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. Santilopez10

    I What is the value of the integral for higher order poles in the real axis?

    Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole. It is easy to prove that if the pole is of simple order, the value of the...
  2. Adesh

    How to solve a complex equation to get the current?

    I was reading The Feynman Lectures on physics http://www.feynmanlectures.caltech.edu/I_23.html chapter 23, section 4. In it he derives the equation for current when inductor, resistor and capacitor is connected in series with an alternating voltage source, he derives this equation:-...
  3. Demystifier

    A Complex Numbers Not Necessary in QM: Explained

    [Note from mentor: This was split off from another thread, which you can go to by clicking the arrow in the quote below] Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.
  4. Toolkit

    Help w/ Circuit Theory: Complex Numbers & Voltage

    Hi, I'm working on an assignment for circuit theory, and I'm wondering if someone could let me know if I'm heading in the right direction? 1) I have a voltage value of 120 /_0 (polar form), from this can I assume that Arctan (a/b) =0, so voltage =120 in phase? Therefore, V =120+J0, where V...
  5. D

    Textbook Recommendations: Complex Analysis

    Hello, I was interested in learning more about complex analysis. Also, very interested in analytic continuation. Can anyone recommend a good text that focuses on complex analysis. Also, is there a good textbook on number theory that anyone recommends? Thanks! <mentor - edit thread title>>
  6. F

    B How can complex numbers be elevated to complex powers?

    Hello I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows: ##z^{w}##...
  7. MakVish

    Complex Analysis: Find Analytic Functions w/ |ƒ(z)-1| + |ƒ(z)+1| = 4

    Homework Statement Find all analytic functions ƒ: ℂ→ℂ such that |ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ and ƒ(0) = √3 i The Attempt at a Solution I see that the sum of the distance is constant hence it should represent an ellipse. However, I am not able to find the exact form for ƒ(z). Any help...
  8. Santilopez10

    How Does the Sinc Function Integral Relate to Quantum Collision Theory?

    Homework Statement The following is a problem from "Applied Complex Variables for Scientists and Engineers" It states: The following integral occurs in the quantum theory of collisions: $$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$ where p is real. Show that $$I=\begin{cases}0 &...
  9. M

    I Do complex roots have a physical representation on a curve?

    If we have y=x^2 -4. This is represented by curve intersect x-axis at (-2, 0) and (2, 0) or if we wish to find it algebraically we set y =0 then we solve it. The roots must lie on the curve. when y=x^2+4 the roots are 2i and -2i "complex" consequently there is no intersection with x-axis, so...
  10. P

    I Complex conjugate of an inner product

    Hi everyone. Yesterday I had an exam, and I spent half the exam trying to solve this question. Show that ##\left\langle\Psi\left(\vec{r}\right)\right|\hat{p_{y}^{2}}\left|\phi\left(\vec{r}\right)\right\rangle =\left\langle...
  11. S

    How to write the complex exponential in terms of sine/cosine?

    I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...
  12. Ventrella

    I Iterating powers of complex integers along axes of symmetry

    I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...
  13. Christopher Rourk

    A Is the Fenna-Matthews-Olson complex a quantum dot?

    The FMO complex has a size that is within the typical size range for quantum dots, and absorbs photon energy at what appears to be an effective bandgap between 2-3 eV. While various techniques have been used to investigate the behavior of the FMO complex, such as femto photography or...
  14. iVenky

    I Complex permitivity of good conductors

    We can define complex permitivity of any medium as \epsilon=\epsilon'-j\epsilon'' And the loss tangent as tan \delta = \frac{\omega \epsilon'' + \sigma}{\omega \epsilon'} The question that I have is for good conductors. I read that for good conductors, we are dominated by σ rather than...
  15. T

    I Exploring the Connection between Trigonometric and Exponential Functions

    Hi all: I really do not know what to ask here, so please be patient as I get a little too "spiritual" (for want of a better word). (This could be a stupid question...) I get this: eiθ=cosθ+isinθ And it is beautiful. I am struck by the fact that the trig functions manifest harmonic...
  16. binbagsss

    Complex function open set, sequence, identically zero, proof

    Homework Statement Hi I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere. ##f: u \to C##Please see attachment...
  17. M

    I Singular matrices and complex entries

    Hi PF! Let's say we have a matrix that looks like $$ A = \begin{bmatrix} 1-x & 1+x \\ i & 1 \end{bmatrix} \implies\\ \det(A) = (1-x) -i(1+x). $$ I want ##A## to be singular, so ##\det(A) = 0##. Is this impossible?
  18. S

