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. S

    I Principal difference between complex numbers and 2D vectors revisited

    I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
  2. I

    Expectation value of an angular momentum with a complex exponent

    I am struggling to figure out how to calculate the expectation value because I am finding it hard to do something with the exponential. I tried using Euler's formula and some commutator relations, but I am always left with some term like ##\exp(L_z)## that I am not sure how to get rid of.
  3. B

    B Complex number inequality question

    Z can be any point on the argand diagram so if z molous is less than 2 , is that somehow giving us the distance from origin? But how i assumed mod sign only makes things positive therefore its not sqrt( (x+yi)^2 ) = distance ??
  4. Math Amateur

    MHB Complex Function Theory: Explaining Example 1.5, Section 1.2, Chapter III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  5. D

    Rigid body simulation of complex mechanical systems

    Greetings, I'd like to simulate a complex mechanical automaton with lots of gears, cams, levers and springs. Most of the parts are going to be 3D printed except for some metal springs, rods and bearings. I want to make sure everything fits together and works as expected. Here is just a small...
  6. Math Amateur

    MHB Complex Square Root Function: Qs from Bruce P. Palka's Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  7. Math Amateur

    MHB Differentiating Complex Square Root Function: Bruce P. Palka, Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...
  8. Math Amateur

    MHB Complex Derivatives .... Palka, Examples 1.1 and 1.2, Chapter III, Section 1.2 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ... Examples 1.1 and 1.2, Section 1.2...
  9. Math Amateur

    MHB Diiferentiability of Functions of a Complex Variable .... Markushevich, Theorem 7.1 .... ....

    I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ... I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows: At the start of the above proof by Markushevich we read the following: "If f(z)...
  10. kuruman

    Insights SOHCAHTOA: Seemingly Simple, Conceivably Complex

    Continue reading...
  11. T

    What does it mean to have an complex transmission angle?

    I am getting a complex number for my transmission angle in part (c) but I do not know what that means. Am I even doing this correctly? Any help will be greatly appreciated. Thanks!
  12. F

    Geometric sum using complex numbers

    Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...
  13. arcTomato

    I Complex Fourier transform (represented by Σ)

    Dear all. I can't understand how to derive Eq.(2.3a). Fourier coefficients, ##A_j## and ##B_j## are described by summation in this paper as (2.2).　I think this is weird. Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms". and I understand...
  14. UFSJ

    Courses Post doctoral opportunity in Complex Systems

    I'm looking for research projects in the study of complex system, meanly applied in social and biology problems. I would be so grateful with the indication of researchers in the theme. Thanks!
  15. Physics lover

    A tricky question with complex numbers

    All i was able to think was that i have to find a point (x,y) such that sum of its distances from points (0,0),(1,0),(0,1) and (3,4) is minimum.I tried by assuming the point to be centre of circle passing through any of the above 3 points,But it didn't helped me.
  16. Math Amateur

    MHB Verify Gamelin's Remark: Complex Square and Square Root Functions

    I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...
  17. Math Amateur

    MHB Complex Derivative .... Remark in Apostol, Section 16.1 .... ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ... I need help in order to fully understand a remark of Apostol in Section 16.1 ... The particular remark reads as follows: Could someone...
  18. S

    Solve polynomial using complex number

    I can do question (a). For question (b), I can not see the relation to question (a). Can we really do question (b) using result from (a)? Please give me little hint to relate them Thanks
  19. e101101

    Complex representation of a wave

    Homework Statement: Hi there, I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an...
  20. Troxx

    I Trouble with infinity and complex numbers

    Summary: Trouble with infinity and complex numbers, just curious. I'm not too familiar with set theory ... but <-∞, ∞> contains just real numbers? Does something similar to <-∞, ∞> exist in Complex numbers? My question, is it "wrong"?
  21. fresh_42

    I Necessity of Complex Numbers in Quantum Mechanics

    Summary: Which properties of ##\mathbb{C}## are actually necessary? The following is speculative as well as a honestly meant question about the way QM is modeled. I don't want to create a new theory, just understand the necessities of the old one. Physicists use complex numbers for QM. But...
  22. jisbon

    What Is the Smallest Positive Argument for the Sum of Complex Roots of Unity?

    --Continued-- 7) Let ##\sum_{k=0}^9 x^k = 0## Find smallest positive argument. Same thing as previous question, but I guess I can expand to ##z+z_{2}+z_{3}+...+z_{9}=0## ##z=re^{iθ}## ##re^{iθ}+re^{2iθ}+re^{3iθ}+...## What do I do to proceed on? Cheers
  23. jisbon

    How Do You Solve Complex Number Equations in Quadratics and Exponentials?

    Hello all! Thanks for helping me out so far :) Really appreciate it. I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :) 4) Take ##3+7i## is a solution of ##3x^2+Ax+B=0## Since ##3+7i## is...
  24. jisbon

    Help Checking a Complex Numbers Problems

    Hello, here with some complex number questions which I need some assistance in checking :) 1) z=3+5i1+3iz=3+5i1+3i Find Re(z) and Im(z) My answer is 9595 and −25−25 respectively. Checked by Wolfram 2) Find principal argument of the complex numberz=−5+3iz=−5+3i and express it in radians up to 2...
  25. Math Amateur

    MHB Understanding Differentiability and Continuity in Complex Analysis

    I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ... The two books are as follows: "Functions of a Complex Variable...
  26. D

    I What is the Topological Interpretation of Orders of Poles in Rational Functions?

