Compute the maximum of $|f(z)|$ when $|z| \leq 1$ and $f(z)=\sin (z)$  So since $f$ is holomorphic on  $|z| \leq 1$, we know we'll find the max of $|f(z)|$ on $|z|=1$.  So:  $|f(z)|=|\sin(z)|=|\frac{e^{iz}-e^{-iz}}{2i}|=|\frac{e^{iz}}{2i}-\frac{e^{-iz}}{2i}|\leq |\frac{e^{iz}}{2i}|+|\frac{e^{-iz}}{2i}|=\frac{|e^{iz}|}{2}+\frac{|e^{-iz}|}{2}=\frac{e^{|iz|}}{2}+\frac{e^{|-iz|}}{2}=e$   I'm not so certain about this solution. Is there another way? Is using the triangle inequality safe when looking for a maximum?
Maximum of $|\sin(z)|$ as $\{z: |z| \leq 1 \} $  So according to the Maximum Principle, the maximum is when $|z|=1$. I tried using the fact that $\sin z= \dfrac {e^{iz}-e^{-iz}}{2i}$, but didn't know how to continue from there. Setting $z=e^{it}$ doesn't help much.  I saw this question was answered before, however the answer uses hyperbolic functions ($\sinh, \cosh$) which I haven't actually studied.  Thanks in advance for any assistance!
I was setting up up android-studio referring to . While installing oracle-java7-installer, this error is displayed:  sudo apt-get install oracle-java7-installer [sudo] password for cyan:  Reading package lists... Done Building dependency tree        Reading state information... Done oracle-java7-installer is already the newest version (7u80+7u60arm-0~webupd8~1). The following package was automatically installed and is no longer required:   snap-confine Use 'sudo apt autoremove' to remove it. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] y Setting up oracle-java7-installer (7u80+7u60arm-0~webupd8~1) ... Downloading Oracle Java 7... --2017-11-27 08:40:47--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving download.oracle.com (download.oracle.com)... 223.223.158.138, 223.223.158.136 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz [following] --2017-11-27 08:40:47--  https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving edelivery.oracle.com (edelivery.oracle.com)... 23.58.58.93, 2600:140f:7:18e::2d3e, 2600:140f:7:18d::2d3e Connecting to edelivery.oracle.com (edelivery.oracle.com)|23.58.58.93|:443... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 [following] --2017-11-27 08:40:48--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2017-11-27 08:40:50 ERROR 404: Not Found.  download failed Oracle JDK 7 is NOT installed. dpkg: error processing package oracle-java7-installer (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  oracle-java7-installer E: Sub-process /usr/bin/dpkg returned an error code (1)   How do I fix this error? I did a fresh installation of Ubuntu 16.04. How do I list all available shell builtin commands? We know bash supports number of  buitin commands  like:  $ type type cd help command type is a shell builtin cd is a shell builtin help is a shell builtin command is a shell builtin   I want to get list of all available shell builtin commands. How do I do that by command-line? I am new and struggling with java and I need assistance with code. Value keeps resetting to default Show that any prime ideal from such a ring is maximal. Let R be a commutative ring with an identity such that for all $r\in$ R, there exists some $n&gt;1$ such that $r^n = r$. Show that any prime ideal is  maximal. (Atiyah and MacDonald, Introduction to Commutative Algebra, Chapter 1, Exercise 7.)    Any hints? Can I visit other Schengen countries on a long-term Schengen visa or residency permit? Is it grammatical to omit the subject from "I did my work"? Where can I get the original Unix? Where can I get the original Unix (from the year 1969)? I would like to look at the source code of the original Unix. Free GIS workshops, tutorials, and applied learning material This Q&amp;A lists free as in $0 workshops in GIS and related fields that have their material available to view or download online. The list is not limited to FOSS GIS, as GISers usually use a combination of open and closed source tools to accomplish their tasks. Some of these tools are easier than others and some are better documented. This list will be a great benefit to the community especially for new comers to the field. General GIS   from   from   from   from   Open-source GIS   from   from   from   from   from   ESRI Products   from   from   Transportation These links include either theoretical or applied transportation knowledge in transportation planning          Disaster Management    Databases   from   Geo Statistics    from Recently, there was some news that said that the researchers at NASA have come across some impossible kind of space engine which does not require any fuel. I have read at a few places like ,  and . They all say that this engine seems to violate the laws of physics including conservation of momentum.  The original paper is .  Is the statement that it violates conservation of momentum true? Why didn't Thor lose his power? In Thor: Ragnarok, Valkyrie told Thor this about Hela:  Her power comes from Asgard the same as yours  Thor's plan was to unleash Surtur who destroys Asgard, which weakens Hela allowing Surtur to kill her. But why didn't that weakened Thor as well? We saw him later in Avengers: Infinity War very powerful. Why don't Jedi (or Sith) use the Force to snap their foes' necks? I understand that the Jedi use the Force for self defence, but still, couldn't they just use the Force to snap their foes' necks and be done with it? Like when Jaina Solo was hunting down Darth Caedus. Or even when Caedus was duelling Grand Master Skywalker. If he just used the Force to snap Skywalker's neck (like he did to poor Lieutenant Tebut), it will all be over very fast.  Is there a canon reason why Jedi and Sith don't use their powers to directly harm their opponents? How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? Impact of denied entry to Croatia on trying to enter a Schengen country with a valid visa let $n\gt1$  be a fixed natural number, $S:=${$A $: $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$   $d(A,B)$:=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and $GL(n,\mathbb R)$ is set of all $n\times n$ non singular matrix how to prove that $GL(n,\mathbb R)$ is  not connected and open subset of $S$ ?  thanks in advance What are those zigzagged folded paper things you see in supernatural anime? In a lot of supernatural anime, like Shin Sekai Yori, or Tokyo Ravens there often appear these large ropes with folded paper things hanging from them:    As far as I can tell, they serve to seal or mark places that are sacred or in some way supernatural, like shinboku, or holy trees:    What are these folded paper things, and what purpose do they really serve? How do I convert a Word file's references to .bib format How do I convert a Word file's references to .bib format? Is any tool available  to convert it to .bib format? Install brother dcp195c, receiving data but nothing happens, why? Can someone please help me to install brother dcp195c drivers on Ubuntu v12?  Using the default driver, the printer says that it is receiving data but nothing happens. Let $G$ pe a $p$ group. I have to show that         the number of nonnormal subgroups is divisible by $p$       the number of subgroups differs from the number of normal subgroups by a power of $p$.   Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don't know how to apply them - if those are the theorems I need. (Does this theorem also hold for infinite groups $G$ ?) Mark my own question as favorite? How do I tell Maven to use the latest version of a dependency?