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. R

    Finding value in a complex set region

    Homework Statement The largest value of r for which the region represented by the set { ω ε C / |ω - 4 - i| ≤ r} is contained in the region represented by the set { z ε C / |z - 1| ≤ |z + i|}, is equal to : √17 2√2 3/2 √2 5/2 √2 Homework Equations complex number = a + ib where a,b ε R The...
  2. L

    Fortran Can complex numbers be used in Fortran array constructors?

    Hi, so I need to write a fortran code with 2, 2x2 matrices. These matrices are in the form of B=(1 exp(i)(theta) 0 0) and D=(0 0 exp(i)(theta) 1) where i is sqrt of -1 and theta is an angle between 0 and 2pi. I've expanded the exponential so it reads cos(theta)+isin(theta) and let theta=pi/2...
  3. I

    Superposition: Find V(t) in complex (AC) circuit

    Homework Statement Homework Equations phasor forms voltage division current division The Attempt at a Solution Using superposition, considering only the varying voltage source. Z (L) = 4j Z (C) = 5j Total impedance: 4 is parallel with 5 = 2.44 + 1.95j series with 1 + 4j Total...
  4. K

    2nd order pole while computing residue in a complex integral

    Hello, I am trying to understand how to get the residue as given by wolfram : http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2 The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...
  5. N

    Complex Analysis: Open Mapping Theorem, Argument Principle

    Homework Statement In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample. (a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...
  6. T

    Finding root of complex equation

    Homework Statement Good day, I've been have having difficulties finding the roots of this: Find the roots of 3ix^2 + 6x - i = 0 where i = complex number i = sqrt(-1) Homework Equations quadratic formula (apologies for the large image) The Attempt at a Solution...
  7. I

    Wavelet coefficient based QRS complex classifier

    Dear all, I am new to Wavelet field and I wanted to ask you for a help for an idea. I am supposed to create QRS complex (certain part of ECG signal wave) wave morphology classifier based on Wavelets in other words, I am supposed to create classifier which will separate waves with similar wave...
  8. N

    Repeated complex conjugate roots for Cauchy-Euler

    Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation. This is incorrect, but I think it is close: X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2] I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?
  9. George Zucas

    Complex System Boundary Conditions

    Edit: Sorry about the vague title, it was intended to be complex beam system boundary conditions but somehow it turned out like this. Hello, I am trying to learn complex beam system designs and I sometimes struggle to assign boundary conditions. For example I am trying to design the lifting...
  10. DennisN

    Complex organic molecules discovered in protoplanetary disk

    (@mfb posted an article about this here, I think it deserves an own thread, and I did not find one, so I start one :smile:) The comet-like composition of a protoplanetary disk as revealed by complex cyanides Karin I. Öberg, Viviana V. Guzmán, Kenji Furuya, Chunhua Qi, Yuri Aikawa, Sean M...
  11. C

    Complex number inequality graph

    Homework Statement How would Re(z)<0 be graphed? Homework Equations Re(z) is the real part of z The Attempt at a Solution It looks similar to y>x, but only shaded in the third quadrant, how can this be explained? not relevant anymore
  12. B

    Finding the centroid of a triangle using complex numbers

    Hi all, I'm preparing for a deferred exam this semester after falling ill last year. Just looking over my course notes and have a question. I understand how this works in the big picture scheme. What I don't understand however is how my instructor simplified the original equation. 1. Homework...
  13. TESL@

    Parametrizing a Complex Curve on a Torus Surface

    Hello, I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the resulting vectors are page long expressions and the reason for it is because I have four...
  14. ognik

    MHB Complex Matrices, please check my working

    Hi, just starting with complex matrices, would appreciate checking if I'm on the right track. 1) Three angular momentum matrices satisfy the commutation relation: [Jx , Jy]= i.Jz If 2 of the matrices have real components, show the elements of the 3rd must be pure imaginary. (I assume pure...
  15. X

    Circuit analysis complex numbers

    Homework Statement I'm going crazy. I've done this problem nearly 20 times and keep getting the same answer. I've read my textbook so many times too! What am I doing wrong? Homework Equations Zcapacitor = 1/(jwC) Zinductor = jwL Zresistor = R The Attempt at a Solution Z for the...
  16. C

