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.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Conservation of Noether charge for complex scalar field

    Homework Statement Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time. Homework Equations ##\pi=\dot{\phi}^{*}## The Attempt at a Solution...
  2. S

    A Complex scalar field - commutation relations

    I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...
  3. caters

    Complex zeros to polynomial

    Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...
  4. C

    What are the properties of nonzero complex numbers satisfying z^2 = i\bar{z}?

    Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework EquationsThe Attempt at a Solution I have...
  5. S

    MHB Understanding Complex Number Math: |iz^2|

    Complex numbers If z=rcis(theta) FIND: |iz^2| I am confused about how I incorporate the i into the absolute value. I can't remember what it means. Please help and show exactly how I complete the workings. I can easily find the absolute value of z^2 I just really don't understand how to put the...
  6. W

    A Existence of Complex Structures and Characteristic Classes

    Hi, Just curious if someone knows of any Characteristic class used to determine if a manifold allows a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n} yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs...
  7. D

    I Complex conjugate and time reversal operator

    Hi. I'm confused about the action of the complex conjugate operator and time reversal operator on kets. I know K(a |α > ) = a* K | α > but what is the action of K on | α > where K is the complex conjugation operator ? What is the action of the time reversal operator Θ on a ket , ie. what is Θ...
  8. V

    I Can a CW complex exist without being a Hausdorff space?

    I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without...
  9. M

    Can I solve this complex numbers equation? Finding values for z

    Homework Statement ask to find all the values in z to the equation to be true[/B]Homework Equations [/B]The Attempt at a Solution this is my atemp of solution i don't know what else do, because the problem ask for z values well i must add that i am thinking about x=0 and y= pi/2 will solve...
  10. K

    I Understanding Complex Coefficients in Quantum Mechanics

    Ok. I will try to do my best to explain you what is my doubt. I'm not a native English Speaker and the book I was reading is not in English Lang, but I've translated it to English. In a spin-1/2 Stern Gerlach-like experiment, we can express the ket representing the spin component of the...
  11. terhje

    Complex numbers. write equation on form "a+bi"

    Homework Statement Write this complex number in the form "a+bi" a) cos(-pi/3) + i*sin(-pi/3) b) 2√2(cos(-5pi/6)+i*sin(-5pi/6)) Homework Equations my only problem is that I am getting + instead of - on the cosinus side.(real number) The Attempt at a Solution a) pi/3 in the unit circle is 1/2...
  12. Marcin H

    Complex Numbers (Exponential/Rectangular Form)

    Homework Statement Homework Equations Theta = arctan (y/x) The Attempt at a Solution Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A...
  13. Destroxia

    Complex Analysis - Exponential Form

    Homework Statement Write the given numbers in the polar form ##re^{i\theta}##. ## \frac {2i} {(3e^{4+i})} ## Homework Equations ## z = re^(i\theta) ## ## \theta = Arg(z) ## ## r = |z| = \sqrt { x^2 + y^2 } ## The Attempt at a Solution I'm not really sure how to go about the exponential...
  14. Graham1874

    Forces/Moments on Complex Beam System

    To give a bit of context, I am doing my final year university project on micro-mechanical interactions between an AFM probe and a sample surface. I do not have notes for a system this complicated, as we always considered our systems to be rigid bodies. I was always relatively clueless at...
  15. Graham1874

    Mechanical-Structural Engineering: Forces/Moments on Complex Beam

    Homework Statement To give a bit of context, I am doing my final year university project on micro-mechanical interactions between an AFM probe and a sample surface. I do not have notes for a system this complicated, as we always considered our systems to be rigid bodies. I was always...
  16. P

    MHB Understanding Complex Geometric Sequences: A Revision Question

    need a hand with a revision question, I don't quite understand how to go about solving it question is attached below
  17. B

    Analysis Readability of Rudin's Real and Complex Analysis

    So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its...
  18. Zeeree

    Finding an upper bound for a contour integral (Complex)

    C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...
  19. Carlos de Meo

    Complex dielectric constant -- metals, insulators and Reflections

    Hi everyone Can anyone help me understanding the physical meaning for the complex dielectric constant? Assuming a electromagnetic wave from air to a conductor, the following equation is valid R= ((n-1)2+k2)/((n+1)2+k2) where K is the extinction coefficient (the complex part of the complex...
  20. N

    I Help evaluating complex function in form m+ni?

