I am trying to solve:  $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$  I translate it to its complex equivalent:  $$\lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz - \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$  where $C$ is a half circle with radius $R$, above the real axis, and $\gamma$ is only the curved (top part) of the half circle.  I am trying to show that $$\int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$ tends to 0 as $R \rightarrow \infty$.  I have that:  $$\left| \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz \right|\leq \pi R \cdot \sup_{\xi \in \gamma} \mid \frac{z\sin{z}}{z^2 + a^2} \mid$$  That is, the integral is upper bounded by the length of the contour multiplied by the supremium of the contour.   This leads me to:  $$\pi R \cdot \sup_{\xi \in \gamma} \left| \frac{z\sin{z}}{z^2 + a^2} \right| \leq \pi R \cdot \mid \frac{R}{R^2 + a^2} \mid$$  as $\sin{z}$ has a maximum of 1. But what other inequalities can I produce to show that this tends to 0 as $R \rightarrow \infty$?  Thanks.
I am trying to solve:  $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$  and show that it equals $\pi e^{-a}$ for all $a &gt; 0$.  I translate it to its complex equivalent:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz - \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$  where C is a half circle with radius R, above the real axis, and $\gamma$ is only the curved (top part) of the half circle.  I've shown that $$\int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$ tends to 0 as $R \rightarrow \infty$.  So now:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz$$  By residue theorem,   $$\int_{C}\frac{z\sin{z}}{z^2 + a^2}dz = 2\pi i \cdot res_{ai}f$$  as $ai$ is the pole of $f$ that lies above the real axis.  $$2\pi i \cdot res_{ai}f = 2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z sin{z}}{z^2 + a^2}$$  By L'Hospital's rule, this is equivalent to:  $$2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z^2\cos{z} + 2z\sin{z} - zai\cos{z} - ai\sin{z}}{2z} = 2\pi i \cdot \frac{\sin{ai}}{2} = \pi i \sin{ai}$$  But how is $\pi i \sin{ai}$ equivalent to $\pi e^{-a}$? Did I make a mistake?
why __getitem__ cannot be classmethod? Render an InputText based on Selected Option In my VF page I have an apex:selectList with a List of Vendors (e.g. Alcatel,Cisco ec..), the last option is "Other".   When the user select "Other" an apex:inputText should be rendered  where he can writes the Vendor. I'm not able to render the inputText Field   My VF page:  &lt;apex:outputPanel id="vendor2"&gt; &lt;p&gt;&lt;b&gt; Vendor: &lt;/b&gt;&lt;/p&gt; &lt;apex:selectList value="{!vendor}" multiselect="False" size="1"&gt;     &lt;apex:actionSupport event="onchange" action="{!specifyVendor}" rerender="otherPanel"/&gt;     &lt;apex:selectOptions value="{!vendors}"/&gt; &lt;/apex:selectList&gt;&lt;p/&gt; &lt;/apex:outputPanel&gt; &lt;br/&gt; &lt;apex:outputPanel id="otherPanel"&gt;     &lt;apex:inputText value="{!otherVendor}" rendered="{!vendor=='Other'}"&gt; Other: &lt;/apex:inputText&gt; &lt;/apex:outputPanel&gt;     &lt;br/&gt;&lt;br/&gt;    The custom controller:      public with sharing class ExcelPageController {          public String vendor {get; set;}//Vendor Name         public String otherVendor {get; set;}//Vendor Name          public List&lt;SelectOption&gt; getVendors(){             System.debug('getVendors');                 List&lt;SelectOption&gt; options = new List&lt;SelectOption&gt;();             options.add(new SelectOption('Cisco','Cisco'));             options.add(new SelectOption('Alcatel','Alcatel'));             options.add(new SelectOption('Tibco','Tibco'));             options.add(new SelectOption('Other','Other'));             return options;         }     public PageReference specifyVendor(){         System.debug(vendor);         return null;     } } How do ideal quotients behave with respect to localization? Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When does localization at $S$ commute with taking quotients, i.e. when does the equality $$(S^{-1} J : S^{-1} I) = S^{-1} (J:I)$$ hold? More generally, when is it true that $$S^{-1} \text{Ann}_{R} (M) = \text{Ann}_{S^{-1} R} (S^{-1} M)$$ for an $R$-module $M$? I am developing firefox extension and wanted to count total number of text box of webpage so how to count text box? I believe this to be fraud — hired, then asked to cash check and send cash as Bitcoin I have been "hired" by Nissan Motor as a key account specialist.  