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. K

    Complex Analysis/Radius of Convergence question.

    Homework Statement Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less. Homework Equations The Attempt at a Solution I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am...
  2. Arman777

    I Can Complex Numbers Be Ordered?

    Can we order Complex Numbers ? I searched a bit most places says it can but not like the real numbers. I am confused a bit.And I am not sure abouth the truth of those sources. Thanks
  3. K

    Complex Analysis. Laurent Series Expansion in region(22C).

    <Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
  4. A

    Linear Algebra - what is Re and Im for complex numbers?

    Homework Statement http://prntscr.com/eqhh2p http://prntscr.com/eqhhcg Homework EquationsThe Attempt at a Solution I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even? As far as I'm concerned I am has to do wtih...
  5. PhotonSSBM

    Complex Analysis-Maximum Modulus of ze^z

    Homework Statement Let ##f(z) = ze^z## be bounded in the region where ## |z| \leq 2##, ##Im(z) \geq 0##, and ##Re(z) \geq 0## Where does it achieve it's maximum modulus and what is that maximum modulus? Homework Equations N/A The Attempt at a Solution A theorem states that any function...
  6. F

    Synthetic pathways to produce the provided complex

    Question Describe the color changes that occur when ##NH_{3(aq)}## is gradually added, with stirring to ##[CuCl_4]^{2-}_{(aq)}## until the ##NH_{3(aq)}## is in excess. Identify the three compounds or ions responsible for the new colors. Now the marking scheme shows that somehow...
  7. F

    Complex integration on a given path

    Homework Statement Calculate the following integrals on the given paths. Why does the choice of path change/not change each of the results? (a) f(z) = exp(z) on i. the upper half of the unit circle. ii. the line segment from − 1 to 1. Homework Equations ∫γf(z) = ∫f(γ(t))γ'(t)dt, with the...
  8. I

    A Penetration Depth of General Complex Conductivity

    Hi all, I'm working through chapter 2 of Michael Tinkham's Introduction to Superconductivity. On page 40, he asserts that the skin-depth for a general complex conductivity is (In Gaussian units) $$\delta = \frac{c}{\sqrt{2\pi\omega\left(|\sigma| + \sigma_2\right)}}$$ where $$\sigma = \sigma_1...
  9. Macykc2

    Can I Use Antiderivatives to Evaluate this Complex Integral?

    Homework Statement I need to evaluate the following integral using the antiderivative: $$\int log^2(z) \, dz$$ I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from ##π## to ##i##, not crossing the non-positive x-axis...
  10. D

    Huge and complex experiments -- validity

    How blindly theorits trust the data comming from huge and complex experiments such as the LHC CERN? Is it possible for one person to understand the whole experiment mechanics and still be able to come up with theoretical freamworks describing the data behaviour? Is it possible even to...
  11. Delta31415

    Newton's method and complex roots

    1) the problem I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function (...
  12. M

    How Do You Determine if an Operator is Unitary, Hermitian, or a Projector?

    Homework Statement Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi> A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1| So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and...
  13. javii

    Finding the polar form of a complex number

    Homework Statement Homework Equations r=sqrt(a^2+b^2) θ=arg(z) tan(θ)=b/a The Attempt at a Solution for a)[/B] finding the polar form: r=sqrt(-3^2+(-4)^2)=sqrt(7) θ=arg(z) tan(θ)=-4/-3 = 53.13 ° 300-53.13=306.87° -3-j4=sqrt(7)*(cos(306.87+j306.87) I don't know if my answer is correct...
  14. needved

    Help with found Fourier complex series of e^t

    Homework Statement i have this function \begin{equation} f(t) = e^t \end{equation} Homework Equations [/B] the Fourier seria have the form \begin{equation} f(t) = \sum C_{n} e^{int} \end{equation}The Attempt at a Solution } [/B] so i need to find the coeficients $c_{n}$ given by...
  15. D

    What is the Method for Finding the Magnitude of a Complex Vector?

    Homework Statement Let a is a complex vector given by a = 2π K - i ρ / α^2 , where ρ is a two dimensional position vector and K is the corresponding two dimensional vector in the Fourier space. In order to find magnitude of this vector, i found that it is 4π^2 K^2 + ρ^2 / α^4 . The logic...
  16. M

    B Complex Number Solutions for |z+1| = |z+i| and |z| = 5

    This is a question from a competitive entrance exam ...I just want to check whether my approach is correct as i don't have the answer keys . here is the question : How many complex numbers z are there such that |z+ 1| = |z+i| and |z| = 5? (A) 0 (B) 1 (C) 2 (D) 3 My approach : let z = x+iy...
  17. cg78ithaca