    Engineering Solving a Circuit with a Complex Source

    Homework Statement An image of the problem is attached. I need to solve for ic(t) and vc(t) by adding a complex source. Homework EquationsThe Attempt at a Solution I don’t know where to start here. I don’t understand the question, and I can’t find the information I need in my notes. Can...
  19. R

    I If the wave function is complex and the measurement is real

    Would not any real measurement taken on a complex state logically require that the results of the measurement have less information than the state? Although I’m just beginning in QM, it appears to me unsurpring that a real measurement on the complex wave function seems to collapse the wave...
  20. e0ne199

    Engineering Problems about Zin in complex circuit analysis

    1. Homework Statement the problem is my answer for question (a) is not the same as the answer provided by the question, i get 2.81 - j4.49 Ω while the answer demands 2.81 + j4.49 Ω Homework Equations simplifying the circuit, details can be seen below The Attempt at a Solution...
  21. A

    Online app which plots F(z) in the complex plane

    I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given. It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z. One possible...
  22. S

    Finding a Complex Number Given Arg and Modulus

    Homework Statement If ##\text{arg}(w)=\frac{\pi}{4}## and ##|w\cdot \bar{w}|=20##, then what is ##w## of the form ##a+bi##. Homework EquationsThe Attempt at a Solution The only way for the argument of ##w## to be ##\frac{\pi}{4}## is when ##a+bi## where ##a=b \in \mathbb{Z}## right?
  23. Asawira Emaan

    MHB Solving Complex Number With Negative Fractional Exponent: i^(-21/2)

    Kindly help me with this. Solve i^(-21/2) Note: i means iota.
  24. A

    I Equating coefficients of complex exponentials

    I have an equation that looks like ##i\dot{\psi_n}=X~\psi_n+\frac{C~\psi_n+D~a~\psi^\ast_{n+1}+E~b~\psi_{n+1}}{1+\beta~(D~\psi^\ast_{n+1}+E~\psi_{n+1})}## where ##E,b,D,a,C,X## are constants. I have the ansatz ##\psi_n=A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast}##, ##x## and ##A_n,B_n## are complex...
  25. karush

    MHB Ap1.3.51 are complex numbers, show that

    $\textsf{ If $z$ and $u$ are complex numbers, show that}$ $$\displaystyle\bar{z}u=\bar{z}\bar{u} \textit{ and } \displaystyle \left(\frac{z}{u} \right)=\frac{\bar{z}}{\bar{u}}$$ok couldn't find good example on what this is and I'm not good at 2 page proof systemsso much help is mahalo
  26. E

    Finding z component of center of mass of a complex shape

    Homework Statement The rigidly connected unit consists of a 2.5-kg circular disk, a 2.8-kg round shaft, and a 4.2-kg square plate. Determine the z-coordinate of the mass center of the unit.Homework Equations ∑zm/∑m The Attempt at a Solution Circular disk: mass = 2.5 kg z = 0 zm = 0 Round...
  27. H

    MHB Are Non-Ordered Numbers More Than Complex Numbers?

    1. The complex number are not ordered. Which else number are not ordered? 2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?
  28. Math Amateur

    MHB Complex and Real Differentiability .... Remmert, Section 2, Ch. 1 .... ....

    I am reading Reinhold Remmert's book "Theory of Complex Functions" ...I am focused on Chapter 1: Complex-Differential Calculus ... and in particular on Section 2: Complex and Real Differentiability ... ... ...I need help in order to fully understand the relationship between complex and real...
  29. Math Amateur

    I Complex & Real Differentiability ... Remmert, Section 2, Ch 1

    I am reading Reinhold Remmert's book "Theory of Complex Functions" ... I am focused on Chapter 1: Complex-Differential Calculus ... and in particular on Section 2: Complex and Real Differentiability ... ... ... I need help in order to fully understand the relationship between complex and real...
  30. N

    Does a Circular Capacitor with a Dielectric Radiate an Electromagnetic Field?

    Hi guys, Consider a circular capacitor with a disk of radius a and plate separation d, as shown in the figure below. Assuming the capacitor is filled with a dielectric constant epsilon and the capacitor is fed by a time harmonic current I0 (a) Find the magnetic field distribution inside the...
  31. H

    MHB Why is ln(k) a Complex Number When k is a Positive Integer?

