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. Math Amateur

    MHB Complex Integration - Conway - First Example on page 63 .... Section 1, Ch. IV .... ....

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding the first example on page 63 ... ... The the first...
  2. A

    MHB A complex numbers' modulus identity.

    I am searching for a shortcut in the calculation of a proof. The question is as follows: 2.12 Prove that: $$|z_1|+|z_2| = |\frac{z_1+z_2}{2}-u|+|\frac{z_1+z_2}{2}+u|$$ where $z_1,z_2$ are two complex numbers and $u=\sqrt{z_1z_2}$. I thought of showing that the squares of both sides of the...
  3. Math Amateur

    MHB Complex Valued Functions BV: John B. Conway Prop 1.3 Explained

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding another aspect of the proof of Proposition 1.3...
  4. Math Amateur

    MHB Understand Proposition 1.3 in Conway's Functions of Complex Variables I

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its...
  5. A

    A Complex wavenumber of Lamb waves in lossy materials

    Dear all, I have a question related to acoustic propagation in isotropic lossy media, more specifically generation of Lamb waves at fluid-solid interfaces. There goes the question: I am trying to obtain the Lamb wave velocity and attenuation dispersion curves of viscoelastic materials...
  6. D

    MHB Complex Residue Calculation at a Specific Point

    My residue is wrong. What is the solutions and the steps to achieve it ?
  7. Math Amateur

    MHB Smooth Paths in Complex Analysis .... Palka Example 1.3, Section 1.2 in Chapter 4 .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ... I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ... Example 1.3, Section 1.2, Chapter 4...
  8. W

    Can Jordan's Lemma be applied to clockwise contours?

    Homework Statement My notes state the Lemma as shown above. I believe one of the underlying conditions is that the arc we integrate over must be +ve oriented (anti-clockwise) in the Upper and Lower half of the Complex Plane. However my notes doesn't mention whether or not the result holds...
  9. R

    Modulus of a complex number with hyperbolic functions

    Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...
  10. W

    Contour Integrals: Working Check

    Homework Statement Hi all, could someone help me run through my work for these 2 integrals and see if I'm in the right direction? I'm feeling rather unsure of my work. 1) Evaluate ##\oint _\Gamma Z^*dz## along an anticlockwise circle of radius R centered at z = 0 2) Calculate the contour...
  11. A

    MHB Complex number geometrical problem

    Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$ I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.
  12. A

    MHB Mapping of a Circle in the Complex Plane

    I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings: a) w=e^(ipi)z b) w = 2z c) w = 2e^(ipi)z d) w = z + 2 + 2i I have managed to complete the question for a square and a rectangle as the points are easy to map as they are...
  13. Math Amateur

    MHB Continuity of Complex Functions ....Palka, Example 1.5, Chapter 3 .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 3: Analytic Functions ... I need help with some aspects of Example 1.5, Chapter 3 ... Example 1.5, Chapter 3 reads as follows: In the above text from Palka Chapter 3, Section 1.2 we...
  14. Adgorn

    B Square root of a negative number in a complex field

    Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...
  15. M

    Help with these two problems in complex analysis

    Homework Statement What is the argument of -4-3i, and -4+3i? Homework Equations tantheta=opposite/adjacent side The principle of argument is that the argument lies between -pi and pi (not including -pi). The Attempt at a Solution arg(-4-3i) = -pi + arctan(3/4) arg(-4+3i) = pi - arctan(3/4)...
  16. Math Amateur

    MHB Limits of Complex Functions .... Final Remark from Palka in Section 2.2 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects regarding an example in Palka's final remarks in Section 2.2 Limits of Functions ... Palka's final remarks in...
  17. Math Amateur

    MHB Limits of Complex Functions .... Example from Palka ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of a worked example in Palka's remarks in Section 2.2 Limits of Functions ... Palka's remarks in Section 2.2 which...
  18. Math Amateur

    MHB Continuity of Complex Functions .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of the proof of Lemma 2.4 ... Lemma 2.4 and its proof reads as follows: My questions are as follows: Question 1...
  19. Math Amateur

    MHB Parametrization Of Complex Curves .... Mathews And Howell, Example 1.22 .... ....

    I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 1.6 The Topology of Complex Numbers ... I need help in fully understanding a remark by M&H ... made just after Example 1.22 ... Example...
  20. T

    How Can You Simplify the Calculation of a Complex Number Raised to a Power?