    Hey all, I need the complex version of the sigmoid function in standard form, that is to say $$f(\alpha) =\frac{1}{1+e^{-\alpha}} , \hspace{2mm}\alpha = a+bi , \hspace{2mm} \mathbb{C} \to \mathbb{C}$$ in the simplified form: $$f = m+ni$$ but found this challenging, for some reason i assumed...
  21. M

    Finding the Limit of a Complex Expression

    Homework Statement limit (5/(2+(9+x)^(0.5))^(cosecx) x-->0 attempt: tried applying lim (1+x)^(1/x) = e. x->0 couldn't get anywhere.
  22. david102

    Find the argument of the complex number.

    Homework Statement If modulus of z=x+ iy(a complex number) is 1 I.e |z|=1 then find the argument of z/(1+z)^2 Homework Equations argument of z = tan inverse (y/x) where z=x+iy modulus of z is |z|=root(x^2+y^2) The Attempt at a Solution z/(1+2z+z^2) = x+iy / 1+2(x+iy)+( x+iy)2 ...
  23. D

    MHB Complex number as a root and inequality question

    Question 1: (a) Show that the complex number i is a root of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 (b) Find the other roots of this equation Work: Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...
  24. B

    Analysis Supplements to Complex Analysis of Rudin-RCA?

    Dear Physics Forum friends, I will be doing a reading course in the complex analysis starting on this Fall Semester. The assigned book is Rudin's Real and Complex Analysis. From my understanding, Rudin treats complex analysis very elegantly, but very terse. I am curious if you could suggest...
  25. P

    Complex analysis integral

    Homework Statement calculate ## \int_{-\infty}^{\infty}{\frac{2}{1+x^2} e^{-ikx}dx}##, where k is any positive number Homework EquationsThe Attempt at a Solution So first consider the closed contour ##I= \int{\frac{2}{1+z^2} e^{-ikz}dx}## We can choose the contour to be along the real axis ##...
  26. S

    A Rank 3x4 Complex Matrix Constraints

    I am dealing with a 3x4 complex matrix M that relates a vector d to another vetor c. That is: c = [M]*d d is 4x1 and c is 3x1. I want to introduce a new line (constraint) into M, say d(1) = d(2). However, I would like to only apply the constraint to the real or only the imaginary parts. Is...
  27. The Bill

    I Radius of the largest ball inside a complex set.

    I've been thinking about notions like the following: "How far can one be from the nearest road while in a particular country." "What's the 'maximum thickness' of a subset of \mathbb{R}^n?" "What mountain range has the biggest circular region entirely within it?" These sorts of questions lead...
  28. L

    MHB Complex numbers simultaneous equations

    Hi all, I have spent a couple of hours on this perplexing question. Solve the simultaneous equations: z = w + 3i + 2 and z2 - iw + 5 - 2i = 0 giving z and w in the form (x + yi) where x and y are real. I tried various methods, all to no avail. I have substituted z into z2 , I got the wrong...
  29. The Bill

    A What is meant by "locally like a simplical complex?"

    What is the name for a toplogical space that is everywhere "locally like a simplical complex" in that every point has at least one neighbourhood which is either a topological manifold, or can be countably decomposed by surgery into a set of topological manifolds which intersect along...
  30. K

    I Complex Analysis, holomorphic in circle.

    Hi, I have a question regarding corollary 2.3. in the uploaded image. it looks very trivial to me becauese Cauchy's theorem states "if f(z) is holomorphic, its closed loop integral will be always 0". Is this what the author is trying to say? what's the necesseity of the larger disk D' at here...
  31. P

    Studying Problem solving of complex physics problem

    What kind of steps do you use when you need to solve a complex physics problem? What kind of questiono do you ask yourself? Richard Feynman said that you have to write down the problem and then think hard, but what thoughts and questions can help yourself find a creative solution? Is creativity...
  32. Rectifier

    Complex numbers - factors

    The problem The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions? The attempt This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...
  33. M

    Complex integration, possibly branch cut integral

    Homework Statement The integral I want to solve is $$ D(x) = \frac{-i}{8\pi^2}\int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r^2\sin\theta$$ which I've reduced to $$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$ by integrating over ##\theta##. However, I...
  34. S

    Anyone on PF involved in complex systems research?

    Hi everyone! I have been perusing PF for some time, and what's struck me is how little discussion there has about research in complex systems & complexity, such as the research conducted in places like the Santa Fe Institute or the Center for the Study of Complex Systems at the University of...
  35. G

    Engineering Circuit analysis -- Find the complex apparent power of I(g2)

    Homework Statement Given the circuit of sinusoidal current (attachment 1) with given data: \underline{Z_3}=200(3-j4)\Omega, \underline{Z_4}=100(3+j20)\Omega, \underline{Z_5}=100(3+j4)\Omega, \underline{Z}=100(2+j5)\Omega, \underline{I_{g2}}=-10(2-j)mA After the switch is closed, the increment...
  36. T

    Which Option Should I Choose for the Complex Ion Based on the Given Information?