    The Unique Limit of a Complex Function

    Homework Statement I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained. Homework Equations 0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...
  17. Ondina

    Complex derivative of x: ((x)^(1/x))'

    <<Moderator note: Remember that filling in the complete homework template is mandatory in the homework forums. This thread has not been deleted due to containing relevant replies.>> 1. Homework Statement ((x)^(1/x))' Homework Equations This probably isn't overly dificult, but it has got me...
  18. N

    Complex Analysis: Theorem Name

    Hi, In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional...
  19. M

    An Interesting Complex Number Problem

    Hey all. I am giving Panhellenic exams this year, and part of my preparation comes from solving previous' years questions. This problem was a complex number based one, in the 2013 Panhellenic exams. It goes as follows: Consider 3 complex numbers, α0,α1 and α2 which belong in the line given by...
  20. D

    Engineering Complex arithmetic for circuit equation

    Hi, I am trying to find the current of a circuit using mesh analysis so far I have; My voltages V1 = 415 ∠ 90° or 0 + j415 V2 = 415 ∠ 0° or 415 + j0 impedances Z1 = j4 Z2 = j6 My formula is; -V1+Z1*I+Z2*I+V2=0 Which equates to; -0+j415+j4*I+j6*I+415+j0=0 Could...
  21. L

    Why complex reps of gauge group for chiral theory?

    Why must the gauge group be in a complex representation so that chirality of the fermions is respected? thanks
  22. K

    MHB Showing that equality of complex numbers implies that they lie on the same ray

    This problem has been on my mind for a while. ---------- **Problem:** Show that **if** \begin{equation} |z_1+z_2+\dots+z_n| = |z_1| + |z_2| + \dots + |z_n| \end{equation} **then** $z_k/z_{\ell} \ge 0$ for any integers $k$ and $\ell$, $1 \leq k, \ell \leq n,$ for which $z_{\ell} \ne 0.$...
  23. K

    MHB Uniform convergence of a complex power series on a compact set

    I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...
  24. rayne1

    MHB Eigenvector of 3x3 matrix with complex eigenvalues

    Matrix A: 0 -6 10 -2 12 -20 -1 6 -10 I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of: 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 So, how do I find the nonzero eigenvectors of the...
  25. Q

    Do Complex Objects Have Equations?

    For example, if someone with an architectural modeling software designed a beautiful mansion or other type of intricate and ornate structure, is there some equation in the computer or some type of mathematical structure to what was designed? Are there potential mathematical blueprints for all...
  26. J

    Engineering Circuit Analysis using Phasors

    Do I have the right idea? I'm not getting what is in the solutions manual but it's been wrong on past chapter and I'm beyond frustration.
  27. baby_1

    Find theta angle in complex form

    Homework Statement Here is my equation that I want to find theta angle Homework EquationsThe Attempt at a Solution I try to set different value of cos (theta) to find theta but it failed , I want to know the main solving strategy Thanks
  28. 2

    Finding the nth root of a complex number?

    Homework Statement Find the solutions to z^{\frac{3}{4}}=\sqrt{6}+\sqrt{2}i Homework Equations de Moivre's theorem The Attempt at a Solution z^{\frac{3}{4}}=2\sqrt{2}e^{\frac{\pi i}{6}}=2\sqrt{2}e^{\frac{\pi i}{6}+2k\pi}=2\sqrt{2}e^{\frac{\pi +12k\pi}{6}i} z=4e^{\frac{4}{3}{\frac{\pi...
  29. M

    MATLAB Optimizing Plotting for Complex Functions with Large Numbers

    Hi PF! I am trying to run the following plot: k = .001; figure; hold on [X,Y]=meshgrid(-4:0.01:4); a = 5.56*10^14; b = .15/(2*.143*10^(-6)); for n = 1:8 k = k*2^(n-1); Z = a./(X.^2+Y.^2).*exp(b.*(X-sqrt(X.^2+Y.^2)))-k; contour3(X,Y,Z) end which works great if a = b = 1. But now...
  30. D

    Real and complex vector spaces

    I've just encountered the following theorem : If T is a linear operator in a complex vector space V then if < v , Tv > =0 for all v in V then T=0 But the theorem doesn't hold in real 2-D vector space as the operator could be the operator that rotates any vector by 90 degrees. My question...
  31. Khronos

    Optimisation - Critical Numbers for Complex Functions.

    Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...
  32. P

    Equivalent resistance in a complex circuit

    Homework Statement In the circuit shown, R1 = 0.810 Ω, R2 = 8.10 Ω, R3 = 81.0 Ω, and R0 = 810 Ω. See attachment Calculate the equivalent resistance of the circuit when a 7.70 V power supply is connected between points A and C. Calculate the current through R0 and R1 Homework...
  33. Astronuc

    Bárðarbunga volcanic complex - Baugur crater

    Geologists Climb Into Iceland Volcano, Come Out With Stunning Images http://news.yahoo.com/geologists-climb-iceland-volcano-come-stunning-images-132015005.html Talk about a hot job. I hope they get samples for isotopic analysis. There are "small vents of blue sulfur-dioxide gas rise from the...
  34. N

    Complex Analysis: Series Convergence

    Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...
  35. V

    Complex Gaussian Integral - Cauchy Integral Theorem

    Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...
  36. N

    Complex Analysis: Identity Theorem

    Homework Statement Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique? (a) f(1/n)=4 for n=1,2,\dots (b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots A side question: Is corollary 1 from my...
  37. Digitalism

    Smoky 3D complex network analysis that evolves in time

    Does anyone know of any tools/graphs that show relationships in networks in a more "smoky" form that evolves in time? Preferably that correlates with spatial location though not necessarily to a map. This is a "normal" picture and not what I am looking for, rather something for which the...
  38. W

    MATLAB Matlab summation of a complex function

    Hi, I need to plot the last function of this: But I don't know how to generate the sum. I know the for loop is totally wrong, but I can't go any further. This is what I have: Can someone fix the summation loop part for me? Thanks in advance
  39. N

    Complex Wave Orientation Correction

    The configuration and dimensions of any experiment are important in determining wave amplitudes. Then why are the orientations of complex waves not considered when they are added? For example in two dimensions; To find a resulting wave at a point P1 from two paths R1 & R2 we have...
  40. N

    Can Complex Numbers Bridge the Gap Between Algebra and Geometry?

    This really isn't a homework question but I wasn't sure where to post it. I was watching a video by numberphile about complex numbers and the professer being interviewed said the most important thing about complex numbers is that they help bring algebra and geometry together. What did he mean by...
  41. W

    Sum of a geometric series of complex numbers

    Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...
  42. N

    Complex Analysis: Special Power Series

    Homework Statement Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex]. Homework Equations R is the radius of convergence and the derived series is the term by term...
  43. J

    Complex number epsilon-delta problem

    Homework Statement Prove that \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 Homework Equations It is specifically mentioned in the text that the epsilon-delta relation should be used, |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta . Where...
  44. ddd123

    Why are Ramanujan sums the same as the complex Zeta values?

    Possibly a difficult question, but I've never found a discussion on the topic. Thanks
  45. Math Amateur

    MHB Rotations, Complex Matrices and Real Matrices - Proof of Tapp, Proposition 2.2

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding the proof of Tapp's Proposition 2.2. Proposition 2.2 and its proof read as...
  46. Math Amateur

    MHB Rotations, Complex Matrices and Real Matrices

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding what Tapp is saying in this section regarding the function \rho_n \ : \ M_n...
  47. gimak

    The second one is a PDF file so you will need Adobe Acrobat Reader to read it.

    Homework Statement y=[(1+y^2)^1.5]/[2(y+sqrt 3)^2]; solve for y Homework Equations see above The Attempt at a Solution I tried to use algebra to solve it, but I can't. The textbook says it can be solved numerically or by iteration. By numberically I think it means algebraically. But I don't...
  48. M

    Complex Analysis: Largest set where f(z) is analytic

    Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...
  49. A

    Complex Power Q: Solved w/ My Friend & I

    Homework Statement [moderator note: Image inserted to be visible in problem statement] Homework EquationsThe Attempt at a Solution My friend and I are trying to figure out how this works. This is the questions and the solution manual. We understand where the real power comes from but not at...
  50. S

    Find the maximum value of this complex function

    Homework Statement Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1 Homework Equations |f(z)| yields the maximum value The Attempt at a Solution f(z) = exp(x) ( cosy + i siny) Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric...
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