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. B

    MHB Complex function that satisfies Cauchy Riemann equations

    Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!
  2. D

    What is the polar complex form of a wave with amplitude and phase?

    Homework Statement What is the amplitude and phase of the complex function? f(t) = (1-2i)e^(iwt) Homework Equations None/unknown Normal Polar Form = Real*e^imaginary i = e^pi/2*i The Attempt at a Solution [/B]I am trying to bring this into a normal polar form to easily see the phase and...
  3. K

    MHB Showing the “only if” direction of equality in a complex equality

    *How can I finish off the "only if" direction? I am just unable to prove the only if direction! Using the induction hypothesis and the triangle inequality is confusing me for some reason.* Show that \begin{equation} |z_1+z_2+\dots+z_n| = |z_1| + |z_2| + \dots + |z_n| \end{equation} if and only...
  4. M

    Question regarding complex numbers

    1)If a= cosα + i sinα and the equation az2 + az +1 =0 has a pure imaginary root, then tanα=? 2) cosα+isinα=eiα , quadratic formula 3) What i tried to do was,i put a constant real number and tried to solve it and used demoivres theorem, although the answer is getting weirder and weirder.
  5. Ahmad Kishki

    Linear Algebra Linear algebra with complex numbers

    Recommend a self study book for linear algebra with complex numbers
  6. C

    Exploring the Limits: Evaluating a Complex Fraction

    Homework Statement Find lim_{x->- \infty} \; \frac{(x^6+8)^{1/3}}{4x^2+(3x^4+1)^{1/2}} Homework Equations N/A The Attempt at a Solution Factoring out \frac {(-x^6)^{1/3}}{-x^2} leaves me with \frac{(-1-8x^{-6})}{-4+(3+x^{-4})^{1/2 }} Taking the limit at infinity gives me...
  7. C

    Numerical Solution to Complex DiffEQ?

    I've been trying to figure out a way to get an approximation to a complex DiffEQ. dx/dt = c1 / (c2 + c3*x*t) Does anyone have any input on wether this problem can be approximated? Thank you.
  8. N

    What is the Theorem for Differentiability in Advanced Calculus?

    Homework Statement This isn't a standard homework problem. We were asked to do research and to find a theorem of the form: If something about the partial derivatives of u and v is true then the implication is ##D(u,v)## at ##(x_0,y_0)## exists from ##R^2## to ##R^2##Homework EquationsThe...
  9. Y

    Complex Variables - principal argument

    Homework Statement Find the principle argument Arg z when z = (sqrt(3) - i)^6 Homework EquationsThe Attempt at a Solution I'm sorry to say that I'm not sure how to solve this problem. It's my understanding that what this question is basically asking me to do is find theta such that...
  10. nmsurobert

    Partial derivatives and complex numbers

    Homework Statement show that the following functions are differentiable everywhere and then also find f'(z) and f''(z). (a) f(z) = iz + 2 so f(z) = ix -y +2 then u(x,y) = 2-y, v(x,y) = x Homework Equations z=x+iy z=u(x,y) +iv(x,y) Cauchy-Riemann conditions says is differentiable everywhere...
  11. B

    Solving Complex Electrical Circuit: V1, V2, & V3

    Homework Statement Homework EquationsThe Attempt at a Solution I assumed there to be two supernodes. The first one was between V1 and V2. The second supernode was between V2 and V3. This is what i tried and i am not getting the right answers. Please can someone help me out! Apologies...
  12. LachyP

    Calculate Complex Conjugate of Ψ(x,t) for x=4, t=9

    I'm just starting this, but what would the complex conjugate of Ψ(x,t) in the equation : |Ψ(x,t)|^2= Ψ(x,t)* Ψ(x,t) be.. Let's just say, for example, that x is 4 and t is 9... Please help if you can.. Could you please help me out with the steps to completing this, because I really want to...
  13. Logan Land

    Expressing complex numbers in the x + iy form

    Homework Statement ((1-i)/(sqrt2))^42 express in x+iy form Homework Equations z1/z1=(r1/r2)e^(i(theta1-theta2)) The Attempt at a Solution Ive found that (1-i) has r=sqrt2 so since r is sqrt2 and x=1 y=-1 so the angle is 7pi/4 so then I have (sqrt2e^(-i7pi/4)/sqrt2)^42 now from here is where I...
  14. G

    Linking Fourier Transform, Vectors and Complex Numbers

    Homework Statement Homework EquationsThe Attempt at a Solution I tried to attempt the question but I am not sure how to start it, at least for part (i). My biggest question, I think, is how does the multiplication of a random complex number to a Fourier-Transformed signal (V(f)) have an...
  15. G

    Independence of Complex Fields?

