The exercise asks to find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(z+n)^2+a^2}$$where $a$ is a complex number.  I know that $\frac{\pi^2}{\sin^2(\pi z)}=\sum_{n=-\infty}^{\infty}\frac{1}{(z+n)^2}$, but don't how to develop this fact. Is this some kind of a shift of this series?  Thank you!
I want to express $$\sum_{n=-\infty}^\infty \dfrac{1}{(z+n)^2+a^2}$$ in closed form. What comes to mind is the formula $$\pi\cot\pi z = \dfrac{1}{z}+\sum_{n\ne 0}\left(\dfrac{1}{z-n}+\dfrac1n\right)=\dfrac{1}{z}+\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}$$  and also $$\dfrac{\pi^2}{\sin^2\pi z}=\sum_{n=-\infty}^\infty \dfrac{1}{(z-n)^2}.$$ But neither of these gives the term $(z+n)^2+a^2$ that we want. Perhaps we can adjust somehow?
Identifying a short story about advertising Proving that either $2^n-1 $ or $ 2^n+1$ is not prime How do I get attention for one of my own questions without a good answer? I asked a question almost as soon as I got my account. I got a few decent answers, but none of them helped solve my problem. The question has long since become buried by newer questions, and there's still no accepted answer. Unfortunately, because there are quite a few answers with votes, to a casual observer it probably looks like the question was answered and I just neglected to mark an answer as accepted, and the system does not consider it "unanswered".  Or, I asked a question that was a bit more difficult to answer. After waiting for some time, I received no answers, little or no comments, and only a few views. The question does show up in the "unanswered" list, but no one has answered my question.  In either case, what's the protocol for me to try to get this question answered?  I see several options:   Post the question a second time, possibly linking to the original, to get attention for the original question. This is encouraged on sites like Reddit.  However, it's probably not appropriate here, since we're encouraged to check for duplicates and not submit a question that's already been asked by someone else, so this probably falls under that heading. Wait for some kind of wiki-like talk page to be implemented, then post a plea on the talk page for input and hope that activity on the talk page bumps up the question. Edit the question, but do nothing further on the Stack Exchange site. If I edit my post to say that I haven't accepted any of the current answers but am looking for more, then the question might get answered eventually by people browsing through old questions to find something they can answer. Post the question on other sites, but link to my original question on Stack Exchange. People will either answer here, or they'll answer on the other site and I can post the accepted answer here myself.  This might be considered spamming, so it might be better to do this without the link.      For more information, see the "" page in the . Is there a quick way to reset a .blend files windows and settings to default Whenever I download a .blend from someone all of the windows and settings are the way they have left them as it is saved in the file. It takes way to long to rearrange every window and toolbar to the way it is by default. Is there any tool or shortcut that can reset the UI to blenders default and just leave the actual contents of the file? The only way I could do this is by creating a new file and using link/append but this is not the best option. Could not load 'vboxdrv' after upgrade to Ubuntu 16.04 (and I want to keep secure boot) I upgrade from Ubuntu 15.10 to 16.04 and since then VirtualBox 5.0.18 isn't starting my VMs anymore. It complains that 'vboxdrv' isn't loaded. So I try to load it and get the following error:  $ sudo modprobe vboxdrv modprobe: ERROR: could not insert 'vboxdrv': Required key not available   I believe it is related to secure boot which I use and which I want to continue using. Actually with Ubuntu 15.10 secure boot and VirtualBox were working just fine.  Also I tried $ sudo apt-get --reinstall install virtualbox-dkms which built the kernel module successfully but didn't solve this issue.  Any idea how to get vboxdrv loaded while keeping secure boot enabled?  Update 2: Also I tried executing sudo mokutil --disable-validation. When executing this command, during the next boot I get prompted to disable secure boot, add a key or hash from disk. Since I don't want to disable secure boot, it seems that this doesn't solve my issue either. Also I want to keep UEFI activated for a parallel Windows installation.  Note: If you don't mind disabling secure boot, see  instead. I have been given a Bode plot of a low pass filter, and tasked with essentially reversing the effects by using a high pass filter. I have solved for the transfer function which is the first equation, and I am assuming the inverse transfer function will give me the information I need to construct the high pass filter. I know the values for \$z\$ and \$p\$. Is there a method given any transfer function to realise the circuit elements required? $$H(s) = 0.17\frac{s-z}{s-p}$$ $$H(s)^{-1} = 5.88\frac{s-p}{s-z}$$ How can I write a spiral symbol? How can I get a spiral like    There seems to be a spiral-like unicode codepoint:   I tried  \documentclass[varwidth=true, border=2pt]{standalone} \usepackage[ngerman]{babel}  \usepackage[utf8]{inputenc} % Color \usepackage{color} \definecolor{lightgray}{gray}{0.95}  \DeclareUnicodeCharacter{43612}{\spiral}  %document \begin{document}{\textcolor{lightgray}{a}}\spiral{\textcolor{lightgray}{b}} \end{document}   but I got  ! Undefined control sequence. l.13 ...document}{\textcolor{lightgray}{a}}\spiral                                               {\textcolor{lightgray}{b}}   edit: I know . None of the answers gave me a spiral symbol. Is $7$ the only prime followed by a cube? How to manually parse a floating point number from a string Is the set of all possible binary strings countable? Determine whether the given sets are countable or uncountable. Justify your answers either with bijections or using results/methods. The set of all possible binary strings. Note a binary string is a sequence of 0s and 1s.  