    Why ln(k) when k is a possitive integer, ln(k) is a complex number?
  32. T

    How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?

    Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...
  33. N

    I Understanding what the complex cosine spectrum is showing

    The complex exponential form of cosine cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t) The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right? The...
  34. G

    Is the Integral Zero for Closed Paths in Complex Analysis?

    Hey, I have been stuck on this question for a while: I have tried to follow the hint, but I am not sure where to go next to get the result. Have I started correctly? I am not sure how to show that the integral is zero. If I can show it is less than zero, I also don't see how that shows it...
  35. Measle

    Complex Analysis - sqrt(z^2 + 1) function behavior

    Homework Statement Homework Equations The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi] The Attempt at a Solution The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
  36. Measle

    I Principal branch of the log function

    I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables. On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
  37. V

    Complex Kinematics and Dynamics

    Homework Statement Two pucks (5 kg each) made of Teflon are on a long table, also made of Teflon. Puck A is sitting at rest on the left end of the table. Puck B is 15 m away at the right hand end of the table, and is travelling toward Puck A with an initial speed of 0.5 m/s. A person on the...
  38. V

    Complex Kinematics and Dynamics

    Homework Statement Two pucks (5 kg each) made of Teflon are on a long table, also made of Teflon. Puck A is sitting at rest on the left end of the table. Puck B is 15 m away at the right hand end of the table, and is travelling toward Puck A with an initial speed of 0.5 m/s. A person on the...
  39. Safder Aree

    Contour Integration over Square, Complex Anaylsis

    Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
  40. K

    Solving Complex Equations: z2+2(1-i)z+7i=0

    Homework Statement So it is pretty straight forward, solve this. z2+2(1-i)z+7i=0 Homework Equations z2+2(1-i)z+7i=0 (-b±√(b2-4ac))/2a The Attempt at a Solution So what I would do first is solve 2(2-1)z, I get (2-2i)z=2z-2iz we now have z2-2iz+7i+2z=0 Now I don't really know what to do because...
  41. J

    Problem with Complex contour integration

    Homework Statement I want to compute ##I=\int_C \dfrac{e^{i \pi z^2}}{sin(\pi z)}##, where C is the path in the attached figure (See below). I want to compute this by converting the integral to one whose integration variable is real.Homework Equations There are not more relevant equations. The...
  42. K

    Is f(x) = (x-iy)/(x-1) a Continuous Function?

    Homework Statement Determine if the following function is continuous: f(x) = (x-iy)/(x-1) Homework Equations How do find out if a function is continuous without graphing it and without a point to examine? I know I've learned this, probably in pre-calculus too, but I'm blanking The Attempt at...
  43. Mr Davis 97

    I Complex exponential to a power

    Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##. However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for...
  44. D

    I Solving Trig Integrals with Residue Theorem

    Hi. I have looked through an example of working out a trig integral using the residue theorem. The integral is converted into an integral over the unit circle centred at the origin. The singularities are found. One of them is z1 = (-1+(1-a2)1/2)/a It then states that for |a| < 1 , z1 lies...
  45. C

    MHB Plus or minus question for the complex log

    Dear Everybody, I am having troubles figuring out why the plus or minus sign in this problem. The question is: Solve the equation $\sin\left({z}\right)=2$ for $z$ by using $\arcsin\left({z}\right)$ The work for this problem is the following: $\sin\left({z}\right)=2$...
  46. maistral

    I  Evaluation of a certain complex function

    Hi. I would like to ask regarding this function that keeps on cropping up on my study (see picture below). What I did is simply substitute values for A and b and I noticed that it ALWAYS results to a real number. If possible, I would like to obtain the "non imaginary" function that is...
  47. D

    I Complex analysis - removable singular points

    Hi. I have 2 questions regarding removable singular points. 1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is...
  48. F

    Complex numbers sequences/C is a metric space

    Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
  49. thecastlingking

    Complex exponential Fourier series coefficients?

    Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?
  50. C

    MHB Set of points on the complex plane

    Dear Everybody, I am wanting to check the solution to this question: Sketch the set of points determined by the given conditions: a.) $\left| z-1+i \right|=1$ b.)$\left| z+i \right|\le3$ c.)$\left| z-4i \right|\ge4$ work: I know (a.) is a circle with radius 1 and its center at (-1,1) on the...
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