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. D

    Complex numbers: Solve ##Z^2\bar{Z}=8i##

    Solve ##Z^2\bar{Z}=8i## i am confused on how to proceed i have tried to substitute ##z=a+ib## solve the conjugate and the square, then separate the real from the imaginary and put all in a system, but becomes too complicated
  2. Antarres

    A A question about a complex integral

    I was trying to calculate an integral of form: $$\int_{-\infty}^\infty dx \frac{e^{iax}}{x^2}$$ using contour integration, with ##a>0## above. So I would calculate a contour integral with contour being a semicircle that goes along the real axis, closing it in positive direction in the upper...
  3. C

    MHB Solving a complex value equation

    Dear Everyone, I have a question about how to solve for x near the end of the problem: \[ 1+2\sinh^{2}(z)=0 \] Here is the solution and work: \[ 1+2\sinh^2(z)=0 \\ \sinh^2(z)=\frac{-1}{2}\\ \sqrt{\sinh^2(z)}=\pm \sqrt{\frac{-1}{2}}\\ \sinh(z)=\pm i\frac{1}{\sqrt{2}}\\ \] Then we can split...
  4. e_mts

    Real and Complex representations of an oscillation equation

    I've been trying to continue my education by self-teaching during quarantine (since I can't really go to college right now) with the MIT Opencourseware courses. I landed on one section that's got me stuck for a while which is the second part of this problem (I managed to finish the first part...
  5. Remixex

    Contour integration around a complex pole

    $$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$ So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise. The generalized complex integral would be $$\int_{C} \frac{e^{-i \alpha...
  6. A

    Complex numbers: convert the exponential to polar form

    Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a] 4e^(-j)=4 ∠-180° c=4, 4=(√a^2 +b^2) solving for a : a=(√16-b^2) θ=tan^(-1)[b/a]= -1 b/(√16-b^2)=...
  7. M

    Mathematica Find the real zeros of a complex equation

    Hi PF! Here's my equation 0.5 Sech[0.997091 Sqrt[ 2.4674 + \[Xi]\[Xi]^2]] (-1. ((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2) Cosh[ 0.997091 Sqrt[2.4674 + \[Xi]\[Xi]^2]] - 1. \[Sqrt](((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2)^2 Cosh[ 0.997091 Sqrt[2.4674 +...
  8. B

    Choosing proper coordinates in a complex 2 pulley system

    FBD Block 1 FBD Block 2 FBD Pulley B I'm mainly concerned with the coordinate system direction in this problem, but just to show my attempt, here are the equations I got from the system. ##-T_A + m_1g = m_1a_1## ##T_B - m_2g = m_2a_2## ##T_A - 2T_B = 0## Using the fact that the lengths...
  9. kolleamm

    Is it possible to polarize a complex 3d shape?

    I have this idea for LED eyes. Basically its an LED screen behind a half spherical shape. I want the user only to see what's on the display if they are looking directly at it. So for that I would need to polarize the half sphere. The half sphere can be made out of anything, the problem is how do...
  10. D

    Finding Domain for Complex Numbers: |y-x|<=2, |x|<=2

    i have to find such domain z=x+iy , y,x∈ℝ , |y-x|<=2, |x|<=2 i'm confused with |y-x|<=2, how should i proceed ? with abs of x i am ok.
  11. D

    Analyzing a Complex Line Integral Using Substitution and Logarithmic Properties

    if ## \gamma (t):= i+3e^{2it } , t \in \left[0,4\pi \right] , then \int_0^{4\pi} \frac {dz} {z} ## in order to solve such integral i substitute z with ##\gamma(t)## and i multiply by ##\gamma'(t)## that is: ##\int_0^{4 \pi} \frac {6e^{2it}}{i+3e^{2it}}dt=\left.log(i+3e^{2it}) \right|_0^{4...
  12. anemone

    MHB Finding Real Part of $z$ for Complex Numbers

    Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
  13. C

    I Rewriting a complex number for use in an analytic computation

    Consider an equation, $$\tilde{x_0} = \ln(X+ i\delta),$$ where X may be positive or negative and ##0< \delta \ll 1##. Now, if ##X>0## this evaluates to ##\ln(X)## in some limiting prescription for ##\delta \rightarrow 0## while if ##X<0##, we get ##\ln(-X) + i \pi. ## Now, consider...
  14. M

    MATLAB Plotting a 3D image from a matrix for a complex domain

    Hi PF! Each element of an ##n\times m## matrix is complex valued. In the following code, I call this "domain". There is also an ##n\times m## matrix that is real valued, below I call this "f". I'd like to plot a 3D image where the ##x-y## plane is the complex plain given by the coordinates...
  15. M

    A Eigenvalue problem: locating complex eigenvalues via frequency scan

    Hi PF! Here's an ODE (for now let's not worry about the solutions, as A LOT of preceding work went into reducing the PDEs and BCs to this BVP): $$\lambda^2\phi-0.1 i\lambda\phi''-\phi'''=0$$ which admits analytic eigenvalues $$\lambda =-2.47433 + 0.17337 I, 2.47433 + 0.17337 I, -10.5087 +...
  16. blazh femur

    Is randomness real or the inability to perceive hyper complex order?