    Are a field and its complex conjugate independent? It seems like they're not, as one is the complex conjugate of the other, so if you have one, you know the other. However, it seems in path integrals, you integrate over the field and its conjugate, so they can take on values that are not the...
  16. J

    Number of complex calculations in FFT and inverse FFT

    Homework Statement Calculate the total number of compex multiplications required for the calculation in (b) when FFTs are used to perform the Discrete Fourier Transforms and Inverse Discrete Fourier Transforms.[/B] There were two FFT multiplied together and one inverse FFT of that product to...
  17. L

    Complex Representations: Real vs. Complex Lie Algebras

    When do we call a representation complex? What are examples of complex representations? Also, when we say real and complex forms of Lie algebras, is that related to real and complex representation classification? I read that spinors are complex representations of SO(3), because their...
  18. Calpalned

    Calculating Arc Length for Parametric Equation x = e^t + e^-t and y = 5 - 2t

    Homework Statement The question involves finding the arc length of the parametric equation x = e^t + e^-t and y = 5 - 2t Homework Equations Arc length of a parametric equation ∫√(dy/dt)^2 + (dx/dt)^2 dt limits are from 0<t<3 The Attempt at a Solution Taking the derivative of both x and y...
  19. S

    Complex number problem with trig functions

    Homework Statement Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B] Homework Equations 1. z=a+bi 2. re^itheta The Attempt at a Solution I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...
  20. Spinnor

    Massless complex field+assumption(s)=massive field. Doable?

    Suppose we have a massless complex field in 3+1 spacetime where E^2 = P^2. Suppose that the only excitations that are possible are those that in some rest frame consist of an excitation of a pair of states p1 and p2 such that p1 = -p2 and ιp1ι = ιp2ι = mc^2 = (+or-)E, and the pair of states p1...
  21. THE HARLEQUIN

    Real number calculus vs complex calculus

    I have started studying complex integration recently and i just can't seem to get the things in my head . the biggest problem i am facing is that : when solving real number integrals the area under the curve of the function is what integration means ... but i can't seem to find an analogy...
  22. nmsurobert

    Complex conjugate proof, i think

    Homework Statement prove that sqrt2|z| greater than or equal to |Rez| + |Imz| Homework Equations |z|^2 = x^2 + y^2 Rez=x, Imz=yThe Attempt at a Solution so far I've worked it down to this. 2(x^2 + y^2) greater than or equal to x^2 + 2xy + y^2 I've used a few different values for x and y and...
  23. M

    Set of Points in complex plane

    Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...
  24. R

    Complex fraction in numerator help?

    1. Evaluate the limit http://www4a.wolframalpha.com/Calculate/MSP/MSP64511d2754i3f4iaefab00001fa62g875680a1ia?MSPStoreType=image/gif&s=44&w=125.&h=45. 2. No formulas 3.The answer is -1/9. I have tried multiplying the top by the conjugate but that seems wrong as there are no square roots...
  25. A

    How to deal with this sum complex analysis?

    Homework Statement Homework Equations Down The Attempt at a Solution As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the beginning n=1 the residue...
  26. anemone

    MHB Challenge for Polynomial with Complex Coefficients

    Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region $|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.
  27. neosoul

    Complex numbers and differential equations for physics

    How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...
  28. P

    Why were complex numbers introduced in physics?

    hello can you tell me please why we introduced complex numbers? what was the problem that we couldn't express with rest of algebra and we introduced complex numbers? I am basically interested in why we introduced complex number to describe and analyze AC circuits, like voltage, current and...
  29. J

    Wave equation complex problem

    Homework Statement Ytt = 1 Yxx with initial conditions of yT(x,0) = 0 y(x,0) = \begin{cases} 1 & \text{if } x \geq 0 \ & \text{if } x \leq 1 \\ 0 & \text{if } otherwise \end{cases} Sketch the solution of this wave equation for 5 representative values of t, when the solution of the wave is...
  30. M

    A complex equilibrium question

    Kc1 = (5.8*2/5)^2 / (14/5)(1.4/5) = 6.865 6.865 = [HI]^2 / (45/253.8/100)(0.5/2/100) [HI] = 5.52x10^-3 M mass no of HI = [HI] x (126.9+1) x 100 = 70.6g is it correct? And how to do 3bii I GOT 3bi Kc2 = [HCl(g)]^2/ / Kc(1) [H2(g)] [Cl2(g)]
  31. Elroy

    Qubits, 2 complex numbers, forcing one to real

    Hi All, I'm working out a program to emulate a quantum computer (definitely in a nascent stage), and I'm struggling with a piece of the math. I looked at the math sections in these forums, but thought this might be more appropriate to post it. I'll try to conceptually outline the problem, and...
  32. PcumP_Ravenclaw

    Solution to a complex cubic equations

    Homework Statement Solve the equation ## z^3 + 6z = 20 ## (this was considered by Cardan in Ars magna). Homework Equations Please see the 2nd attachement. The Attempt at a Solution I want to know if my solution is correct because the book (2nd attachment) says that there should only be 3...
  33. G

