I am asking myself why for a complex number $z$ and real numbers $a \le b$ holds $$\bigg\lvert \int_a^b f(z) dz \ \bigg\rvert \le \int_a^b \bigg\lvert f(z) \bigg\rvert dz$$  I know that this holds for real numbers, but I do not see how we can transfer this to the real and imaginary part of $f$, but I do not see how. Could you help me?
I've been given a proof that shows the following:  If $f:[a,b]\to \mathbb C$ is a continuous function and $f(t)=u(t)+iv(t)$ then $$\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$$ The proof begins by letting $\theta$ be the principle argument of the complex number $\int_a^b f(t)dt$ and there is one step in the proof I don't understand; can anyone explain to me why $$\int_a^b e^{-i\theta}f(t) dt=\Re\bigg(\int_a^be^{-i\theta}f(t)dt\bigg)$$  Would this not imply that the imaginary part of the left hand side is equal to $0$? I'm not sure why the LHS and RHS would be equal here.
Was Grand Moff Tarkin under Darth Vader's control? In A New Hope, we see Grand Moff Tarkin ordering Darth Vader around. Also, it was . It's uncertain whether he knew .  But considering that Darth Vader was a Sith Lord, and a Jedi before that, are we entirely sure he wasn't secretly controlling Tarkin through a Jedi Sith mind trick? How to add a matrix to a LaTeX document How do we add a matrix to a LaTeX document? Performance of try-catch in php What kind of performance implications are there to consider when using try-catch statements in php 5?   I've read some old and seemingly conflicting information on this subject on the web before. A lot of the framework I currently have to work with was created on php 4 and lacks many of the niceties of php 5. So, I don't have much experience myself in using try-catchs with php. Expectation inequality for a set and its subset. Given that a function $g(x)$ is a monotone increasing function. Its domain is an interval [c,d]. There are two sets $a_j \in A, j= 1,...,J$ and $b_k \in B,k=1,...,K$  on this interval, which satisfy $c_z \in C = B$\A and $c_z \geq a_j$ for all $a_j \in A$. For example, the interval $[c,d]=[0,8]$,$A=\{0,1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6,7\}$. In this case, does the following inequality hold? If yes, could you give me some hints to prove that?  $$\frac{1}{K} \sum_{k=1}^{K}g(b_k) \geq \frac{1}{J} \sum_{j=1}^{J}g(a_j)$$ I could not install my modem driver so I have to update and install applications offline. For this I want to use keryx but to install keryx, Python and wxPython have to be installed first. So how do I install Python and wxPython offiline on Ubuntu 13.04?  Note : I don't have internet connection in Ubuntu. So I'll have to download the required file from my Windows 7. I flagged a post and (choose one)   my flag was marked as helpful, yet nothing was done my flag was marked as declined, yet the post was deleted/edited/closed   Why? What if I offer a bounty and don't get any good answers? I have started a bounty:   It expires in 3 days. I don't have an appropriate answer to accept, and the only answer which it has gotten has no votes.  So am I going to simply lose 100 reputation without getting a satisfactory answer?  Or should I accept the lone inappropriate answer so that I will not lose my right to accept answers to any questions which I might ask in the future, which is what the bounty rules claim? What is wrong in this proof: That $\mathbb{R}$ has measure zero Consider $\mathbb{Q}$ which is countable, we may enumerate $\mathbb{Q}=\{q_1, q_2, \dots\}$. For each rational number $q_k$, cover it by an open interval $I_k$ centered at $q_k$ with radius $\epsilon/2^k$.  The total length of the intervals is a geometric progression that sums up to $\epsilon$.  Each real number is arbitrarily close to a rational number since $\mathbb{Q}$ is dense in $\mathbb{R}$. Thus, each real number is in one of the open intervals.  Thus the entire real line is covered by the union of the $I_k$, thus $\mathbb{R}$ is a null set with measure zero.  Clearly there is something wrong in the above proof, however I am not sure where is it?  Thanks for any help. Can we have a notification when the answer to any of my question got edited? If a user asked a question and got an answer which was not helpful.  After sometime the answerer edits the answer with some good content. How the OP will get to know that?  Is it necessary to add comment check my edited answer?  We can say that the OP should refresh the page to find the helpful answer.  Consider if the OP finds the answer as helpful and accepted that. After sometime if the answerer/anyother user edit the answer to change the content which may not solve the issue.   