I'm to solve an exercise which supposes that $ f: U \rightarrow \mathbb{C}$ is holomorphic on an open subset U $\subset \mathbb{C}$ and that $f'(a) \neq 0$ for some $a \in U$.  I'm asked to show that $f$ is injective on some disk around $a$, and then having done so that it's inverse $g$ on such a disk is given by  $$ g(w) = \frac{1}{2 \pi i}\int_{\gamma} \frac{zf'(z)}{f(z) -w}dz$$  where $\gamma (t) = a + re^{it}$, $t\in [0, 2\pi]$ is the path around the boundary of that disk.  I've managed to show that it's injective in the disk, but I have no idea how to show the inverse is of that form.  The section of the textbook I'm using this is in has covered the argument principle, Rouche's theorem, the open mapping theorem, and the inverse function theorem, but I can't see any obvious way to apply any of them to show $f$ is injective on some disk.  I thought of using the inverse function theorem now that I've established $f$ is injective, but this just tells me that $g'(w) = \frac{1}{f'(g(w))}$, and I can't see how to make that of the desired form.   I'd really appreciate any help you could offer, thanks in advance.
I am asked to prove this theorem:  If $f:U \rightarrow C$ is holomorphic in $U$ and invertible, $P\in U$ and if $D(P,r)$ is a sufficently small disc about P, then   $$f^{-1}(w) = \frac{1}{2\pi i} \oint_{\partial D(P,r)}{\frac{sf'(s)}{f(s)-w}}ds$$     The book says to "imitate the proof of the argument principle" but I am not seeing the connection.
This might be a duplicate of  but that question is 5 years old and unanswered.  When I'm in drive.google.com and I open a G Suite file (either double-click or select &amp; press enter) in Firefox it opens in a new window whereas in Chrome it opens in a new tab.   Why do they do that? Is this a browser-specific quirk or a Google setting (that I haven't found)? Is there a way to control the behavior? I'd like them both to open in tabs, but presumably the same answer would be relevant in the opposite direction too. I have recently started reading and watching Naruto. I was wondering in what aspect does the clones made by different techniques differ? They all have the same potential and capabilities. Is it (if at all) possible in Blender 2.6+?   If not, will future versions of Blender support it?  I see some old posts ( and ) saying that it is not possible. However, I found  referring to operators under  that allowed me to add sensors and controllers from python.  bpy.ops.logic.sensor_add(type='ALWAYS', object=my_obj.name) bpy.ops.logic.controller_add(type='LOGIC_AND', object=my_obj.name)   However I am not able to find any way to link them from python. I tried following (but did not work):  new_controller = my_obj.controllers[-1] new_sensor = my_obj.sensors[-1]   It throws AttributeError saying that object has no attribute 'controllers' :-/ Amazon RDS I/O requests - 1 query = 1 billable I/O? I have an InnoDB database that I'd like to migrate onto Amazon RDS.  My current implementation, hosted on my own server, shows roughly 8M queries per month.   say that  I/O Rate    $0.10 per 1 million requests  Does 1 I/O = 1 query?  ie, will I be billed $80/mo for this amount of usage + the RDS fees? Intuitive understanding of the elements in the stress-energy tensor There is an image in the Wikipedia about the :    I have a rough understanding of the stress tensor: you imagine cutting out a tiny cube from the fluid and form a matrix out of the forces on each side of it: it's not hard to see that the forces that push the faces outwards so the x direction of the force on the x side, the y direction of the force on the y side, etc. is the pressure, while other forces directions are the shear stress (these can be non-zero is some jelly-like substance).  Now the first question: Why is the lower half of the spatial part is the momentum flux, while only the upper one is the shear stress on the image? Is the image wrong? Isn't the spatial part describe the classical stress tensor?   The second question is: what's the intuition behind the temporal part?  So if I have a Minkowski spacetime and cut out a cube of it, then it's quite hard to see why "pressure" on the temporal face means density, also why "shear stresses" on the temporal face translate to momentum density, not speaking of the "temporal force components" the spatial faces...  Also stresses are measured in Pascals, mass density is measured in $kg/m^3$ while momentum density is in the units of $Ns/m^3$. These units doesn't seem to be compatible, but still they are in the same matrix, why?  EDIT: It has been pointed out that momentum flux refer to the entire blue part, it indeed have a lighter frame I didn't notice. Then yet another subquestion: what's the difference between the momentum density and momentum flux? I used python add in to make a tool bar in Arcmap that edits features on a map. It works on a file geodatabase, but not on an ArcSDE version one. I've been getting an FDO error: 0. The solutions that I've found all involve opening a workspace, but I'm getting an error doing that as well. Here is the relevant portion of my code:          outFolderPath = r"U:\tempStuff"         outName = "Stuff.sde"         databasePlatform = "SQL_Server"         instance = "SQL Server"         arcpy.CreateDatabaseConnection_management(outFolderPath, outName,                                                   databasePlatform,                                                   instance)         print "Centroid of buidling:(%r,%r) Side length: %r" % (xCoordinate,                                                                 yCoordinate,                                                                 bob)         print "workspace is %r" % outName #outName is workspace          #Start edit session with an undo/redo stack then start         #edit operation         print "Starting editing"         edit = arcpy.da.Editor(outName) #outName is workspace Line 86         edit.startEditing(True, True)         edit.startOperation()          #stuff happens          cursor = arcpy.da.InsertCursor( fc, ["SHAPE@"])         cursor.insertRow([stuff])             #stop the edit operation and then save changes         edit.stopOperation()         edit.stopEditing(True)          del cursor         print "Building should be drawn"   This is the exact error I've been getting:  Traceback (most recent call last):   File "C:\Users\153289\AppData\Local\ESRI\Desktop10.3\AssemblyCache\{0BEF4E61-6332-4BCA-A908-959CA3C0B4E7}\leetScripts_addin.py", line 86, in onClick     edit = arcpy.da.Editor(outName)#outName is workspace RuntimeError: cannot open workspace   What's am I doing wrong in my code? On the one side, the USA is just one country. Logic says it should be, then, singular, just like the United Kingdom is. Example:     The USA owns this domain.   On the other side, if I however expand "the USA" to "the United States of America", I'd tend towards using plural &mdash; the noun the verb agrees with, "States," is definitely plural. Example:     The United States of America own this domain. → The USA own this domain.   What form should I prefer? I have a custom search page for use in Customer Portal.  VF Page:    &lt;form name="frmSearch"&gt;      &lt;input type="text" name="keyword" id="keyword" value="{!portalSearchModel.searchTerm}" onkeypress="return noenter(event);" style="width:25%; height:25px;margin:0; padding: 0px 6px 0px;" placeholder="What are you looking for?" /&gt;      &lt;input type="button" id="btnSearch" name="btnSearch" value="Search" onclick="searchTerm()" class="go-we3" /&gt;   &lt;/form&gt;   &lt;script type='text/javascript'&gt;       function searchTerm(){         var searchTerm = document.getElementById("keyword").value;          var url="/apex/PortalSearch?s="+searchTerm;          window.location = url;          return false;     }          &lt;/script&gt;   Is there a way to call this apex method in Controller on button click along with the JavaScript?      public void logSearchTerm(){         portalSearchModel.searches();         Search_Log__c sl = new Search_Log__c();          sl.search_term__c = portalSearchModel.searchTerm;         sl.user__c = UserInfo.getUserId();              insert sl;  } Grub menu at boot time... "holding shift" not working I read here at  that holding Shift during boot can bring up the GRUB menu. However I try holding one or the other Shift keys, tapping, tapping then holding, nothing works. Am I missing something? How does one access the GRUB menu during boot time?   No dual boot here, just standard Lubuntu 14.04 install. I'm trying to install CentOS 7 on a HP DL120 gen 9 server with a b140i controller. I have 4 1TB HDD setup in a raid 1+0 and the CentOS installer doesn't seem to recognize the logical drive created from the storage utility.  From my understanding i need some drivers for CentOS 7 to recognize the B140i controller. Also, the driver seems to be closed source by HP.  What is my approach here? Is there any other free linux OS that i can use on this server and that would recognize the raid controller?  Thank you. How to get the last date of a particular month with JodaTime? Two Circles and Tangents from Their Centers Problem Let $\Gamma_1$ and $\Gamma_2$ be two non overlapping circles with centers $O_1$ and $O_2$ respectively. From $O_1$, draw the two tangents to $\Gamma_2$ and let them intersect $\Gamma_1$ at points $A$ and $B$. Similarly, from $O_2$, draw the two tangents to $\Gamma_1$ and let them intersect $\Gamma_2$ at points $C$ and $D$.  Prove that $AB=CD$.  I've done some extensive angle chasing on this but have been unable to make any real progress. Can't decide whether $ABCD$ is supposed to be rectangle (as in my diagrams), a parallelogram or even a trapezium. There is a homothety taking one circle to the other but as far as I can see this doesn't help as we don't have a clearly defined center to this.  Any help/hints would be greatly appreciated. How to recover a removed file under Linux? By accident, I used rm on a file I didn't want to delete. Is there a way that I can get it back under Linux? 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I've got some lenses of brand X, but I've just got a new camera of brand Y. How can I tell if it's possible to use my lenses on my camera? I accept that I may lose a lot of automatic functionality like autofocus, aperture control and metering when using an adapter to connect the two. Can someone please verify my proof?     Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?   No. Let $X = \mathbb{R}$.  Clearly, $\{x\} \in \tau_\infty$ for all $x \in X$.  Set $\displaystyle{B = \cup_{x \neq 0} \{x\}}$. Clearly, $X - B = \{0\}$, but $\{0\} \notin \tau_\infty$. Since $\tau_\infty$ is not closed under arbitrary unions, it does not define a topology on $X$. How to integrate mv command after find command? I am searching for files which name which contain AAA within their path using following command:  find path_A -name "*AAA*"   Given the output showed by the above command, I want to move those files into another path, say path_B. 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