    Solving a Complex Homework Question with Inverting Amp Equation

    Homework Statement Hi Guys, I am trying to solve this question, please look at the attached picture Homework Equations The general equation for a inverting amp is -Rf/R1 * Vin = Vout The Attempt at a Solution Well as the question says the two resistors, R2 and R must be treated as parallel...
  34. E

    Rotation formula Complex numbers

    Homework Statement If arg(\frac{z-ω}{z-ω^2}) = 0, \ then\ prove \ that\ Re(z) = -1/2 Homework Equations ω and ω^2 are non-real cube roots of unity. The Attempt at a Solution arg(z-ω) = arg(z-ω^2) So, z-ω = k(z-w^2) Beyond that, I'm not sure how to proceed. Using the rotation formula may also...
  35. A

    MHB Evaluating a logarithmic integral using complex analysis

    Hello, I am evaluating: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using the following contour: $R$ is the big radius, $\epsilon$ is small radius (of small circle) Question before: Which $\log$ branch is this? I asked else they said, $$-\pi/2 \le arg(z) \le 3\pi/2$$ But in the...
  36. A

    Complex Contour Integral Problem, meaning

    Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...
  37. L

    What makes complex numbers so special?

    Is there in a nutshell an explanation or even a single reason why complex numbers have so many fascinating consequences and give rise to so much deep stuff like analytic functions (with all its stunning properties), Riemann surfaces, analytic continuations, modular forms, zeta function, its...
  38. A

    MHB Complex Contour Keyhole Integration Methods

    This is an interesting complex analysis problem; **The figure on the bottom left is what is being referred to,Fig7-10.** **Firstly: (1)** How is the branch point $z=0$ at $z=0$?? We have $f(0) = 0$ that is not a discontinuity is it? **Secondly:(2)** It says that: $AB$ and $GH$ are coincident...
  39. A

    MHB Complex Contour integration of rational function

    Hello, Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ We know that because $f(x)$ is even:$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = \frac{1}{2} \cdot \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Consider a complex function, with $z = x + iy$ $$f(z) =...
  40. B

    Unitary Operation On A Complex Matrix

    Hello everyone, Let ##A = (\alpha_{ij})## be an $n \times n# complex matrix. Define ##\hat## acting on ##A## as producing the matrix ##\hat{A} = (\alpha_{ij} I_n)##. I don't understand what this is saying. Isn't ##I_n## the identity matrix, and therefore the product of it with any matrix...
  41. P

    Why is the principal square root of a complex number not well-defined?

    Within the context of real numbers, the square root function is well-defined; that is, the function ##f## defined by: ##f(x) = \sqrt{x}## Refers to the principal root of any real number x. Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##...
  42. I

    Systems of Equations with Complex Numbers

    Mod note: This thread was moved from a technical math section, so doesn't include the homework template. I know this has been asked before, but none of the other posts have helped me. I cannot for the life of me figure out how to solve a system of equations with complex numbers. Here is a very...
  43. D

    Differentiate x to a complex power?

    Is it possible to differentiate xa+bi where a and b are real ? if so what is the answer ?
  44. PcumP_Ravenclaw

    Doubt about condition solutions of complex line equation

    Dear All, Please help me clear some doubts about Theorem 3.3.1 in the 1st attachment. The condition ## |a| = |b| ## has only 8 cases right? ## { x+iy. x - iy, -x + iy, -x - iy, y + ix, y - ix, -y + ix, -y - ix } ## so for the condition ## |a| = |b| ## and ## b \bar c = \bar a c...
  45. C

    Complex Equilibrium Problem

    Homework Statement A uniform 300-kg, 6.0 M long, freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.0 M long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at...
  46. dwn

    Navigating a Complex Plane Curve: A Homework Guide

    Homework Statement Attached Image Homework Equations this is not a simple plane curve or a close plane curve so I use the formula: ∫ F ⋅ dr/dt dt The Attempt at a Solution From the point (0,0) to (2,4) Direction Vector v(t) = <2-0, 4-0> Parametric Equation: r(t) = (2t + 0) i + (4t + 0) j...
  47. F

    Functional Analysis vs. Complex Analysis?

    I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...
  48. R

    Why Do We Use the Complex Conjugate of Velocity to Calculate Acoustic Intensity?

    Hello All, I would like to know why do we multiply the complex conjugate of velocity (not just the velocity) with the complex pressure to obtain the complex acoustic intensity. Could someone please help me with this? Regards, Radel...
  49. C

    Shoud I take Ring/Field Theory or Complex Analysis?

    Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...
  50. B

    Integrating a Complex Function Over a Contour

    Homework Statement ##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx## Homework EquationsThe Attempt at a Solution Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the...
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