    Hi. If I look at the function ## (z^2+z-2)/(z-1)^2## it appears to have a double pole at z=1 but if I factorise the numerator I get ##z^2+z-2 = (z+2)(z-1)## and it is a simple pole. Is it wrong to say it is a double pole ? If I overestimate the order of the pole in this case as 2 and...
  27. Math Amateur

    MHB Limits of Complex Functions .... Zill & Shanahan, Theorem 3.1.1/ A1

    I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ... I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ... The statement of Theorem 3.1.1 (A1) reads...
  28. jisbon

    Complex numbers: if (u+v)/(u-v) is purely imaginary, show that mod(u)=mod(v)

    I'm kind of stuck over here at one part, but something is telling me that it might be wrong too :( Do assist, thanks.
  29. D

    I How to know if a complex root is inside the unit circle

    Hi. I have been trying to calculate the real definite integral with limits 2π and 0 of ## 1/(k+sin2θ) ## To avoid the denominator becoming zero I know this means |k|> 1 Making the substitution ##z= e^{iθ}## eventually ends up giving me a quadratic equation in ##z^2## with 2 pairs of roots...
  30. D

    Is this complex function analytic?

    ## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ## ## u_y = -6xy## and ## v_x = -6xy## To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x## Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0 However...
  31. D

    I Division of a complex number by zero

    Hi I know that division of a real number by zero is not defined. I just came across the following in a textbook on Complex Analysis by Priestley , " we are allowed to divide a complex number by zero as long as the complex number ≠ 0 " Is this correct ? What happens if the complex number is...
  32. alaspina

    Representing a complex oscillating system

    Hello, I have an equation relating the angular acceleration (d2Θ/dt2) of an undamped system to a forcing function and the an angular term (Θ). The system in question is an inverted pendulum. I know that such an oscillating system can be represented by the following function: The problem is...
  33. T

    Help with a complex integration in QFT

    N/A
  34. hilbert2

    Complex polynomial on the unit circle

    So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...
  35. E

    Help Proving a Complex Laplace Transform

    So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...
  36. X

    Computing AC voltage in a complex filter

    Problem Statement: Computing AC voltage Relevant Equations: cut off frequency, ac voltage. Hey guys, Amazing to be in this group ! Please find attached the diagram of the problem. E = 10kV at 60Hz, C1 = 60pF and R1 = 1k. I have to compute the ac voltage in reference to ground at point [1]...
  37. Eclair_de_XII

    Showing that multiplication by a complex number is a linear transform

    If I had to guess what the complex matrix would look like, it would be: ##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix} a+bi & 0 \\ 0 & -b+ai\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}## I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover...
  38. E

    Why do snowflakes freeze into complex geometric patterns?

    Why do snowflakes freeze into complex geometric patterns?
  39. R

    Argument of a complex expression

    Problem Statement: What is the correct way of computing the argument of the following equation? Relevant Equations: I am trying to compute the argument ##\Phi## of the equation $$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$ which using Euler's equation...
  40. Pierson5

    Complex impedance and phase angle of a circuit

    I've attached my work below. The numbers seem odd to me though. Are my equations correct? Is the phase angle really (0/12)? If so, what are the implications of that?
  41. Robin04

    Calculating a complex integral

    As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?
  42. Robin04

    Calculating the residue of a complex function

    The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
  43. amjad-sh

    Using Simpson's method to integrate a complex function

  44. S

    I Running through a complex math derivation of plasma frequency

    Background of problem comes from Drude model of a metal (not necessary to answer my problem but for the curious): Consider a uniform, time-dependent electric field acting on a metal. It can be shown that the conductivity is $$\sigma = \frac{\sigma_0}{1-i\omega t}$$ where $$\sigma_0 =...
  45. K

    How can I use spherical coordinates to simplify the Fourier transform equation?

    By applying the Fourier transform equation, and expanding the dot product, I get a sum of terms of the form: $$V(k)=\sigma_1^x\nabla_1^x\sigma_2^y\nabla_2^y\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|}e^{-ik(r_2-r_1)} =...
  46. R

    A straight line in the complex plane

    sz+tz*+r=0=say w so w* = s*z* + t*z + r*=0 Now , w+w* = (s+t*)z + (t+s*)z* + r+r* = 0 = p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w) which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number. Now, again...
  47. fresh_42

    Algebra Chevalley Eilenberg Complex

    Does anybody know a good book about especially the Chevalley Eilenberg complexes of arbitrary Lie algebras, i.e. not automatically semisimple Lie algebras, and where the Whitehead Lemmata are more an example than the main subject. @lavinia, @A. Neumaier perhaps?
  48. Haorong Wu

    How to diagonalize a matrix with complex eigenvalues?

    Homework Statement Diagonalize the matrix $$ \mathbf {M} = \begin{pmatrix} 1 & -\varphi /N\\ \varphi /N & 1\\ \end{pmatrix} $$ to obtain the matrix $$ \mathbf{M^{'}= SMS^{-1} }$$ Homework Equations First find the eigenvalues and eigenvectors of ##\mathbf{M}##, and then normalize the...
  49. Ventrella

    A All complex integers of the same norm = associates?

    Are all complex integers that have the same norm associates of each other? I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...
  50. V

    How to find the value of a complex number with high exponent

    Homework Statement Find the value of (-√3 + i)43/243 Homework EquationsThe Attempt at a Solution I do not know how to really go about this problem. I know that i0=1, i1=i, i2=-1, i3=-i, and I tried to use that to help but I got to no where, I also tried to break up the exponent into...
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