Complex Definition and 1000 Threads

Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...

I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...

Does it make sense to write ##r^c## where ## r\in R/e ## and ##c\in C## ?

I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0## I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0## I...

so I have already done most the work in finding the following things M1A1=M1g-T1, M2A2=M2g-T2 T2=2T1 M2A2=M2g-T2 and then through further working out(use of the invariance of length of L1 and L2 the strings attached to M1 and M2 then finding A1=-2A2) A1=2g(2M1-M2)/(4M1+M2) and...

Mentor note: Edited to fix LaTeX problems ##f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy## ## f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy) ## ## \Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}##

Can you help me with this two questions I am really struggle on how to do it Please help me Thank you in advance

Given (a,b) is the coordinate just like (x,y). Find equation Zo and coordinate (a,b) ?Please help me Thank you in advance.

1.4.1 Miliani HS Find all complex numbers x which satisfy the given condition $\begin{array}{rl} 1+x&=\sqrt{10+2x} \\ (1+x)^2&=10+2x\\ 1+2x+x^2&=10+2x\\ x^2-9&=0\\ (x-3)(x+3)&=0 \end{array}$ ok looks these are not complex numbers unless we go back the the...

I have an extremely fast (f/1.05) night vision optic that I scavenged from a old night vision unit which is faster than current production lenses (f/1.23). However due to the design of newer night vision tubes, the lens will not focus to infinity on the night vision tube. This is because the old...

Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$

I assumed the angular velocity of the center of mass of the two discs about z axis to be w1 note that angular velocity of center of mass of both discs and center of anyone disc about z axis is same, you can verify that if you want, me after verifying it will use it to decrease the length of the...

The true answer is : The voltage between P and Q is 1.5 V. I got stuck finding the total resistance. My question is: Is B parallel to both A and C? Is C parallel to D ? I tried many ways to find the total R but failed! My first attempt: I say the system ABD is in serie with C...

I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)? Thanks

I read this in the wiki article about Wick rotation: Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no...

Hey guys...Im in 11th currently... actually meeting new people on internet used to inspire me but now it makes me all jealous...i am suffering from this complex from 5 months...like there are people of class 9th-10th who are doing UG courses and am such a slow learner i am just self studying...

To find √(-2)√(-3). Method 1. √(-2)√(-3) = √( (-2)(-3) ) = √(6). Method 2. √(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6. Why don't the two methods give the same answer? Thanks for any help.

What are the practical purposes of studying complex systems found in nature? And applying statistical methods to them etc.

I am looking for books that have sections or even chapters devoted to complex random variables, or random variables that can take on complex values (NOT probabilities that are valued in the complex range, in this regard). On the other hand, if someone does know any books that contain material on...

A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity ω in a circular path of radius R. A smooth horizontal groove AB of length L(<<R) is made on the surface of the table. The groove makes an angle θ with the radius OA of the circle in which the...

Hello, I have this (I am solving scholarship exams)math problem and I don't quite know what to do with it , Could You please help? The exercise is about complex numbers and it says: Calculate in the algebraic form(a+bi) I thought on applying substitution since -1=i^2 and z is the real part but...

Dear Everyone, This post is not a homework assignment... I want to use the quartic formula. In one step is to solve the resolvent cubic. I know that there is 3 real solutions this particular resolvent cubic. I want to know how Bombelli got his answers before the discovery of the trigonometric...

I want to study the coherence transfer of the excitation in the FMO complex, so I have to solve the Lindblad master equation. Can I treat my system as a two level system?

So I've been absolutely stumped on this one. I've tried about a million different things but none of them have seemed right. A combination of the odd layout of the circuit, and a combined voltage/current source are making things really complicated for me. I'm somewhat sure the RTh is 15kΩ due to...

Is Non-equilibrium statistical physics and complex systems a good area of study to go into? Is it a well respected field? Thank you

I’m coming at this question with a physics application in mind so apologies if my language is a bit sloppy in places but I think the answer to my question is grounded in math so I’ll post it here. Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z =...

A Construction is stiffened by a bottemplate with welded ribs. A flange welded to the inletpipe is bolted at the botomplate. I want to simulate the current situation. Therfore I need a handcalculation of the stresses in the ribs of the constuction. The forces and moments are working in the worst...

Hey! :giggle: Question 1: If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get...

Hi PF community, I'm reading about complex numbers and i have some questions about the argument of a complex number that i can't solve with Google or reading again the same page. Well, my first doubt is about , i can't understand where come this and why there is some random integer, i...

Assume a transformer as above, with 230V L-N, and I want to work out the L-L voltage. A phasor diagram will show me that the voltages are 120° out of phase. (230∠0°) + (230∠120°) = (230cos0 + j230sin0) + (230cos120 + j230sin120) = 230 + (-115 + j199.2) 115 + j199.2 = 230∠60 What I’m looking...

