$\tan(\alpha)=\alpha$ has not complex roots? By the Strum theory it is easy to see that the equation $\tan(\alpha)=\alpha$ has not complex roots and it's real roots occur at zero and near $(2n+1)\pi/2,\ n\in Z^{+}$.  I am interesting to know that is there any other way to see that the above equation has not complex roots?
Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified? I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained  $$\begin{eqnarray} \frac{\sin z}z&amp;=&amp;\frac{\sin(x+iy)}{(x+iy)}\\ &amp;=&amp; \frac{\sin(x)\cosh(y)+i\cos(x)\sinh(y)}{x+iy}\\ &amp;=&amp; \frac{x\sin(x)\cosh(y)+y\cos(x)\sinh(y)}{x^2+y^2}+i\frac{x\cos(x)\sinh(y)-y\sin(x)\cosh(y)}{x^2+y^2}. \end{eqnarray}$$  Before calculating it, I hoped I could use this in a solution to a certain problem, but I would need to differentiate the real and imaginary parts of $\dfrac{\sin z}z$ with regard to $x$ and $y$. But what I'm getting is very complicated. I understand that it could be that that's just the way it is. But maybe there are some simplifications to be done either in the functions or in their derivatives? I've never really used hyperbolic functions before so I'm not sure I'm not missing something.  Added. The problem I'm working on is the following.     Prove that the solutions of the equation $$\tan z=z$$ are all real.   I will sketch the solution I know here.  It can be calculated that  $$\tan(x+iy)=\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}+i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}.$$  Plugging this to the equation, we get  $$\begin{eqnarray} x&amp;=&amp;\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}\\ y&amp;=&amp;\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}. \end{eqnarray}$$  From this we can get   $$\sin(2x)\cdot2y=\sinh(2y)\cdot2x.$$  By plotting the functions $\sin,$ $\mathrm{id}$ and $\sinh$ (and proving the right inequalities we can see from the plots), we can see that this is only possible when $xy=0$. But when we plug $x=0$ to the second equation, we can show that the only solution is $y=0$, which ends the proof.  What strikes me in this solution is the fact that $\dfrac{\sin z}z$ is used. (Not explicitly in my statement.) This function has another connection to the problem:     The non-zero solutions of the equation $\tan z=z$ are exactly the zeros of the derivative of $\dfrac{\sin z}z.$   This is easy to prove. I wondered if there shouldn't be a solution to the problem which uses this fact, given that $\dfrac{\sin z}z$ appears in the solution I know.
What is the difference between the dashed and solid attack bars? I was flipping through my Pokemon list to see which kind of attacks they had, and I notice there was a small different between the attacks. Some attacks contained a dashed blue bar where as others contained a solid blue bar.   What is the difference and significance between the two? Explain proof by contradiction. Where do I ask where to ask? If I have a specific question I want to ask about, but I don't know which Stack Exchange site would be on topic for asking that question, where can I ask where to ask that question?  Is there any generic place where we can ask for advice? Is it here?  This is not about a specific question but in general.  For example: "Where should I ask for advice about fishing?", or if I need to know where I should ask "What would be the best fishing rod to fish for salmon?" (regardless of how good/bad these examples are) I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. Our 3 months old baby is very keen on rolling over and being on his stomach. Most of the time he automatically rolls over in a few minutes after we put him on his back, including during bed time. Is this dangerous? We are particularly worried that   his nose maybe blocked when facing down it is said that SIDS risk increases when a baby sleeps on his stomach   If it is dangerous, how can we stop this, especially during bed time or other times when we can't watch him continuously? Multiple (diamond) inheritance compiles without "virtual", but doesn't with Can I cast a Warlock spell with another (higher) spell slot from a different multiclass? Reviewing an answer to a recent question I posted, I had a thought about Armor of Agathys. In the event of a Sorcerer 18 / Warlock 2 build, would the character be able to cast Armor of Agathys with a 9th level spell slot, thus getting 45 Temporary Hit Points and dealing 45 damage when hit?   Is the spell in any way limited to being cast using Warlock spell slots or to a maximum of a 5th level spell slot? Blogs link from the not existing community page refers to 404 The “blogs” link from the , e.g:        refers to another not existing page:        Looks like blogs link should refer to the same page as for “blog” link in the footer: I am looking for a closed form (ideally expressed as elementary functions) of the function $\exp_n(x) = \sum_{k=0}^n x^k / k!$.  