Let $f(z)=\sum\limits_{n=0}^{\infty}{a_nz^n}$ is an univalent analytic function on the open disk $\Delta(0,1)=\{z:|z|&lt;1\}$, such that $G= f(\Delta(0,1))$. If $A$ is the area of the region $G$, prove that:  $$A=\pi\sum\limits_{n=0}^{\infty}{n|a_n|^2}$$  I found this question from the book "Fundamentals of Complex Analysis", written by Liao Liangwen. I have no idea about this. So I want to get some help.  Thanks.
I have the following problem.      Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set $A\subseteq \mathbb C$ and $G=f(D).$ Prove that $$l_2(G)=\pi\sum_{n=1}^\infty n|c_n|^2.$$   After a long struggle I managed to come up with the following formula $$l_2(G)=\iint_D |f\,&#39;(z)|^2dxdy.$$ It looks like the right thing to use because $$f\,&#39;(z)=\sum_{n=1}^\infty nc_nz^{n-1},$$ which is similar to what I have to prove. I understand that the $\pi$ will appear when I integrate something over the angle $\phi$ in polar coordinates. But I don't know how to find the square of the absolute value of the right-hand side to even start integrating...  EDIT: The function is supposed to be  (one-to-one). (I'm not perfectly sure I'm translating the term correctly. The (Polish) word in the statement of the problem was "jednolistna". I have not met it before.)
"An hour" or "a hour" Which indefinite article should precede hour — a or an?        an hour     a hour      Does the usage of an vs a depend on the pronunciation — a history, a hobby, but an hour, an honor? Uncountable well ordered set in $Z^-$ theory. I want to build an uncountable well-ordered set within the theory $Z^\textbf-$. So, I take $A=\omega$ (exists by infinity axiom) and define $$W:=\{(X,R)\in\mathcal{P}(A)\times\mathcal{P}(A\times A):\langle X,R\rangle\ \text{is a well-ordered set}\}$$  With this I consider the set $T:=W/\cong$ (where $\cong$ is isomorphism relation). Note that $W$ and $T$ are sets by power set axiom. So I define for each equivalence class $[x],[y]\in T$ the order  $$ [x]\leq_T[y]\Leftrightarrow \text{type}(x, R_x)\leq \text{type}(y,R_y) $$  Here, $R_x$ means the order of $x$ that make it belongs to $[x]$.   So, $T$ is well ordered by $\leq_T$ and it is uncountable.  Question: I don't pretty sure if I can build $\leq_ T$ without replacement axiom. Another doubt is, Can I take the $R_x$ without AC? How to explain in layman’s terms the significance of the difference of qubits of the D-Wave and IBM QX? So, @AndrewO mentioned recently that he has had 'encounters' with people wondering why D-Wave has a lot more qubits than IBM. Of course, this comparison is faulty, since the IBM and D-Wave's machine may both exploit quantum effects to a certain degree, IBM's machine matches the thing the TCS people call a 'Quantum computer' a bit more than D-Waves's alleged quantum annealer.   How do you explain to a novice why IBM is still reaching important milesstones, even though the D-Wave has a lot more 'qubits'. I understand that an easy answer is, 'well, you're comparing apples and pears', but that is merely relying on your possible authority and simply doesn't explain anything!   How can you explain that those devices are different, how can you dispell the myth that the number of qubits is not the only metric to judge quantum devices? (preferably to a layman, but assuming basic (undergrad?) physics knowledge is ok, if needed) How To check Windows device is rooted/jailbrake or not programatically? Solve the integral $\int_0^\infty x/(x^3+1) dx$ I'm new here!  The problem: integrate from zero to infinity x over the quantity x cubed plus one dx. I checked on wolfram alpha and the answer is that the indefinite integral is this:  $$\int \frac{x}{1+x^3} dx = \frac{1}{6}\left(\log(x^2-x+1)-2 \log(x+1)+2 \sqrt{3} \arctan((2 x-1)/\sqrt{3})\right)+\text{constant}$$  and the definite integral is this:  $$\int_0^\infty \frac{x}{1+x^3} dx = \frac{2 \pi}{3 \sqrt{3}}\approx 1.2092$$  I am trying to figure out all the steps in between. I see that there are logs, which are equivalent to the ln variant that i am more familiar with, which means it was integrating $1/x$ at some point; I also see an inverse tangent in there.  I started with long division to simplify and I got (which could be wrong because I am very tired right now) $x/(x^3+1) = x^2+ 1/(x^3+1)$ which seems to be a step in the right direction. Wolfram Alpha thinks I definitely did that step wrong. The two equations do not evaluate as equal.  Then I cheated and took the wolfram alpha factorization of $(x^3+1) = (x+1)(x^2-x+1)$...I probably should have known that but didn't offhand. Now it is looking like $x^2$ plus the partial fraction decomposition $1/(x^3+1) = A/(x+1)+(Bx+C)/(x^2-x+1)$. Am I heading in the right direction with this? At this point do I just plug in and crunch? Is there a way to target the arrow from an OP bow via command block? I am trying to make an effect trail for a specific weapon in MC, a bow with power 120.  I want to use a command block to target any arrow with the damage it should be doing (63) and summon a trail of splash potions.  I also want to kill any arrow entities that have inGround=True  Is there any way to do this or do I have to come up with something else?  Attempt:  /execute @e[type=Arrow,damage=63,inGround=0] ~ ~ ~ summon ThrownPotion ~ ~ ~ {Potion:{id:potion,Damage:8197}} /kill @e[type=Arrow,inGround=True]   Each unfortunately just affects all arrows.  Is this even possible? Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! How can I upgrade Mac OS without the administrator password? I recently received an old iMac secondhand.  I have no way to reach the original owner. 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Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?   $$(1) \ S=f(X_1,X_2...X_n)$$  $$(2) \ s=f(x_1,x_2...x_n)$$  I haven't been able to get any clarification for this and I've seen the term statistic describe both situations How to check for empty xml files when parsing through multiple xml files Is the op-amp inverting or non-inverting? I am studying the function and characteristic of an op-amp to build a line-following robot with the schematics below. However, in many tutorials, the way to differentiate an inverting and non-inverting op-amp is by looking at the connection to ground via the inverting or non-inverting terminal.  However, in the schematics, the op-amp is considered to have connections to ground, thus in this case, how do I differentiate it then?  I may have understood the whole concept wrongly. Pls help!    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My "network reputation" graph does not accurately reflect my accounts I'm trying to display opportunities associated with an OpportunityLineItem on a Visualforce page on the OpportunityLineItem object   I have a problem in the select I do not know how to retrieve the OpportunityLineItem name in this request I have hard code the OpportunityLineItem name  public List&lt;OpportunityLineItem&gt; getOpp() {     return [SELECT TotalPrice, Opportunity.Name, Opportunity.StageName,                     Opportunity.Account.Name, Opportunity.id, Product2.Name              FROM OpportunityLineItem               WHERE Product2.Name='GenWatt Diesel 10kW'     ]; }