirmed (for wording) by Jeremy Crawford in .   How do you know which part of the damage is reduced?   This could actually make a difference to the amount of damage taken if, for example, the creature taking damage is resistant to one or more of the damage types involved.       Cutting Words: ... When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature's roll.   Using this ability to expand upon the examples found in  and :     Barny, an Ice Devil (MM, p.  75) hits Bobby, rolling 9 slashing damage and 11 cold damage.      Bobby's ally, Vera, is a level 10 bard of the college of Lore, and uses her Cutting Words ability on the attack's damage roll. She rolls a 9, which is subtracted from the Barny's damage roll.      Bobby is wearing his Boots of the Winterlands (DMG, p.  156), giving him resistance to cold.   We know that "resistance and then vulnerability are applied after all other modifiers to damage" (PHB, p.  197) and both slashing damage and cold damage are treated as "one big damage roll" (see the JC tweet), so is all the 9 slashing damage removed by cutting words, and then the cold damage halved for 5 total damage, or is 9 of the cold damage removed by cutting words, leaving 2 to be halved for a total of 1 + 9 = 10 damage taken by Bobby?   Who makes this decision?  When an effect (like cutting words) reduces damage without specifying type from a source containing multiple damage types, what part of the damage is reduced? Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied . I am using an USB port as a power supply. Is there a way I can take the weak 5V of a regular laptop USB and transform if to 100V+ while begin in the smallest package possible. I've heard of boost converters but they are usualy too big. I want something that has the size of like a normal USB key.   What I want to do is to charge up around 5 capacitors at 100+V and then discharge them later. Power efficiency not a concern for me.  It would be helpful to include a schematic of the circuit.