I am already aware of expressing it in terms of the gamma function. Background / Motivation When counting combinations of objects with generating functions, it is useful to be able to express the partial sum $1 + x + \cdots + x^n$ as $\frac{1-x^{n+1}}{1-x}$. For example, to count the number of ways to pick 5 marbles from a bag of blue, red, and green marbles where we pick at most 3 blue marbles and at most 2 red marbles, we can consider the generating function $f(x) = (1+x+x^2+x^3)(1+x+x^2)(1+x+x^2+\cdots)$. By using the partial sum identity, we can express it as $f(x) = \left(\frac{1-x^4}{1-x}\right)\left(\frac{1-x^3}{1-x}\right)\left(\frac{1}{1-x}\right)$.  Simplify, express as simpler product of series, and find the coefficient of the $x^5$ term. I want to be able to do the same for a generating function in the form $g(x) = \exp_{n_1}(x)^{p_1} \exp_{n_2}(x)^{p_2} \cdots \exp_{n_j}(x)^{p_j}$ The easiest way to extract the coefficient of a given term $x^p / p!$ would be to use a similar closed form expression for $\exp_n(x)$ and a similar technique to $f$. Attempted Solutions Differential equation Recall that the way to prove the identity $1+x+x^2+\cdots+x^n = \frac{1-x^{n+1}}{1-x}$ is to define $S = 1 + x + x^2 + \cdots + x^n$ and notice that: $S - Sx = 1 - x^{n+1}$. Likewise, notice that $y(x) = \exp_n(x)$ satisfies $y - y' = x^n/n!$.  Via SAGE, the solution is $y(x) = \frac{c+\Gamma(n+1,x)}{n!}e^x$.  Our initial condition $y(0) = 0$ so $c=0$.  By , $\Gamma(n+1,x) = n! e^{-x} \exp_n(x)$ so we have $y(x) = \exp_n(x)$. Recurrence Relation Notice that $\exp_n(x) = \exp_{n-1}(x) + x^n/n!$. Using the unilateral Z-Transform and related properties, we find that $\mathcal{Z}[\exp_n(x)] = (z e^{x/z})/(z-1)$. Therefore, $\exp_n(x) = \mathcal{Z}^{-1}\left[(z e^{x/z})/(z-1)\right] = \frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz$. $(z^n e^{x/z})/(z-1)$ has two singularities: $z = 1$ and $z = 0$.  The point $z = 1$ is a pole of order one with residue $e^x$.  To find the residue at $z = 0$ consider the product $z^n e^{x/z} (-1/(1-z)) = -z^n \left( \sum_{m=0}^\infty x^m z^{-m} / m! \right) \left( \sum_{j=0}^\infty x^j \right)$.  The coefficient of the $z^{-1}$ term is given when $n - m + j = -1$. The residue of the point $z=0$ is then $-\sum_{m,j} x^m / m! = -\sum_{m=n+1}^\infty x^m / m!$. Let $C$ by the positively oriented unit circle centered at the origin.  By Cauchy's Residue Theorem, $\frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz = \frac{1}{2 \pi i} 2 \pi i \left(e^x - \sum_{m=n+1}^\infty x^m / m!\right) = \exp_n(x)$. Finite Calculus I've tried to evaluate the sum using finite calculus, but can't seem to make much progress. How to capture Control+D signal? I know the statement:  create table xyz_new as select * from xyz;   Which copies the structure and the data, but what if I just want the structure? $\LaTeX$ and Mathematica What is Uplay? Can I play Ubisoft games without it? Heroes of Might and Magic VI seems to have installed a program called "Uplay", which it uses to launch (in addition to Steam, mind!)  What exactly is Uplay supposed to be? I notice that when I stop playing, it captchalogs the highlights of my past session (achievements earned, etc.) - which is kinda cool, but also kinda creepy and big-brotherish. Given that I value my privacy, is there anyway to play Uplay games without using Uplay?  Is this just Ubisoft's infamous "always-on" DRM? How should I structure my URLs for both SEO and localization? When I set up a site in multiple languages, how should I set up my URLs for search engines and usability?  Let's say my site is www.example.com, and I'm translating into French and Spanish. What is best for usability and SEO?  Directory option:  http://www.example.com/sample.html http://www.example.com/fr/sample.html http://www.example.com/es/sample.html   Subdomain option:  http://www.example.com/sample.html http://fr.example.com/sample.html http://es.example.com/sample.html   Filename option:  http://www.example.com/sample.html http://www.example.com/sample.fr.html http://www.example.com/sample.es.html   Accept-Language header:  Or should I simply parse the  header and generate content server-side to suit that header?  Is there another way to do this? If the different language versions don't have different urls, what do I do about the search engines?    UPDATE 2011-12-06  Google has new recommendations for meta tags for explicitly pointing to other language content: .  UPDATE 2012-05-25  Related but not precisely:   UPDATE 2013-06-12  includes discussion of several URL schemes directly relevant to the question. A Basic Question about Gravity, Inertia or Momentum or something along those lines Proof that the Casimir invariant of a representation commutes with everything Here's Prob. 3, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:      If $s_1 = \sqrt{2}$, and $$s_{n+1} = \sqrt{2 + \sqrt{s_n}} \ \ (n = 1, 2, 3, \ldots),$$ prove that $\left\{ s_n \right\}$ converges, and that $s_n &lt; 2$ for $n = 1, 2, 3, \ldots$.    My effort:   We can show that $\sqrt{2} \leq s_n \leq 2$ for all $n = 1, 2, 3, \ldots$. [Am I right?]  Then we can also show that $s_n &lt; s_{n+1}$ for all $n = 1, 2, 3, \ldots$. [Am I right?]  But how to calculate the exact value of the limit? Where does this sequence occur in applications? Is it possible to set DNS for mobile network Can't login with Google using Internet Accounts USB is a popular standard and I was wondering whether there is some way to connect two or more computers using USB ports rather than the Ethernet ports.