What is Complex: Definition and 1000 Discussions

The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) together form the UCL School of the Built Environment, Engineering and Mathematical and Physical Sciences.

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  1. cajer

    I What Lens to Put In Front of an Complex Lens to Push Focal Plane Back?

    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...
  2. siddjain

    I Prove Complex Inequality: $(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|)>=\sqrt{2}$

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

    Existential dilemma on angular velocity of a complex rigid body

    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...
  4. Pouyan

    Complex circuit and its resistance

    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...
  5. H

    Question about the argument in a Complex Exponential

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

    A Wick's rotation on a complex vector space

    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...
  7. Physicistpropeller

    How Can I Improve My Self-Study Technique for Math?

    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...
  8. G

    B Real numbers and complex numbers

    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.
  9. T

    I What are the biggest problems in the study of complex systems?

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

    Prob/Stats Material on complex random variables and exotic probabilities

    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...
  11. V

    Forces on particle in complex motion relative to ground observer

    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...
  12. Purpleshinyrock

    Scholarship exam exercise about complex numbers - Can't solve

    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...
  13. C

    I Simplifying a nested radical that includes a complex number

    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...
  14. A

    A The exciton dynamics in the FMO complex

    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?
  15. S

    Engineering Can a voltage test source help solve for Rth in a complex Thevenin's circuit?

    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...
  16. AllThatJaz22

    MHB Complex Binomial Coefficient

    Howdy! I have a deck of cards I created called Better Backstories. The Basic Deck is made up of 60 cards and each one has a unique title. 38 of the cards have a chart of 10 suggestions, and the remaining 22 have flavor text that could reasonably include 3 suggestions. So, the total number of...
  17. T

    Physics Non-equilibrium statistical physics and complex systems

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

    I Expansion of a complex function around branch point

    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 =...
  19. H

    How to calculate the maximum stress in a rib of a complex construction

    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...
  20. M

    MHB Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?

    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...
  21. Santiago24

    I Questions about the arg of complex numbers

    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...
  22. G

    Using complex numbers to model 3 phase AC

    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...
  23. WhiteWolf98

    Using Complex Conjugates to Decompose a Fraction

    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...
  24. K

    Tension force of a thread in a complex structure of six masless rods

    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...
  25. I

    I Solving a complex linear system with parameters

    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...
  26. F

    Show that the real part of a certain complex function is harmonic

    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...
  27. Tony Hau

    I The derivative of the complex conjugate of the wave function

    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...
  28. Kaguro

    RC circuit using complex numbers

    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}##...
  29. S

    I Curve of zeta(0.5 + i t) : "Dense" on complex plane?

    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...
  30. Arman777

    I Writing Complex Roots of Negative Numbers

    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...
  31. dRic2

    I Exploring Complex Poles in Functions and Their Consequences

    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...
  32. mcastillo356

    B Principal square root of a complex number, why is it unique?

    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...
  33. S

    B Is complex analysis really much easier than real analysis?

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

    I Any Good Lecture Series on Complex Analysis?

    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.
  35. J

    Understanding Complex Circuit Configurations

    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...
  36. mcastillo356

    What is the argument of 1+i in the complex number 1/(1+i)?

    ##\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!
  37. mcastillo356

    Why Euler spoke of them as "complex" numbers?

    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...
  38. M

    Help me with this Algebra problem please (quotient of complex numbers)

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

    Find the greatest value of argument- complex numbers

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

    Finding the Equation System of a Complex Mechanism: Using Lagrange Formalism

    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...
  41. FEAnalyst

    Complex frames - analytical calculations

    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...
  42. Y

    How to simplify this complex expression?

    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.
  43. R

    I Is Complex Differentiation Defined for Linear Transformations?

    Let z = [a b]^T be in the 2-dimensional vector space over real numbers, and T a linear transformation on the vector space. Consider $$\lim_{z'\rightarrow \mathbf{0}} \frac{T(z+z')-T(z)}{z'}$$ I argue this could be an alternative definition for complex derivative. To illustrate this, z as a...
  44. S

    How Do You Get From 1's and 0's to complex software and digital info?

    How do simple Architectural basic computing binary code signaling create a diversity of information and software properties from information signals of existence (=1's) and non-existence (=0's)? After reading some Theoritical and Analytical Science papers recently about the debate on...
  45. C

    I Simplest self-contained numeral system for complex numbers?

    Anyone know what the simplest possible self-contained numeral system for complex numbers would be, analogous to signed ternary for integers? My guess would be quarter-imaginary base (https://en.wikipedia.org/wiki/Quater-imaginary_base.)
  46. L

    Engineering How does complex multiplication work?

    is this right Q) Determine this voltage in its simplest complex number form. v = (2xj6)(3-j8) 2x3=6 2x-j8=-16 j6x3=j18 j6x-j8=-j48 v=6 +(j18-j16) - J(^2)48 (j^2 = -1) v=6 +j2 +48 V=54 + j2
  47. I

    I Complex Conjugate of Wave Function

    I've been studying quantum mechanics this semester in school and have ran into an issue I can't find an answer for. I understand why we take the complex conjugate of the wave function, such as when calculating expectation values. I'm a little confused though as to why we take the complex...
  48. JD_PM

    Minimal substitution on the Lagrangian of the complex KG field

    a) I think I got this one right. Please let me know otherwise We have (let's leave the ##x## dependence of the fields implicit :wink:) $$\mathscr{L} = N \Big(\partial_{\alpha} \phi \partial^{\alpha} \phi^{\dagger} - \mu^2 \phi \phi^{\dagger} \Big) = \partial_{\alpha} \phi^{\dagger}...
  49. G

    I What is an (almost) complex manifold in simple words

    I try to understand (almost) complex manifolds and related stuff. Am I right that the condition for almost complexity simply is that the metric locally can be written in terms of the complex coordinates ##z##, i.e. ##g = g(z_1, ... z_m)## (complex conjugate coordinates must not appear)? These...
  50. N

    Using Complex Impedances in these RLC Circuit Calculations

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