So the original question is from Control Theory, and the topic is the inverse z-transform. This is a part from the solution I just can't understand. The reason it has to be in this form (##z^{-1}##) is because that's the form used in the z-transform table. The question essentially is, how do you...

At first I tried solving the problemteh following way: Due to symmetry let the rods connected to the green rod have tension forces in magnitde T1 => mg = 2T1cos(a), where a is half the angle formed by the two rods. From tere I got an expression from the longer rods in the force projected by them...

I have to study the solutions of the following system of three equations and three unknowns upon variation of parameters k and h. ix1+kx2-x3 = 1+i (k+i)x1+(1-i)x2-(ik-1)x3 = h kx1+(4+2i)x2-(k-3-3i)x3 = 1-i Obviously i is the imaginary unit. And as stated k and h are the parameters . I can't...

Hello, I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane. This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...

It is a rather simple question: In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$ $$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...

The impedance Z = R -j/wC + ##\frac{1}{\frac{1}{R} - \frac{\omega C}{j}}## But,1/wC=R So, solving this, I find: Z= 3R/2(1-j) |Z| =##\frac{3R}{\sqrt 2}## I =##\frac{V_i \sqrt 2} {3R}## Vi - IR-IXc =Vo Solving this, ##Vo = V_i -\frac {V_i \sqrt 2}{3} - \frac{V_i \sqrt 2}{3R} \frac{-j}{wC}##...

This is a discussion on MathOverflow where a conjecture is discussed that the curve of ##\zeta(0.5+it)## is "dense" on the complex plane. https://mathoverflow.net/questions/73098/negative-values-of-riemann-zeta-function-on-the-critical-line From a couple of sources, e.g...

Let us suppose I have a number ##x## such that ##x<0##. If I want to write the roots of the ##x^{1/n}##. How can we write the roots of this number. I thought we can write $$|x|^{1/n}e^{i\pi\theta}$$ for ##\theta = \frac{2l + 1}{n}## and ##l = 0,1,2## etc. Is this correct ? Similary If I...

Suppose I have a function $$f(x) = \lim_{\eta \rightarrow 0} \int_{-\infty}^{\infty} d \zeta \frac {g(\zeta)}{x - \zeta + i \eta}$$ and suppose ##g(\zeta)## is a continuous (maybe even differentiable) function. Can ##f(x)## have complex poles of the form ##a + ib## with ##b## not an...

This is a quote from "Calculus", by Robert A. Adams. It's a translation from spanish: "Roots of square numbers If ##a## is a positive real number, there exist two different real numbers whose square is ##a##. They are ##\sqrt{a}\;## (the positive square root of ##a##) ##-\sqrt{a}\;## (the...

This author seems to say so: https://blogs.scientificamerican.com/roots-of-unity/one-weird-trick-to-make-calculus-more-beautiful/

Can anyone suggest a good lecture series on Complex Analysis on YouTube? I have already searched on YouTube myself, and there are a few. But I wanted to know if any of you would recommend some particular lecture series which you consider to be good.

Having more difficulty understanding the concept, thus I am not showing values. What is causing me confusion is the line in the middle. The first aR and bR are obviously in parallel, but the second aR and bR confuse me. I tried calculating the equivalent resistance from the first aR and bR and...

##\dfrac{1}{1+i}=\dfrac{1-i}{1-(-1)}=\dfrac{1}{2}-\dfrac{1}{2}i##. But the argument of ##\dfrac{1}{1+i}##? I mean, why is that of ##1+i##? Why ##1+i\Rightarrow tg(\alpha)=\dfrac{1}{1}=1##? Greetings!

Hi PF, this is just for fun...Or not; I don't know. In 1777 Euler set up the notation ##i## to identify any roots of ##x^2-1##, which are indistinguishable, and verified ##i^2=-1##. This way, the set of real numbers grew larger, to a bigger set called complex numbers. This is a translation made...

Below is the problem and the correct answer for this algebra problem is 7√2. But I cannot get to the correct answer.

since ##|z|≤3## →##z=0+0i##, therefore we shall have centre##(0,0)## and radius ##3##, find my sketch below,

Summary:: equation system of complex mechanism need to be done. Lagrange Formalism could be helpfull to do it but small errors could all destroy .. Hi everyone, I tried to find the equation system of the joined Mechanism (4 DoF). I think there is a mistake but I am not sure where is it . I...

Hi, I came across some interesting frames recently. Here they are: I wonder if all these frames can be solved analytically. If yes then how to do it ? Examples a) and d) are planar frames with diagonal members while b) and c) are spatial frames: b is subjected to uniformly distributed load...

I don't know how to start with the factorization. $$\frac{(-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}{(-1)^{2/9}- (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}$$ Any hints would be nice. Thank you.