    Homework Statement Homework EquationsThe Attempt at a Solution There is 0.2 mol of Cl- ions so therefore this leaves us with a 1:1 ratio for the chloride ion to chromium ion in the complex ion. This leaves us with option B or D. However, I am not sure how to choose between them. Can someone...
  37. Johnny_Sparx

    A Numerical Solution for Complex PDE (Ginzburg-Landau Eqn.)

    I am looking to numerically solve the (complex) Time Domain Ginzburg Landau Equation. I wish to write a python simulator to observe the nucleation of fluxons over a square 2D superconductor domain (eventually 3D, cubic domain). I am using a fourth order Runge Kutta solver for this which I made...
  38. T

    B Product of complex numbers

    If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression: ## E = AB - B^*A^*## I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no...
  39. Q

    I Schrodinger equation in terms of complex conjugate

    I know there's a similar post, but i didn't understand it. Why the derivative respect to t in terms of the complex conjugate of ψ is: instead of being the original S.E in terms of ψ* or the equation in terms of ψ with the signs swapped
  40. K

    I Complex Analysis Radius of Convergence.

    Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...
  41. A

    B Partial derivative of the harmonic complex function

    For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation! Sorry, I...
  42. F

    MHB Helpful Tips on Solving Complex Equations

    Hi! I have problems with this demonstration Let $z= x+iy , x,y \in \mathbb{R} $ then $|x|, |y| \leq{|z|} \leq{\sqrt[ ]{2}} $ , $max \{ |x|, |y| \} $
  43. K

    I Complex Analysis holomorphic condition

    I understood the holomorphic condition this way. For a vector field F(x1, x2 . . ., xm) = <y1(x1, x2, x3 . . . , xm), y2(x1, x2, x3 . . . , xm), y3(x1, x2, x3 . . . , xm) . . . ,yn(x1, x2, x3 . . . , xm)> In a real analysis, its derivative is expressed as a Jacobian matrix because each...
  44. P

    MHB Question via email about complex numbers

    We should note that we can write any complex number as $\displaystyle \begin{align*} z = r\,\mathrm{e}^{\mathrm{i}\,\theta} \end{align*}$ where $\displaystyle \begin{align*} r = \left| z \right| \end{align*}$ and $\displaystyle \begin{align*} \theta = \textrm{arg}\,\left( z \right) + 2\,\pi\,n ...
  45. avikarto

    I Associated Legendre polynomials: complex vs real argument

    I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6, $$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$ $$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu...
  46. Jianphys17

    Which Complex Analysis Textbook Should I Use?

    Hi, i need advice, I'm studying complex analysis on which book would you recommend to do it on that of Ahlfors or T.W.Gamelin's book?
  47. P

    Contour integral- Complex variables

    Homework Statement evaluate ##\int \frac{sinh(ax)}{sinh(\pi x)}## where the integral runs from 0 to infinity Homework EquationsThe Attempt at a Solution consider ##\frac{sinh(az)}{sinh(\pi z)}## Poles are at ##z= n \pi i## So I'm considering the contour integral around the closed contour from...
  48. P

    MHB Effie's question via email about Complex Numbers

    First let's write this number in its polar form. $\displaystyle \begin{align*} \left| z \right| &= \sqrt{\left( -2 \right) ^2 + 2^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\,\sqrt{2} \end{align*}$ and as the number is in Quadrant 2 $\displaystyle \begin{align*} \textrm{arg}\,\left( z...
  49. P

    MHB Sava's question via email about solving complex number equations

    $\displaystyle \begin{align*} z^3 + 1 &= 0 \\ z^3 &= -1 \\ z^3 &= \mathrm{e}^{ \left( 2\,n + 1 \right) \,\pi\,\mathrm{i} } \textrm{ where } n \in \mathbf{Z} \\ z &= \left[ \mathrm{e}^{\left( 2\,n + 1 \right) \, \pi \,\mathrm{i}} \right] ^{\frac{1}{3}} \\ &= \mathrm{e}^{ \frac{\left( 2\,n + 1...
  50. T

    B Simplifying the factors of a complex number's imaginary part

    My question boils down to wondering if there is a way to simplify the imaginary part of a complex-valued function composed of n factors if the real and imaginary component for each of the factors is known but the factors may take on the value of their conjugate as well. For example, is there a...
Back
Top