My first assignment is to cash two checks that I have received. Both were written by a customer who has already received the engines. The total is $2500. I am to cash the checks, get the money, take out my bonus and any other related expenses and deposit into a Bitcoin ATM. Then make copy of receipt and electronically send to my "manager".  I contacted a lawyer, was told to talk to the Department of Labor, who said they will not get involved, as this is between me and my employer. Any advice? I'm getting the following error when trying to update a taxonomy field value via REST api:  {   "error": {     "code": "-1, Microsoft.SharePoint.Client.ClientServiceException",     "message": {       "lang": "en-US",       "value": "Cannot deserialize data for type Microsoft.SharePoint.Taxonomy.TaxonomyFieldValueCollection."     }   } }   The request:  https://server/_api/web/GetFileByServerRelativeUrl('/Shared Documents/file.txt')/ListItemAllFields  {   "__metadata": {     "type": "SP.Data.Shared_x0020_DocumentsItem"   },   "Title": "some title",   "Project": {     "results": [       {         "Label": "Term1",         "TermGuid": "897bdf7f-08d6-44b9-a71a-ddc4cf131664",         "WssId": 5       }     ]   } } Is it acceptable to add a duplicate answer to several questions? Is there an MD5 Fixed Point where md5(x) == x? Is there a fixed point in the MD5 transformation, i.e. does there exist x such that md5(x) == x? What is object slicing? What's the difference between the server version and the desktop version? What's the difference between the server version of Ubuntu and the desktop version? Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$? Let us view our deleted comments If moderators can view , can we see comments we've deleted? We can see our deleted answers, but why not our deleted comments? If development on the Stack Exchange iOS/Android apps has ceased indefinitely, what should we do to their bug reports and feature requests? It's been a while since the Stack&nbsp;Exchange team last had an iOS or Android developer, and the development job on the apps have not had any insights for years.  That, however, doesn't deny the fact that there are still many users to them. As a consequence, people are continuing to submit bug reports and sometimes feature requests about them.  If development on iOS/Android apps has ceased indefinitely, what should we do to their bug reports and feature requests?  Are we going to keep them with the hope that development on apps will resurrect one day, or just ignore them and telling users that development has frozen? How best can I discover what is up with my electrical bill? This might be subjective, I don't know. However, it's a serious problem for me. My electrical bill is outrageous. It's three times as large as my next-door neighbor, and four or five times the size of my neighbor's across the street. It's worse in the summer (I live in central TX), but I've had the air conditioner inspected, and while the house is right at the maximum capacity for our unit, the unit should still be able to take care of the load. That also doesn't explain why our winter bills are larger than the comparables.  What I would like to do is check the amount of current drawn by each running appliance and calculate where all my money is going. I just don't know how to go about doing this. What is the meaning of "algebraically indistinguishable" I heard the term couple of times (in Field theory class and book), for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable".  I understand the meaning intuitively, but what does it really mean ? Replace unknown reference (?) with the tag from \ref or \cite While working on a document, it would be helpful if instead of a ? for unresolved references or citations the tag used in the \ref{thisthings} or \cite{Bloke1978} could appear instead.  Is there a package that turns on this behaviour? What are the system requirements for each flavour of Ubuntu Desktop? What are the benefits/detriments to graduate students joining unions? Star rating in AJAX with Ruby On Rails Why does Spring MVC respond with a 404 and report "No mapping found for HTTP request with URI [...] in DispatcherServlet"?