Use the Laurent series of $e^{1/z}$ on the punctured plane $\mathbb{C} - \{0\}$ to compute $1/\pi \int_0^\pi e^{\cos\theta}\cos(\sin\theta-n\theta) d\theta$ for $n=0,1,2,...$   I have been trying to attempt this question from my complex analysis book and I am not sure where to even begin. All I know is that the Laurent series of $e^{1/z}$ is $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!}\frac{1}{z^n}$, for all $|z|&gt;0$.   I have no idea how this relates the the integral though because there is no $e^{1/z}$ only a $e^{\cos\theta}$ in the integral. I thought maybe I have to put the $\cos\theta$ values into Taylor form but I'm not sure if that's correct.  Any help is greatly appreciated! Thanks :)
I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints?
"Install Spotify" can't be opened because Apple cannot check it for malicious software I'm running macOS 10.15 Catalina Beta.  When I try to install Spotify I see the following error:    Is this a Spotify issue or a Catalina issue?  Thanks. Prove the definitions of $e$ to be equivalent How to prove that the following definitions of $e^x$ are equivalent, with as simple tools as possible and without any knowledge of $e$ or logarithms ?  $$\sum_{n=0}^{\infty} \frac{x^n}{n!}=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$  Also preferably, prove that this is $a^x$ for some real number $a&gt;0$. I am using the 2D Animation preset and so far every new scene I made had the same bright white background. Now my two latest scenes (created with 'copy settings') have a light grey background even though they are set to white. There's also no difference in surface, nodes or strength.  There are apparently exist different background shades with an off-white color but selecting the white one doesn't help either I have several covariates in my calculation for a model, and not all of them are statistically significant.  Should I remove those that are not?  This question discusses the phenomenon, but does not answer my question:   There is nothing in the answer to that question that suggests that non-significant covariates be taken out, though, so right now I am inclined to believe that they should stay in.  Before even reading that answer, I was thinking the same since a covariate can still explain some of the variance (and thus help the model) without necessarily explaining an amount beyond some threshold (the significance threshold, which I see as not applicable to covariates).  There is another question somewhere on CV for which the answer seems to imply that covariates should be kept in regardless of significance, but it is not clear on that.  (I want to link to that question, but I was not able to track it down again just now.)  So...   Should covariates that do not show as statistically significant be kept in the calculation for the model?  (I have edited this question to clarify that covariates are never in the model output by the calculation anyway.)  To add complication, what if the covariates are statistically significant for some subsets of the data (subsets which have to be processed separately).  I would default to keeping such a covariate, otherwise either different models would have to be used or you would have a statistically significant covariate missing in one of the cases.  If you also have an answer for this split case, though, please mention it. Prevent questions with movie/tv shows/game spoilers from appearing in the hot questions list I've noticed that occasionally "hot questions" from SciFi and the Movies &amp; TV network contain major spoilers about the movie/show discussed.  The Matrix spoilers:      (just now there's something about Cypher betraying Morpheus in the Matrix, which is a big deal for people who haven't seen the movie).   I like browsing through those but fear that one day one of these questions will spoil a show I'm excited about and yet to finish   Breaking Bad spoilers:      (like the finale of Breaking Bad, imagine a question asking why did one character betray another at the end or something).   Can users posting such questions be asked to label those with a tag (such as 'spoilers') and then somehow have the network hide those from the "hot questions" list? Or possibly format them differently?  Similarly, this could apply to the Arcade network too.  P.S. Love how this question spikes in popularity around the time GoT starts running again. I swear it should be fairly simple technically to ban questions that have a tag that says 'spoilers' from appearing in the Hot Network Questions.  