How can I prove this? I don't have any idea about its proof.  Suppose $f:\mathbb{D}\rightarrow \mathbb{D}$ be holomorphic function and $f(z_0) = 0$. Prove $|z_0| \geq |f(0)|$.
Function $f(z)$ is holomorphic at least within the unit disk.   If for all arguments $|z|&lt;1$ we have $|f(z)| \le 1$, and there exists a zero of order $n$ at $z=z_0$, then show that $|f(0)| \le |z_0|^n$.  How should I tackle this problem? Can it be generalized if for $|z|&lt;R$ we have $|f(z)| \le M$ ?    So far I only understand that there exists a holomorphic $g(z)$ with $g(z_0) \ne 0$ such that $f(z) = (z-z_0)^n g(z)$,then $|f(z)| = |z-z_0|^n |g(z)| \le 1$ and moreover $|f(0)| = |z_0|^n |g(0)|$, but that doesn't seem useful since apparently I know too little about $g(z)$. In particular why should $|g(0)| \le 1$?
declare property as object? When is it acceptable to call GC.Collect? The general advise is that you should not call GC.Collect from your code, but what are the exceptions to this rule?  I can only think of a few very specific cases where it may make sense to force a garbage collection.   One example that springs to mind is a service, that wakes up at intervals, performs some task, and then sleeps for a long time. In this case, it may be a good idea to force a collect to prevent the soon-to-be-idle process from holding on to more memory than needed.   Are there any other cases where it is acceptable to call GC.Collect? Why can't a class extend its own nested class in C#? For example:  public class A : A.B {     public class B { } }   Which generates this error from the compiler:     Circular base class dependency   involving 'A' and 'A.B'   I always figured a nested class behaved just like a regular class except with special rules concerning accessing the outer class's private members, but I guess there's some implicit inheritance occurring between the two classes? Probability Number A greater Number B Given $a \in \{1,2,...,250\}$ and $b\in \{0,1,...,1000\}$  What is the probability that $a &gt; b$?  I know there already is  question with a very similar question, however it's over an interval of $\mathbb{R}$ so I hope there is another way opposed to the Integral used in the formal solution.  How does one go about calculating said probability? Spider Identification Illinois I noticed a really huge, beautiful web on my porch last week, and tonight, I discovered its resident. I'm thinking it's an Araneus diadematus (Cross Orbweaver), but I was hoping for some confirmation.   Can anyone confirm this or offer a better suggestion? How to access the Keyboard Layout Options in 13.10+ Smallest Width for Galaxy S and Galaxy S2 Change email login to another id Getting text from the Resource file in asp.net 4 failed to fetch security undermined error [IP: 91.189.91.38 80] in ubuntu 20.04 I had posted a  last week about customizing a ToolValidator class and got some very good answers.  In working with the proposed solutions, I have created a custom module that performs queries on a db, and will be called by both the ToolValidator class (to provide values for the drop-down lists) and later in the geoprocessing script (to get other parameters based on items selected in drop-down lists).  However, I can't seem to actually call the custom module in the ToolValidator class.  I have been trying to append to path with no luck.  When I try to apply these changes to the script, I get a runtime error:  [Errno 9] Bad file descriptor.  If I comment out import line, no errors.  sys.path.append('my_custom_module_directory') import my_custom_module   Many of you may be asking why don't I just implement a custom tool with ArcObjects.  The reason is that my end users don't have the privleges necessary to register ANY dlls on their computer.  UPDATE:  This was happening to me in ArcGIS 10.  Interestingly enough, I was initially appending to the path inside of the initialiazeParameters function of the ToolValidator class.  If I do the append outside (i.e., on top of) of the ToolValidator class, everything works as expected.  sys.path.append('C:/Working/SomeFolder') import somescript --------&gt;THIS WORKS  class ToolValidator:   """Class for validating a tool's parameter values and controlling   the behavior of the tool's dialog."""    def __init__(self):     """Setup arcpy and the list of tool parameters."""     import arcpy     sys.path.append('C:/Working/SomeFolder')     import somescript -------&gt; THIS DOESNT WORK     self.params = arcpy.GetParameterInfo()   UPDATE 2:  I think I found the true cause of my problem.  In the code snippets in this post, I have been appending what appear to be real paths (ex C:/Working/SomeFolder) to the sys.path.  In my actual ToolValidator class, I was constructing a relative path using os.path.dirname(__file__) +"\my_special_folder...".  I was anticipating that os.path.dirname(__file__) would return the path of the toolbox, since that contains the ToolValidator class.  I have come to find this is not the case.  As far as I can tell, the ToolValidator class is never actually written to a .py file, and I speculate the the this code is passed to the python interpreter in memory, so __file__ is useless, or some temp script is persisted and then execfile(path_to_script) is called, again rendering __file__ useless.  I'm sure there are other reasons I am missing as well.    Long story short, if I use a hardcoded path, the sys.append works anywhere, relative paths don't work so well in the ToolValidator class. Finding a tricky limit I have a homework in real analysis and I'm very confused about it. I would be very thankful, if you could give me any ideas or solutions how to get this task done. The task is as follows:     Find the limit:   $$\lim_{x\to 0} \frac{1}{x} \int_{0}^{x} (1+ \sin(t))^{1/t} dt.$$   A hint that has been given is to use L'Hospital's Rule. Moreover, it must be substantiated why we can use L'Hospital' Rule in this situation. It is also important to mention that $(1 + \sin(t))$ has a removable point of discontinuity at $0$ and therefore the integral can be interpreted as a Riemann integral. "At least one" - singular or plural subject? Plugin or brush for transforming a photo to a fractal in gimp? Folks,  I've been looking all over for a plugin or brush that will help me take a photo and reduce it to that 'electric look' (or fractal?) that i'm seeing in various places. I'd vastly prefer one for Gimp.  Does anyone know how these are being done? I'm seening references to fractalius for photoshop, but what about gimp?  Examples: Auto delete spam email from a domain in Gmail Should I use the first or last bits from a SHA-256 hash? Determine the number of ways to colour a 1 by n grid using the colours red and white if an even number of squares are to be coloured red. I'm working on a small game and i'd like to know how can i write/read a user settings file in ubuntu touch? "invalid use of incomplete type" error with partial template specialization Let $E$ be an empty set. Then, $\sup(E) = -\infty$ and $\inf(E)=+\infty$. I thought it is only meaningful to talk about $\inf(E)$ and $\sup(E)$ if $E$ is non-empty and bounded?   Thank you.