Suppose f(z)=u+iv is analytic. Under what circumstances will g(z)=u-iv be analytic? Hint: Consider the functions f(z)+g(z) and f(z)-g(z).
$f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? is it true if $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? I am not able to  find out what property of holomorphic map I need to apply. please help.Thank you.  $f(z)=u(x,y)+iv(x,y)$, $\bar{f}(z)=u(x,y)-iv(x,y)$
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$ I'm going over old exam problems and I got stuck on this one. Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be the non-zero zeros of $\mathbb{f}$ in $\mathbb{D}$ counted according to multiplicity. Show that  $$ \sum_{n=1}^\infty \left( 1 - \left|a_n \right|\right)\lt\infty $$  I can understand that $\left|a_n \right|$ goes to $1$ since zeros are isolated, but it doesn't help showing the series is convergent. Any help will be appreciated! If $U$ is unitary operator then spectrum $\sigma(U)$ is contained inside the unit circle In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$.   It is important for me to know how I am doing, and I didn't come by a clear explanation so it would be appreciated to have your correcting and guiding.  Attempt: $U^{*}U=UU^{*}=I$ and therefore $U^{-1}=U$. I have $$ \| U x \|^2 = \langle U x, U x \rangle = \langle U^* U x, x \rangle = \langle x, x \rangle = \|x\|^2 $$ and the same goes for $U^*$ and therefore $\| U \| = \| U^{-1} \| = 1$.   Looking at $U-\lambda$ I can take $$ -\lambda U(U^{-1}-{1\over \lambda}I) =U-\lambda I. $$ Clearly, for $\lambda \ne 0$, $-\lambda U$ is defined and is a linear operator. $(U^{-1}-{1\over \lambda}I)$ is invertible if $|\lambda|\ge 1$ and therefore $\sigma (A^{-1})=\sigma (A)\subseteq \{|z|\le1\}$. But from the RHS, $U-\lambda I$ is invertible if $|\lambda|\le 1$ and therefore $\sigma(A)\subseteq \Bbb{S}^1$. I have noticed my claims tend to have holes in them many times which I cannot spot. If it is substantially true, how can I make sure it is completely formal? What is the minimum RAM required for Pokemon GO? While poking around on Pokemon Go's , it lists some specs required for Android and iOS devices. I'm particularly interested in Android devices, but I do not see a minimum/recommended RAM listed on their site.        Android 4.4 to Android 6.0.1  (Android N will not supported until the   official Android release)    Preferred resolution of 720x1280 pixels   (Not optimized for tablet)    Strong internet connection (Wi-Fi, 3G, or   4G)    GPS and Location Services    Intel CPUs are not supported      My device is a , which has a 1.5GB RAM and Quad-Core processor. It runs fine for the most part, but a majority of the features heavily lags and are sluggish while I've seen the same features run smooth on other devices.  What is the minimum/recommended RAM for Pokemon Go to run more smoothly? Do numbers exist independently from observers? Cooperative game vs Multiplayer game Can any one tell what is difference between cooperative game and multiplayer game? Is there a way to set transparency of windows in GNOME? I have a live USB made with the standard Ubuntu disk image creator for 16.04 LTS. In the current state, I can boot to this disk, but it still requires the original USB to run properly at all times.  Attempting to pull the drive and use the system generates a large amount of I/O errors, further suggesting that the USB drive is actually needed for things not already loaded into RAM (For example, Firefox worked without the USB, but I used it once before already. apt, on the other hand, did not.)  Is there a way to copy the entire Ubuntu live disk to RAM, so I can remove the drive and free up a USB slot for other things? How do I split a string so I can access item x? Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"? Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. What is the size of an enum in C? I'm creating a set of enum values, but I need each enum value to be 64 bits wide. If I recall correctly, an enum is generally the same size as an int; but I thought I read somewhere that (at least in GCC) the compiler can make the enum any width they need to be to hold their values. So, is it possible to have an enum that is 64 bits wide? What do you call someone that believes he is the only living being? What do you call someone that believes he is actually the only living being and that everyone and everything that exist (besides him self) is a state of his own mind? I am not referring to a R version 3.1.1 (2014-07-10) -- "Sock it to Me" &gt; bl &lt;- c(140, 138, 150, 148, 135) &gt; fu &lt;- c(138, 136, 148, 146, 133) &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE) Error in t.test.default(fu, bl, alternative = "two.sided", paired = TRUE) :  data are essentially constant   Then I change just a single character in my fu dataset:  &gt; fu &lt;- c(138, 136, 148, 146, 132)   and it runs...  &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE)      Paired t-test   What am I missing here? The topological space $X=\omega_1\cup \{\omega_1\}$ is not first countable but $\omega_1$ is. Here $\omega_1$ denotes the set of all countable ordinals and $\{\omega_1\}$ is the first uncountable ordinal. Cardinality of $\omega_1$ is $\aleph_1.$And also $\omega_1$ is called the limit ordinal meaning it is the limit point of the set $\omega_1.$  Here I want to prove two things. $1)$ $\omega_1$ is first countable. and $2)$ $\{\omega_1\}$ is the limit point of $\omega_1$. But since no sequence converges to $\{\omega_1\}$, using this I could say that our space $X$ is not first countable.  $1)$ The proof of this I got from  with just one question: it says we need to prove that such an $S(y)$ exists,and I'm thinking $S(y)=\cap\{u\in \omega_1:u\gt y\}.$ This is correct$?$ But here is a chance that the countable base around $y$ may be finite and not countably infinite.  $2)$ If possible let us assume that $\omega_1$ is not a limit point of the set $\omega_1.$ $U$ be a nbd of $\omega_1$ that does not intersect $\omega_1.$ Then what are the elements of $U$ like? The ones coming before $\omega_1$  cannot be uncountable because $\omega_1$  is the first. Then they must be countable. This is absurd so no such $U$ exists. Thus $\{\omega_1\}$ is a limit point.  Are my proofs correct? Thank you. Margin on child element moves parent element Uniform convergence of derivatives, Tao 14.2.7. What are the antonyms of "append" and "prepend"? I need help running python in ubuntu  I can run python without a problem in my terminal but when I create a .py file I can't get it to run. Given Newton's third law, why are things capable of moving? I found this very amusing comic on the internet the other day:   The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this true? Where could I find a proof? Is it only true for terms with a $9$ or is it true for any number? If it's only true for $9$, why is that? How long can a temporary suspension be? I ran into a user profile that states:  This account is  to cool down. The suspension period ends on Apr 27 '24 at 7:54  At first, it seemed like a somewhat longish suspension period.  Referring to  reveals:  Depending on the severity of the problem behavior — and at the complete discretion of the moderator — your account will be placed in timed suspension for anywhere from 1 to 365 days.  This indicates that temporary suspensions can last upto a year.  Additionally, the  doesn't give any other information about the timed suspension other than the link to the wiki above. I am keen to know whether the example above (wherein the account has been suspended until 2024) is a one-off case or is it a case of documentation being obsolete?