P.P.S. Oh, and with every fresh release of Star Wars (The Last Jedi being the most recent one)!  P.P.P.S. Another year, another Game of Thrones, another spike in popularity! How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How   $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$   i have tried to do that by the following procedure:  $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - \sum_{i=1}^n\bar X^2)$  =$\frac{1}{n} \sum_{i=1}^n (X_i^2 - \bar X^2)$  Then i have stumbled. Unix Truncate Column in csv file I am having a problem running the CDC change capture job against some of my test databases.  The error message I'm getting is the following:     Executed as user: NT AUTHORITY\NETWORK SERVICE. Cannot execute as the database principal because the principal "dbo" does not exist, this type of principal cannot be impersonated, or you do not have permission. [SQLSTATE 42000] (Error 15517).   How can I fix this? Traveling to the US with new passport, visa on the old one With Ubuntu One shut down, is there a way to sync applications between computers? Since Ubuntu one is no longer it appears that Sync between computers no longer works.  Is there a new method to do this?  I'd like to sync the apps from one machine to another. Every Window in Windows 8 got this icon if it gots the focus. When clicking on it, nothing seems to happen.   How does this icon work? I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In the complex projective line, antipodal points are identified, but in the Riemann sphere they are clearly distinct. Then how can the two be homeomorphic.   I found out the following map from the complex projective line to $S^2$. But how do I prove this is a homeomorphism? Finding the inverse of this functions seems complicated, and then how do I prove that the inverse map of a open subset is a open subset, and vice versa?  $f:\mathbb{C}\mathbb{P}^1 \to S^2 = [z:w] \to \frac{(2Re(w\bar{z}), 2Im(w\bar{z}),|w|^2-|z|^2)}{(|w|^2+|z|^2)} $ What's the name for the property of a function $f$ that means $f(f(x))=x$? I can think of several examples of functions such that twice application of the function is equivalent to no application of it.   Additive inverse Multiplicative inverse Fourier transform Complex conjugation Any group built up from $\mathbb{Z}_2$, applying (one of) the $\mathbb{Z}_2$ parts' operation.   "Idempotent" came to mind, but that's wrong. It means $f(f(x)) = f(x)$, not $f(f(x))=x$.  What is the word for this "flip-flop" property? General integral to find resistance My question is: is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?.  Obviously, if so, what is it, and if not, why?  It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive.  I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation.  To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$.  Single Volume Framework  We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it.  This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating.  Infinite Volume Framework  An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition.  In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$.  Known Algebraic Analogs  The  that I find insufficient is:  $$R = \rho \frac{\ell}{A}$$  Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area.  Obviously, this is a rather simple equation that won't apply to more complicated geometry.  Even more  seem to give equations that fall short of what I'm asking.  For example:  $$R = \rho \int_0^l \frac{1}{A(x)} dx$$  I think it's obvious that an equation such as this is built upon a myriad of assumptions.  For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly.  Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it.  I have some reasons to suspect this might actually be rather difficult.  A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking.  So maybe this will result in two interconnected calculus equations.  Motivation  I had an interest in , and it occurred to me that I can't quickly and simply write down the equation for resistance between two points.  The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating.  However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions.  If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it.  Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light. Sharepoint 2013. Add JS to NewForm.aspx of a list how to avoid ssh asking permission? We are attempting to speed up the installation of oracle nodes for RAC installation. this requires that we get ssh installed and configured so that it doesn't prompt for a password.  The problem is: On first usage, we are prompted for  RSA key fingerprint is 96:a9:23:5c:cc:d1:0a:d4:70:22:93:e9:9e:1e:74:2f. Are you sure you want to continue connecting (yes/no)? yes   Is there a way to avoid that or are we doomed to connect at least once on every server from every server manually? Let $\mathbb{S} = \{ (X,Y,Z) \in \mathbb{R}^3 :X^2 + Y^2+Z^2=1\}$ denote the Riemann sphere. The following problem is taken from Gamelin "Complex Analysis" chapter one:     What is the image of the spherical cap $ C := \{(X,Y,Z) \in \mathbb{S} : A \leq X \leq 1\}$, where $A$ is some fixed real number between $-1$ and $1$, under the stereographic projection $ \pi(X,Y,Z) = \frac{X+iY}{1-Z}$?   Here is my work so far:  ( The case $A=1$ is trivial, so assume $A &lt;1$)  We want to find $\pi(C) = \{z \in \mathbb{C} \mid \exists(X,Y,Z) \in C: \pi(X,Y,Z) = z \}= \{z \in \mathbb{C} \mid \pi^{-1}(z) \in C\}$, where the last equality holds by $\pi$ being a bijection. Write $z=x+iy$.  Here, $ \pi^{-1}(z) \in C\iff A\leq \frac{2x}{1+x^2+y^2} \leq 1 \iff (\star) :A \leq \frac{2x}{1+x^2+y^2}$, since the inequality $2x \leq 1+x^2+y^2$ holds for all $x,y \in \mathbb{R}$.  $A=0$: $( \star)$ reduces to $ 0 \leq x$, so in this case the image is $\{z \in \mathbb{C} \mid \Re(z) \geq 0 \}$.   $A &gt;0$: Since $A \neq 0$ in $(\star)$ we have,  $ \frac{2x}{1+x^2+y^2} \geq A \iff Ax^2-2x+Ay^2+A \leq 0 \iff (x-\frac{1}{A})^2 +y^2 \leq \frac{1}{A^2}-1$. Thus the image is the interior and boundary of the circle centered at $\frac{1}{A}$ with radius $\sqrt{\frac{1}{A^2}-1}$. Note $\frac{1}{A^2}-1 \geq 0$ as $A \in (0,1)$.  $A &lt;0$ : By symmetry and the above, here the image is $ \mathbb{C} / \{ $ interior and boundary of circle centered at  $  \frac{1}{A}$ with radius $\sqrt {\frac{1}{A^2}-1} \}$.  Is my working correct? Many Thanks! Batch file to delete files older than N days I got  question in review. First sentence is:     I know there is a  question, but no answers.   And indeed the two questions are virtually identical. I started to flag as duplicate - but the original question had no answer at the time, so it wouldn't be correct and apparently . (I didn't go so far as to try to submit the flag).  What is the proper action here? How do we want such a question to be handled? Should both (or more) questions stay open indefinitely until either is answered and somebody gets the good idea to do the duplicate linking?   As the new question is essentially a bump to the front page (perhaps with re-tagging), especially in the case where the OP knows that a previous question exist, perhaps something should be done to enable bumping re-emphasizing without writing the new question.  As I wrote this,  actually got an answer. So the old one should now be closed as a duplicate of the new - but my question remains for the case where all known/found duplicate questions have no answer?  Ironically, just before asking this, the "Questions that may already have your answer" gave me ... (Although the suggestion has no answer). So it has been asked before - the comment there even suggest the ability to re-emphasize. why arcane legacy steps to boot from USB with GPT and UEFI? To run the live AVG rescue "disk" on USB for Windows 10, how is it made to work with a GPT and UEFI system?  (Possible solution listed below.)  I enabled the boot menu so that pressing F12 will display a list to select from.  Also, I put everything USB at the top of the boot sequence and Windows and the HDD at the bottom.  However, I hadn't realized GPT and UEFI are so different from a BIOS.  Because the USB flash drive isn't compatible with GPT it's not being picked up?  This  was suggested:   Open the notebook and press quickly F2 to enter the BIOS. Under the MAIN menu, enable the F12 Boot Menu option. Under the BOOT menu, select Legacy boot and press F10 Save &amp; Exit. While the notebook is rebooting, press F2 again to enter the BIOS. Under the BOOT menu, select as your first boot option the USB HDD stick (your USB stick). Press F10 Save &amp; Exit.    Perhaps this will work, but what is the problem that prevents UEFI boot from registering the thumb drive?  Because it's using GPT and not an MBR?  I don't want to switch to legacy, but stay with UEFI.  The laptop: