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. L

    Understanding Complex Numbers in Equations

    Homework Statement So the problem I have is this silly little equation.. $$\frac {z - 7}{z + 3} = i $$Homework Equations This is the thing, I don't think you need anything more advanced than basic algebra to solve this problem. The Attempt at a Solution And I've tried solving it doing the...
  2. J

    Did complex animals evolve at deep sea volcanic vents?

    We see complex animals such as crabs living near deep sea volcanic vents. (Reference: https://ocean.si.edu/ocean-life/invertebrates/hydrothermal-vent-creatures) This is causing speculation that similar life may be living near deep sea volcanic vents on other world such as Europa. Did these...
  3. pintu935

    I What does complex potential energy mean for a particle?

    Griffith says in problem 1.15 the potential energy has an imaginary part. my question is that any real case exists where the part of the potential energy is imaginary?
  4. Felipe Lincoln

    Proving that sin(z1+z2)=sinz1cosz2+sinz2cosz1 in complex plane (Arfken)

    Homework Statement Prove that ## \sin(z_1+z_2) = \sin z_1\cos z_2+\sin z_2\cos z_1## such that ##z_1,z_2\in\mathbb{C}## Homework Equations ##\sin z = \sum\limits_{n=1, \mathrm{ odd}}^\infty (-1)^{(n-1)/2}\dfrac{z^n}{n!} = \sum\limits_{s=0}^\infty (-1)^s\dfrac{z^{2s+1}}{(2s+1)!}## ##\cos z =...
  5. T

    Deflection of wave in dissipative media with a complex refractive index

    Homework Statement A monochromatic plane wave with wavelength 500µm is propagating through a dissipative medium with refractive index 1-0.0002i. It approaching the edge of the medium, and will pass out into free space. If the angle of incidence is not 90°, how much will the wave deflect as it...
  6. hideelo

    A Integrating Gaussians with complex arguments

    The integral I'm looking at is of the form \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right) Where K \in \mathbb{R} and J \in \mathbb{C} The book I am following (Kardar's Statistical Physics of Fields, Chapter 3 Problem 1) asserts that by completing the square this...
  7. Raziel13

    Fortran How Do You Read and Manipulate Complex Number Data in Fortran?

    Hello, I need to read a fortran data with complex numbers and real numbers, the first column is the real numbers, the second and third complex numbers (real, imaginary). I need to read the first 64 lines and then the next 64 lines in separate ways and save in a variable. for example read from...
  8. T

    MHB What is the minimal dimension of a complex realising a group representation?

    This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...
  9. I

    Courses How difficult is complex variables?

    Hello, I am a rising sophomore in Astronomy and Physics. I am taking complex variables next semester and was wondering the effort required to succeed in the class. There are some other classes I'd like to take, however I don't want to overload myself. I have taken up through multivariable calc...
  10. P

    Superposition Theorem with complex numbers

    1. Homework Statement . Figure 1 shows a 50 Ω load being fed from two voltage sources via their associated reactances. Determine the current i flowing in the load by: (a) Thevenin's theorem (b) Superposition (c) Transforming the two voltage sources and their associated reactances into current...
  11. S

    A Derivation of a complex integral with real part

    Hey, I tried to construct the derivation of the integral C with respect to Y: $$ \frac{\partial C}{\partial Y} = ? $$ $$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$ with $$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...
  12. N

    Quantizing the complex Klein-Gordon field

    I'm self-studying QFT and attempting exercise 2.2 on Peskin & Schroeder. First off, I'm a bit confused on the logic the authors use in the quantization process. They first expand the fields in terms of these ##a_{\vec{p}},a_{\vec{p}}^\dagger## operators which, if I understand correctly, is...
  13. T

    MHB Can you help me find the third zero of this complex polynomial?

    Hey, first off, I'm not sure if this is the right section. If another section is better, please let me know and I'll be more careful next time. So, my problem is with a degree 3 complex polynomial. I'm given one zero of the equation, but since it is a complex zero, I can use the conjugate too...
  14. K

    I Using complex numbers or phasor transform to solve O.D.E's

    Hi particular solution only. As an example of what I am talking about, this method works for this DE: $$ 4y' + 2y = 10\cos(x) \\ \\ 10 \cos(x) = \Re( 10 e^{j(x)} ) = \Re(e^{j(x)} \cdot e^{j(0)} ) \rightarrow \text{complex number that captures the amplitude and phase of 10 cos x is} \\ 10...
  15. T

    Complex Analysis prerequisite material review

    Homework Statement Identify the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}##. My pre-caculus is very rusty, so I am not sure if I am doing this correctly. Homework Equations ##x^2 +y^2= r^2## ##\forall z,z'\in\Bbb{C}, \vert zz'\vert =\vert z\vert\vert z'\vert##...
  16. B

    Nyquist Plot vs. Complex Function Plot

    This is not a homework problem, I just am confused a little about the differences between a Nyquist plot and the plot of a complex function. I believe they are the same given the domain of the plot of a complex function is for all real numbers equal to or greater than zero. However, I am having...
  17. alijan kk

    How can you simplify complex division with imaginary numbers?

    Homework Statement (1+2i+3i2)/(1-2i+3i2) answer options : a : 1 b: -i c: i d: 0 Homework Equations what is the most easy method to solve it , The Attempt at a Solution are they conjugate to each other ? if they are than z/zconjugate =1 , but how can...
  18. B

    Find roots of cubic polynomial with complex coefficient

    Homework Statement Find roots of $$ -\lambda ^3 +(2+2i)\lambda^2-3i\lambda-(1-i) = 0 $$ Homework EquationsThe Attempt at a Solution I tried my old trick I tried to separating the 4 terms into 2 pairs and try to find a common factor in the form of ##\lambda + z## between them, $$ -\lambda ^2...
  19. MountEvariste

    MHB Max Complex Matrices of Order $n$: $\lfloor n^2/4\rfloor + 1$

    Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$
  20. N

    Sinusoids as Phasors, Complex Exp, I&Q and Polar form

    Hi, I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form... 1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity? 2. But then...
  21. T

    Is the Shaded Region Outside the Circle in the W Plane?

    Homework Statement [/B] Homework EquationsThe Attempt at a Solution I had no problems with part a and was able to form the equation of the circle and get its centre/radius. It's part b that I'm stuck on. My notes show that for Z < 3 I would shade inside the circle but the mark scheme for...
  22. ubergewehr273

    Problem involving complex numbers

    Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.
  23. pairofstrings

    I What Is the New Dimension in Complex Number Graphs?

    Hi. If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1. 1. Is this "another...
  24. C

    MHB Stuck solving a complex equation for T

    Hi I wonder if anyone can help. I am not even sure I am on the right forum. I cannot solve this equation for t. It is the final sequence of a number of equations in a book about modelling athletic performance using bioenergetics. I had a high school maths education 40 years ago and I’m stumped...
  25. G

    Trouble computing the cosine of a complex number

    Mentor note: Thread moved from technical section, so missing the homework template. Hi all, I have a homework problem which asks me to compute the complex number cos(π/4 + π/4 i). I've been playing around with it for a while now and just can't seem to get the answer I get when using Wolfram...
  26. R

    I Geometric Meaning of Complex Null Vector in Newman-Penrose Formalism

    Reading Chandrasekhar's The mathematical theory of black holes, I reached the point in which the Newman-Penrose GR formalism is explained. Actually I'm able to grasp and understand the usage of null tetrads in GR, but The null tetrads used in this formalism, are very special, since are made by...
  27. W

    Superposition of Plane EM Waves Using Complex Notation

    Homework Statement I have a simple problem relating to the superposition of plane EM waves that I'd to try out using complex notation. Could anyone run through the work to see if my understanding is right? Many thanks in advance! The incident E bit of the wave is $$\vec{E}_I = E_0 \sin(ky -...
  28. CalcExplorer

    A How to Convert a Complex Logarithm to a Complex Exponential

    Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...
  29. binbagsss

    QFT, Noether and Invariance, Complex fields, Equal mass

    Homework Statement Question attached: Hi I am pretty stuck on part d. I've broken the fields into real and imaginary parts as asked to and tried to compare where they previously canceled to the situation now- see below. However I can't really see this giving me a hint of any sort unless...
  30. M

    Mathematica Complex output from a real integral

    Hi PF! I am integrating the following sin\[Theta][x_, \[Alpha]_] := Sqrt[(2 - 2 x^2)/( 3 - 4 x Cos[\[Alpha]] + Cos[2 \[Alpha]])] cos\[Theta][x_, \[Alpha]_] := Sqrt[(Cot[\[Alpha]] - x Csc[\[Alpha]])/( 1 + 2 x Cot[\[Alpha]] Csc[\[Alpha]] - 2 Csc[\[Alpha]]^2)] \[Rho][x_, \[Alpha]_] :=...
  31. nomadreid

    B Mistake in my complex exponentiation: where?

    I am sure I am overlooking something elementary, but playing around with exponentiation (this is not an assignment), I seem to be making a mistake somewhere. Please don't send me a link for a more compact way of getting the correct result; I wish to know what my particular mistake is. Suppose...
  32. N

    I Sinusoids as Complex numbers (multiplication query)

    DSP Guide .com has the highly rated textbook for digital signal processing. Chapter 30 pg 561 on Complex Numbers http://www.dspguide.com/ch30.htm (chapters are free to download) Hes talking about representing sinusoids with a complex number. Author states "Multiplying complex numbers A and...
  33. Rectifier

    Complex logarithm as primitive

    The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...
  34. B

    Calculating Integral Using Residue Theorem & Complex Variables

    Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...
  35. Rectifier

    Solving Complex Equation: $$ \bar{z} = z^n $$

    The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...
  36. Tspirit

    Fourier transform in the complex plane

    Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...
  37. alijan kk

    Complex number multiple choice

    Homework Statement If Z= (1)/(z conjugate) then Z : ? Homework EquationsThe Attempt at a Solution let z= a+bi the z conjugate= a-bi (a+bi)=(1)/(a-bi) (a+bi)(a-bi)=1 a2+b2=1 does it tell from this expresssion that the complex number is a pure real ?
  38. alijan kk

    Modulus of a complex number

    Homework Statement if z=(x-iy)/(x+iy) then modulus of z is : Homework EquationsThe Attempt at a Solution (x-iy)/(x+iy)= (x2-y2-2x(iy))/(x2+y2) i can't get the real part and the imaginary part to take the modulus : but the answer in any way could be = 1 ? the answer in the book is 1 .
  39. alijan kk

    Equation involved complex number

    Homework Statement Value of x and y , when (x+yi)2= 5+4i Homework EquationsThe Attempt at a Solution x2+2x(iy)-y2=5+4i x2-y2=5 -------> (1) 2x(iy)=4i (imaginary part) xy=2 --------> (2) solving the two equations x=2.388 and y=0.838 or x=-2.388 or y=-0.838 is this the right way to solve...
  40. Z

    Fluid Dynamics -- Use the Milne-Thomson circle theorem to show the complex potential for a fluid....

    Homework Statement Two equal line sources of strength k are located at x = 3a and x = −3a, near a circular cylinder of radius a with axis normal to the x, y plane and passing through the origin. The fluid is incompressible and the flow is irrotational (and inviscid). Use the Milne-Thomson...
  41. Matt Chu

    Proving a complex wave satisfies Helmholtz equation

    Homework Statement Consider a harmonic wave given by $$\Psi (x, t) = U(x, y, z) e^{-i \omega t}$$ where ##U(x, y, z)## is called the complex amplitude. Show that ##U## satisfies the Helmholtz equation: $$ (\nabla + k^2) U (x, y, z) = 0 $$ Homework Equations Everything important already in...
  42. G

    Prove that this function is holomorphic

    Homework Statement Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##...
  43. Leandro de Oliveira

    Calculators HP 50G complex numbers with a fraction?

    I have a problem to put the complex number in mode (1000/3, ∠36.87), apparently the division simbol gives some syntax error
  44. DoobleD

    I Fourier series of Dirac comb, complex VS real approaches

    Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...
  45. H

    I Intuitive understanding of Euler's identity?

    I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...
  46. W

    Numerical/Analytical Solution to a Complex Integral

    Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...
  47. kstorm19

    Question about complex power in three phase circuits

    Homework Statement Assume that the two balanced loads are supplied by an 840-V rms 60-Hz line. Load #1: Y-connected with 30+j40 Ω per phase, Load #2: balanced three-phase motor drawing 48 kW at a power factor of 0.8 lagging. Assuming abc sequence, calculate the complex power absorbed by the...
  48. W

    Complex Integral to error function

    Homework Statement I have an integral $$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$ where j and c are complex, which I'd like to write in terms of ## \text{erf}## I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##. 1) As ##p## tends...
  49. P

    How to calculate leverage with complex shaped levers

    1. The problem statement, all variables and given/known Does the shape or profile of a moment arm impact the torque created at the axel or fulcrum point Homework Equations T=fd[/B]The Attempt at a Solution Please see sketch is the torque created at position 1 in position to correct?
  50. SemM

    A Understanding Complex Operators: Rules, Boundedness, and Positivity

    Hi, from the books I have, it appears that some rules for operators, boundedness, positivity and possibly the definition of the spectrum regard real operators, and not complex operators. From the complex operator ##i\hbar d^3/dx^3 ## it appears that it can be defined as not bounded (unbounded)...
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