If a binary string consists only of 0s and 1s, then is it even possible to find a bijection for it? solving diophantine equations using congruences So, a question in Elementary Number Theory by David M. Burton:     Problems 4.4      2(a) Using congruences, solve the diophantine equation 4x + 51y = 9       Hint: $4x \equiv 9(mod \ 51) $ gives $ x = 15 +51t$, whereas $ \ 51y \equiv 9 (mod \ 4)$ gives $y = 3+4s $. Find the relation between $s$ and $t$ From this hint, I substitute $ x = 15 +51t$ and $y = 3+4s$ in the given equation $4x + 51y = 9$ to get the relation $s = -t-1$ and using which I can get the solutions $ x = 15 + 51t, \ y =-1-4t$ But we know that the solutions of $ax \equiv b (mod \ n)$ is equivatent to the solutions of the diophantine equation $ax -ny = b$, so, solutions of $4x \equiv 9(mod \ 51) $ is equivalent to solutions of the equation $4x-51y=9$ whereas, solutions of $51y \equiv 9(mod \ 4) $ is equivalent to solutions of the equation $-4x+51y=9$ Now, my question is: as, the above equations $4x-51y=9$ and $-4x+51y=9$ are different, then what is the point of substituting $ x = 15 +51t$ (solution of $4x-51y=9$) and $y = 3+4s $ (solution of $-4x+51y=9$) in the given equation $4x+51y=9$ to find a relation between $t$ and $s$ ? Isn't it absurd as $ x = 15 +51t$ and $y = 3+4s $ are solutions of two different equations? Can anyone please explain this? PS: I did it like this $4x \equiv 9(mod \ 51) $ gives $ x = 15 +51t$ and by substituting it in $4x+51y=9$ we get $y=-1-4t$ How do I write content to another browser window using Javascript? I've opened a new window with window.open() and I want to use the reference from the window.open() call to then write content to the new window.  I've tried copying HTML from the old window to the new window by using myWindow.document.body.innerHTML = oldWindowDiv.innerHTML; but that's doesn't work.  Any ideas? A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help! Calibre stopped working recently. Tried uninstalling and reinstalling. When I open a terminal and call $calibre I get the following error message: Traceback (most recent call last):   File &quot;/usr/bin/calibre&quot;, line 20, in &lt;module&gt;     sys.exit(calibre())   File &quot;/usr/lib/calibre/calibre/gui_launch.py&quot;, line 73, in calibre     main(args)   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 543, in main     listener = create_listener()   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 514, in create_listener     return Listener(address=gui_socket_address())   File &quot;/usr/lib/calibre/calibre/utils/ipc/server.py&quot;, line 110, in __init__     self._listener._unlink.cancel() AttributeError: 'NoneType' object has no attribute 'cancel'  Do I need to install some dependency that the sudo apt install didn't get? I tried installing via command line and the software installer/updater GUI. I'm on version Ubuntu 20.04.1 LTS. I made sure that my software in general is up to date also. What is a good introductory book on quantum mechanics? I'm really interested in quantum theory and would like to learn all that I can about it. I've followed a few tutorials and read a few books but none satisfied me completely. I'm looking for introductions for beginners which do not depend heavily on linear algebra or calculus, or which provide a soft introduction for the requisite mathematics as they go along.  What are good introductory guides to QM along these lines? How do I print bold text in Python? My roommate put a hunk of beef roast in the slow-cooker for a good 6 hours.  It's a fairly large sized slow-cooker, at least 8 quarts.  The meat was cooked through but very chewy, and I never remember my mom having that issue with a slow-cooker.  My roommate tends to put only enough water to get halfway up the side of the roast, and always uses a slow-cooker 'bag' (another thing my mom never did).  What advice can I offer them to get a better outcome? Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$ Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I couldn't prove it. Any help will be greatly appreciated. Colors and legend in groupplots/barplot In order to compare three items I use the groupplot command. For each item I have n different criteria to compare.    Is it possible to set the color for each criteria itself? Meaning the first bar in every group goes gray, the second pink etc.. The legend than should contain the n compared criteria.  And... how do I manage to let the whole tikz-picture have a width of 0.9\textwith  \documentclass{standalone} \usepackage{pgfplots,pgfplotstable} \usetikzlibrary{pgfplots.groupplots}  \pgfplotstableread{ nr  fii faa foo a   10  15  12 b   12  18  14 c   13  19  15 d   9   13  10 e   7   11  8 f   10  15  12 }\datatable  \begin{document} \begin{tikzpicture} \begin{groupplot}     [group style={columns=3},         ybar,         ymin=0,         ymax=20,         xtick=\empty,         tickpos=both]  \nextgroupplot[xlabel=fii] \addplot [fill=blue] table [x expr=\coordindex, y=fii] {\datatable}; \nextgroupplot[xlabel=faa] \addplot [fill=green] table [x expr=\coordindex, y=faa] {\datatable}; \nextgroupplot[xlabel=foo] \addplot [fill=red] table [x expr=\coordindex, y=foo] {\datatable};  \end{groupplot} \end{tikzpicture} \end{document} I was using , implemented in scikit-learn, for my research work. But I have no good understanding of this.   Can anyone please give a simple, good explanation of one class SVM?