How will it be notified to OP?  currently we are notifying the OP when there is any comment to the queation or any new answers or any comments with the name of OP with @ to any answer.  This will lead the future visitors to confuse and reduce the quality of post.  Can we have a notification to the original poster when the answer to any of his/her question got edited? What happens if you use Polyjuice potion after saving hair or nails? Suppose you cut a five year old's hair, save it for 10 years, and then use it for a Polyjuice transformation. What age do you turn into: The toddler or the teenager? Logistic regression in R resulted in perfect separation (Hauck-Donner phenomenon). Now what? I'm trying to predict a binary outcome using 50 continuous explanatory variables (the range of most of the variables is $-\infty$ to $\infty$). My data set has almost 24,000 rows. When I run glm in R, I get:  Warning messages:   1: glm.fit: algorithm did not converge   2: glm.fit: fitted probabilities numerically 0 or 1 occurred    I've read the other responses that suggest perfect separation might be occurring, but I'm confident that isn't the case in my data (though quasi-complete separation could exist; how can I test to see if that's the case?). If I remove some variables, the "did not converge" error might go away. But that's not always what happens.  I tried using the same variables in a bayesglm function and got the same errors.  What steps would you take to figure out exactly what's going on here? How do you figure out which variables are causing the problems? Arrangements of a,a,a,b,b,b,c,c,c in which no three consecutive letters are the same Q: How many arrangements of a,a,a,b,b,b,c,c,c are there such that  $\hspace{5mm}$ (i). no three consecutive letters are the same?  $\hspace{5mm}$ (ii). no two consecutive letters are the same?  A:(i). 1314. ${\hspace{5mm}}$ (ii). 174.  I thought of using the General Principle of Inclusion and Exclusion along with letting $p_i$ denote a property that.. , and doing so, I will evaluate $E(0)$, which gives us the number of arrangements without any of the properties.   How do I go about doing that? I am trying to use the method stated above in solving this question, but I am unable to generalize the properties $p_i$. Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? Proof that every number ≥ $8$ can be represented by a sum of fives and threes. $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$ I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to  $\mathbb  Z_5$.   The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \}  \to  \mathbb N$ defined as $N(a+bi)=a^2+b^2$, then I can express any element $z$ in $\mathbb Z[i]$ as $z=(1+2i)q+r$ with $N(r)=1,2,3,4$ or $r=0$.   If I define $f:\mathbb Z[i] \to \mathbb  Z_5$ as $f(x=(1+2i)q_x+r_x)=\overline{r_x}$, then it is clear that $f(x)=0$ if and only if $x \in  \langle 1+2i \rangle$. The problem with this function is that it doesn't satisfy $f(x+y)=f(x)+f(y)$ and it is not surjective for if $r_x=f(x)=3 (5)$, then if $r_x=a+bi$, we have $a^2+b^2=3$, which is absurd.  I don't know what else to do, any suggestions would be appreciated. Is there any way to see the timetable for a  bus route?  On their website I can see the routes map, but I'd me more interested in seeing where the bus is going to stop for a given route.  E.g. if a bus goes from Rotterdam to Amsterdam, I'd like to see each stop. Is there a word for somebody who cares for his/her child too much? I have just finished watching the 3rd season of .  I am wonder whether there is a word for somebody who cares for his/her son/daughter/child too much, like Dr. Cal Lightman? What new capabilities do user-defined literals add to C++? According to preliminary estimates, the project should be completed by 2020 at a cost of about RUB 40 bn.    or      According to preliminary estimates, the project should be completed by 2020 at the cost of about RUB 40 bn.    Are both options okay, or is the first better?   We are introducing the sum to the reader, so logically it should be the first option. But there's a lot of google-hits for "at the cost of" too. I have a current view set up that displays a feed of stories and is sorted in descending order. I now need to have the first 4 stories display as a different style then the remaining ones. I have pagination set to 10. I only want the first 4 to show differently on page 1, and then the current style will show on the remaining pages. Is there a way to do this?