    A Modeling diffusion and convection in a complex system

    I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...
  18. H

    A Uncertainty Propagation of Complex Functions

    Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...
  19. J

    Complex periodic functions in a vector space

    Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t)Homework EquationsThe Attempt at a Solution f(t) = e(i*w0*t)) g(t) =e(i*w0*t...
  20. C

    Derivative in the complex plane

    Homework Statement f(z)=2x^3+3iy^2 then it wants f '(x+ix^2) The Attempt at a Solution So I take the partial with respect to x and i get 6x^2 then partial with respect to y and I get 6iy, then I plug in x for the real part and x-squared for the imaginary part, then I get f '...
  21. javii

    Find the modulus and argument of a complex number

    Homework Statement Find the modulus and argument of z=((1+2i)^2 * (4-3i)^3) / ((3+4i)^4 * (2-i)^3 Homework Equations mod(z)=sqrt(a^2+b^2) The Attempt at a Solution In order to find the modulus, I have to use the formula below. But I'm struggling with finding out how to put the equation in...
  22. Mister T

    Solving for ##\theta## in a Complex Grinding Problem

    Homework Statement Solve for ##\theta##: ##\cot \theta \sin \beta + \rho \csc^2 \theta = \cos \beta## where ##0^\circ<\beta<90^\circ, \ 0^\circ<\theta<90^\circ##, and ##0<\rho<1##. Homework Equations ##\cot^2 x +1 = \csc^2x##, the quadratic formula. The Attempt at a Solution ##\cot \theta...
  23. F

    Where is f(z) = e-xe-iy differentiable and holomorphic?

    Homework Statement Suppose z = x + iy. Where are the following functions differentiable? Where are they holomorphic? Which are entire? the function is f(z) = e-xe-iy Homework Equations ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x The Attempt at a Solution f(z) = e-xe-iy I convert it to polar form: f(z) =...
  24. M

    Solution to complex valued ODE

    Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...
  25. Adgorn

    Proving a function is an inner product in a complex space

    Homework Statement Prove the following form for an inner product in a complex space V: ##\langle u,v \rangle## ##=## ##\frac 1 4####\left| u+v\right|^2## ##-## ##\frac 1 4####\left| u-v\right|^2## ##+## ##\frac 1 4####\left| u+iv\right|^2## ##-## ##\frac 1 4####\left| u-iv\right|^2## Homework...
  26. Poetria

    Complex functions with a real variable (graphs)

    Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...
  27. C

    Does the Limit in the Complex Plane Approach Infinity?

    Homework Statement lim as z--> i , \frac{z^2-1}{z^2+1} The Attempt at a Solution [/B]When we plug in i we get -2/0, so we get division by 0, Does this mean the limit is infinity, I also tried approaching from z=x+i where x went to 0, you get the same answer, I also approached from...
  28. Poetria

    Complex exponentials - homework

    Could you give me a hint how to attack this problem? Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t) I have begun as follows: e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b)) Re e^(z*t)= e^(a*t)*cos(b) What to do now?
  29. Adolfo Scheidt

    I Product of complex conjugate functions with infinite sums

    Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...
  30. C

    Mapping a Circle in the Complex Plane using f(z)=1/z

    Homework Statement What is the mapping of the circle of radius 1 centered at z=-2i under the mappinf f(z)=1/z The Attempt at a Solution I write the circle in polar form -2i+e^{ix} Now we invert it and multiply by the complex conjugate. so we get f(z)=...
  31. MAGNIBORO

    I Question about Complex limits of definite integrals

    Hi, I see a formula of gamma function and i have a question. (1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$ (2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$ (3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$ i understand the formula but...
  32. C

    Graphing a function under a complex mapping

    Homework Statement Illustrate the mapping of f(z)=z+\frac{1}{z} for a parametric line. The Attempt at a Solution the equation for a parametric line is z(t)=z_0(1-t)+z_1(t) so I plug z(t) in for z in f(z), but I don't get an obvious expression on how to graph it, I tried manipulating it...
  33. Vitani11

    Finding the roots of a polynomial with complex coefficients?

    Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?
  34. Eclair_de_XII

    How to relate complex multiplication to Cartesian products?