    How did you find PF?: random Brownian motion Is randomness real or is it simply defined as such due to our inability to perceive hyper complex order? Randomness is a troublesome word. I'd feel better if I knew it was an objective phenomenon and not merely a placeholder description of...
  17. M

    I Know a simple, linear, complex, eigenvalue BVP?

    Hi PF! I'm trying to find a 1D, linear, complex, 2nd order, eigenvalue BVP: know any that admit analytic solutions? Can't think of any off the top of my head. Thanks!
  18. Mayhem

    B Computing a tricky complex math problem - where did I go wrong?

    I stumbled upon the following problem on instagram: $$L = \left (\frac{-1+i\sqrt{3}}{2}\right )^6+\left (\frac{-1-i\sqrt{3}}{2}\right )^6+\left (\frac{-1+i\sqrt{3}}{2}\right )^5+\left (\frac{-1-i\sqrt{3}}{2}\right )^5$$ The idea is to compute it. Using a calculator, it is supposed to equal 1. My...
  19. kaycha

    Complex motion equation (projectile with changing mass and thrust)

    I need an equation to predict the flight path of a changing mass projectile under changing thrust. Any thoughts?
  20. Ssnow

    I Schrodinger equation on the complex disk

    Hi to all member of the Physics Forums. I have this question: it is possible consider the analogue of the Schrodinger equation on the plane with configuration space ##(x,p)\in\mathbb{R}^4## on the complex disk ##\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}##? Ssnow
  21. H

    Is the Proof for a Complex Inner Product Space Correct?

    Summary:: Inner Product Spaces, Orthogonality. Hi there, This my first thread on this forum :) I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill) Chapter 7 - Inner Product Spaces, Orthogonality. Using some particular values for u and v, I...
  22. bob012345

    I Is Quantum Mechanics Infinitely More Complex than Classical Mechanics?

    Please critique this text. It came from a research article* I found but I'm only interested if the sentence is 100% accurate or not and not in the specifics of the article itself. Are they suggesting Hilbert space is always infinite? Thanks. Quantum mechanics is infinitely more complicated than...
  23. P

    Complex Scattered polarization vector? (Conceptual)

    I guess I will show my work for substantiating equation 1 and hopefully by doing so someone will be able to point out where I could generalize. ##\langle \vec{S}_{rad} \rangle = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{rad} \times \vec{B}^*_{rad}\right) = \frac{1}{2 \mu} \mathfrak{R} \left(...
  24. G

    Chemistry Investigating dsp2 & sp3 Configurations in Ni2+ Complexes

    I was expecting Ni2+ to be present also in the answer as it can give dsp2 and sp3 configuration. [Ni(CN)4]2- and [Ni(NH3)4]2+ have dsp2 and sp3 respectively, right?
  25. minimoocha

    MHB Solve Complex Integration: Find 2.36 Area of y=-x/2e+1/e+e & y=e^x/2

    The area of two lines that I need to find is 2.36, however i need this in exact form. The lines are y=-x/2e+1/e+e the other line is y=e^x/2 Since y=-x/2e+1/e+e is on top it is the first function. A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2 If you could please help!
  26. agnimusayoti

    Absolute value of trigonometric functions of a complex number

    So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then 1. Real part: ##\sin x \cosh y## 2. Imaginary part: ##\cos x \sinh y## If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }## How to...
  27. V

    A The map from a complex torus to the projective algebraic curve

    I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...
  28. H

    Complex Analysis - find v given u

    Solution Attempt: \begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} = (x^2+y^2)^{-1} -x (x^2+y^2)^{-2} (2x) = (x^2+y^2)^{-1} - 2x^2 (x^2+y^2)^{-2} \\ \rule{0mm}{18pt} \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} = -x (x^2+y^2)^{-2} (2y) =...
  29. il postino

    Graph complex numbers to verify z^2 = (conjugate Z)^2

    Hello! :smile: I am locked in an exercise. I must find (and graph) the complex numbers that verify the equation: ##z^2=\bar z^2 ## If ##z=x+iy## then: ##(x+iy)^2=(x-iy)^2 ## and operating and simplifying, ##4.x.yi=0 ## and here I don't know how to continue... can you help me with ideas? thanks!
  30. S

    I Derivative of a complex function along different directions

    Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...
  31. C

    Streamlines from a complex potential

    I've been trying this problem for a long time. By operating the lower part of the logarithm and clapping the real and imaginary part of the logarithm, I have come to the conclusion that the correct lines must be those in which it is true that: $ d \ frac {(x ^ 2 + y ^ 2-a ^ 2) ^ 2 + 4y ^ 2a ^...
  32. L

    B Is it possible to find complex numbers Cn, so that both equations are satisfied?

    Is it possible two find complex numbers ##C_n##, so that both equations are satisfied \sum^{\infty}_{n=1}nC_n=0 and \sum^{\infty}_{n=1}|C_n|^2=1 ?
  33. Jonathanos

    Courses Complex Analysis Courses or Complex Variable Courses?

    Hello, My university offers a couple Complex Analysis courses, among them there is one with the following description: Introduction to complex variables: "substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical...
  34. S

    Mathematica Derivative of the Real Part of a Complex Function (Mathematica)

    When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?
  35. e101101

    I Phase Plane Diagram w/ Complex eigenvalues

    Is the spiral I drew here clockwise or counterclockwise ? What’s a trick to know whether it’s going CCW or CW. Thanks!
  36. V

    A Closure of constant function 1 on the complex set

    I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...
  37. Math Amateur

    MHB Proving Tapp's Proposition 2.4: Complex Matrices as Real Matrices

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focused on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4. Proposition 2.4...
  38. nomadreid

    Angles of certain complex numbers

    reducing it to various forms: for example, the one in the title, or 2*pi*k(ln m) = a(ln(n/m)), and so forth. My gut feeling is that it is true (that no such foursome exists), but manipulations have not got me anywhere. Anyone push me in the right direction? I am probably overlooking something...
  39. M

    Taking the derivative of complex functions

    So just based on the cauchy riemann theorem, I think: Ux = 2 = Vy = 2xy, so f(z) is differentiable on xy = 1, and also that Vx = y^2 = -Uy = 0. That doesn't make sense to me because if 0 = y^2, then y = 0, yet that wouldn't satisfy xy = 1, would it? Furthermore, I'm not sure how I would...
  40. jk22

    I Solving the Wave Equation via complex coordinates

    I'm looking for material about the following approach : If one suppose a function over complex numbers ##f(x+iy)## then ##\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial...
  41. r12214001

    Chemistry Complex ion oxidation state (with pic)

    question fig: solution manual: my solution: oxidation state of central cobalt is +6 due to 6 oxygen surrounding it,The other cobalt is +2 due to 2 oxygen surrounding it with NH3 ligand which is no count for oxidation state.
  42. A

    Complex analysis: find contradiction of a relationship

    I have reached a conclusion that no such z can be found. Are there any flaws in my argument? Or are there cases that aren't covered in this? Attempt ##\log(\frac{1}{z})=\ln\frac{1}{|z|}+i\arg(\frac{1}{z})## ##-\log(z)=-\ln|z|-i\arg(z)## For the real part...
  43. M

    Drawing sets of Complex Variables

    I tried saying z = x + iy, then squared both sides so that I would get something that looked like: |z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach. For that...
  44. K

    I Why the integral of a complex exponential can't be equal to zero?

    I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero. On one hand, as we can in principle choose whatever values we like for ##m##...
  45. J

    Solving a Complex Problem: Z_N, p(r), and Averages

    Hi everyone, I want to post this exercise and my attempt to a solution since there are a couple of points I am not entirely sure of and I might need your help. I'll address them while posting my solution 1) ##Z_N = (Z_1)^N##. I can evaluate ##Z_1## integrating with respect of both parts of the...
  46. G

    What is the complex conjugate of this wave function?

    I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0. Thank you
  47. BillTre

    Complex Series of Geologic Processes Generated Seizmic Humming

    I find this interesting. A pretty detailed description, of a complex geological series of events, that can't be directly seen. Here's my summary: In 2018 an usual humming was picked up by seismic equipment an island off Africa, a magma pool drained, flowed up a dyke, when horizontal, and then...
  48. L

    Solving a complex equation (damping/exponential-decay) like this....?

    ##e^{-0.6x}\sin{(5x)}-0.1=0## I have posted my graphical solution to this problem. But, how do I solve this numerically/mathematically without graphing it?
  49. 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...
  50. 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.
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