    Homework Statement "ℝ×ℝ and ℂ are very similar in many ways. How do you realize ℂ as a Cartesian product of two sets? Consider how complex numbers are multiplied; by grouping real and imaginary parts, show how the pattern of complex multiplication can be used to define multiplication in ℝ×ℝ...
  35. M

    Turning Complex Number z into Polar Form

    Homework Statement \frac{z-1}{z+1}=i I found the cartesian form, z = i, but how do I turn it into polar form?The Attempt at a Solution |z|=\sqrt{0^2+1^2}=1 \theta=arctan\frac{b}{a}=arctan\frac{1}{0} Is the solution then that is not possible to convert it to polar form?
  36. Arman777

    B Complex Numbers in a Simple Example that I am Very Confused

    There a simple math example that I am confused ##(\sqrt {-4})^2## Theres two ways to think 1-##\sqrt {-4}=2i## so ##(2i)^2=4i^2## which its ##-4## 2-##\sqrt {-4}##.##\sqrt {-4}##=##\sqrt {-4.-4}=\sqrt{16} =4## I think second one is wrong but I couldn't prove how, but I think its cause ##\sqrt...
  37. C

    What is the polar form of the given complex number without using the argument?

    Homework Statement Write the given complex number in polar form first using an argument where theta is not equal to Arg(z) z=-7i The Attempt at a Solution 7isin(\frac{-\pi}{2}+2\pi n) The weird part about this problem it asks me to not use the argument, The argument is the smallest angle...
  38. I

    Complex numbers De Moivre's theorem

    Homework Statement If $$C = 1+cos\theta+...+cos(n-1)\theta,$$ $$S = sin\theta+...+sin(n-1)\theta,$$prove that $$C=\frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}} cos\frac{(n-1)\theta}{2} \enspace and \enspace S = \frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}}sin\frac{(n-1)\theta}{2}$$...
  39. Vitani11

    Taylor expand (1+z)^n where |z | < 1 and n is any complex #

    Homework Statement Same as title. Homework Equations Taylor expansion. The Attempt at a Solution Okay - what?! I don't even know where to begin. I taylor expanded the function and pretended like n was just some number and that doesn't help. I've never learned this. How? Can you point me in...
  40. Ryaners

    Chemistry Calculating the number of water molecules in trans. metal complex

    Homework Statement [/B] I had an inorganic lab this week which involved making VO(acac)2 from VOSO4⋅xH2O. In order to calculate the percentage yield, I need to work out x, that is, the number of water molecules coordinated with the vanadyl sulfate n-hydrate before the reaction. I'm stuck...
  41. Alettix

    What Are the Loci for Different Values of Lambda in a Complex Equation?

    Homework Statement Consider the relation ## |\frac{z-i}{z*-i}| = \lambda ## where z = x + yi a) For ##\lambda = 1## show that the locus is a line in the complex plane and find its equation. b) What is the locus when ##\lambda = 0##? c) Show that for all other positive ##\lambda## the locus may...
  42. C

    Complex Conjugates in Quadratic Equations: Solving for z

    Homework Statement Solve each equation for z=a+ib z^{*2}=4z where z* is the complex conjugate The Attempt at a Solution I wrote z and z* in terms of x and iy , and tried solving for x and y, but I get quartic terms for y, it doesn't look like it will boil down, It was like over 2 pages of...
  43. K

    Complex Numbers Problem Solution Attempt

    Homework Statement If Z1+Z2+Z3=0 and Z1*Z2 + Z2*Z3 + Z3*Z1=0 and Z1, Z2, Z3 are all complex, what is the value of (|z1|+|z2|+|z3|)/(|z1*z2|+|z2*z3|+|z3*z1|) Homework EquationsThe Attempt at a Solution I tried to multiply the equations by the product of all conjugates and reach some...
  44. Ygggdrasil

    New Findings about the Evolution of Complex Cellular Life

    Humans, other animals, plants, fungi and almost all other forms of complex, multi-cellular life are known as eukaryotes. How eukaryotes evolved from simpler prokaryotic organisms is a major question in evolutionary biology. The current view is that eukaryotes evolved from the fusion between a...
  45. TheChemist_

    Determining graphical set of solutions for complex numbers

    Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...
  46. Rectifier

    Finding anitderivative using complex numbers and Euler

    I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...
  47. anemone

    MHB What is the Solution to the Complex Sum \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}?

    Evaluate \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}.
  48. Rasalhague

    I Simply-connected, complex, simple Lie groups

    I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex...
  49. Deniz

    Complex Power Homework: Is My Solution Right?

    Homework Statement y = 27 Homework Equations The Attempt at a Solution - I calculated the total impedance. - Divide it with the voltage to get the current. - Then I use the load impedance to find the voltage load. - And I calculated the complex power for the load. I am not comfortable...
  50. J

    Applied Books on complex valued functions and solution of PDE

    Hello folks, 1.- In geometry we study for example the conic sections, their exentricity and properties. I was wondering what part of the mathematical science studies the different properties of complex valued distributions. One example are the spherical armonics. I guess mathematicians have...
Back
Top