    Hi I was hoping some of you would give me a clue on how to solve this complex number task. z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2 I think there must be some nice looking way to solve it. My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z| After...
  21. Y

    I Why does intensity mean anything if there's a complex number

    So say a wave is described by Acos(Φ), completely real. Then the to use Euler's Eq, we we say the wave is AeiΦ, which is expanded to Acos(Φ) + iAsin(Φ). We tell ourselves that we just ignore the imaginary part and only keep the real part. And if intensity is |AeiΦ|2, which is (Acos(Φ) +...
  22. Math Amateur

    MHB Theorem 1.8: Sets or Domains in the Complex Plane - Palka Ch.2

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with an aspect of Theorem 1.8 ... Theorem 1.8 (preceded by its "proof") reads as follows...
  23. A

    MHB Well-posedness of a complex PDE.

    I asked my question at math.stackexchange with no reply as of yet, here's my question: https://math.stackexchange.com/questions/2448845/well-posedness-of-a-complex-pde Hope I could have some assistance here. [EDIT by moderator: Added copy of question here.] I have the following PDE: $$u_t=...
  24. D

    What is the solution to the complex cosine equation without using logarithms?

    Homework Statement Solve the equation $$cos(\pi e^z) = 0$$Homework Equations I am not allowed to use the complex logarithm identities. $$ \cos z = \frac{e^{iz}+e^{-iz}}{2} $$ $$e^{i\theta}=\cos\theta+i \sin\theta$$ The Attempt at a Solution All I've gotten is $$\cos(\pi e^z)=0 \iff \pi...
  25. D

    I Complex Conjugates: Replacing i & Taking Alpha's Conjugate?

    Hi. If I have a complex number αe iα where α is complex what is the conjugate ? Usually I just replace i with -i but do i also take the conjugate of all α's ? Thanks
  26. K

    Is uniform continuity related to finding a bound on a complex function?

    Homework Statement Homework Equations $$a^2-b^2=(a-b)(a+b)$$ The Attempt at a Solution $$a^2=\sqrt{1-x_2^2}\,\,\, ,\ \ b^2=\sqrt{1-x_1^2}$$ $$|a^2-b^2|=\left| \sqrt{1-x_2^2}-\sqrt{1-x_1^2} \right|=\left| \sqrt[4]{1-x_2^2} - \sqrt[4]{1-x_1^2} \right|\cdot\left| \sqrt[4]{1-x_2^2} +...
  27. Runei

    I Complex representation of wave function

    When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis. u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right) Here U is the complex form of...
  28. W

    Complex Solutions to Oscillations

    Homework Statement Homework EquationsThe Attempt at a Solution I tried differentiating both sides of 3 and re-arranging it such that it started to look like equation 2, however i got stuck with 2 first order terms z' and couldn't find a way to manipulate it into a function z. I then tried...
  29. D

    Simplifying Complex Fractions, final step

    Homework Statement Please see attachment. Homework Equations I don't know how to get the final product on the ones with the question marks (textbook answers written next to them). I've gotten to the last step (except for # 29 but don't mind that one, I haven't exhausted all ideas). I've...
  30. N

    I Square root of a complex number

    if a is a complex number then sqrt(a^2)=? Is it is similar to Real Number? Help me please
  31. S

    A Complex notation in telegrapher's equations

    Hi everyone, I'm reading about the solution of the telegrapher's equations (e.g. the generalities are here https://en.wikipedia.org/wiki/Telegrapher%27s_equations ). Supposing we are treating only time t and space z, this is a second order partial differential equation on an infinite domain of...
  32. M

    Complex Exponentials Signal processing

    Hello everyone. Iam about to read a course in DSP and I can't get my head why complex exponentials are used as input signals that often? Is it just to simplify the math? If not, what exactly is the imaginary part of a complex exponential? Does it "do" anything special compared to a real valued...
  33. Isaac0427

    B Basic complex number math -- what am I doing wrong?

    For this, f and g are real functions, and k is a real constant. I have ##\psi = fe^{ikx}+ge^{ikx}## and I want to find ##\left|\psi \right|^2##. I went about this two different ways, and got two different answers, meaning I must be doing something wrong. Method 1: ##\psi =(f+g)e^{ikx}##...
  34. M

    B Complex numbers imaginary part

    Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase... Thanks in...
  35. A

    Odd and even in complex fourier series

    Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...
  36. PsychonautQQ

    I Simplicial complex geometric realization 1-manifold

    Prop 5.11 from John M. Lee's "Introduction to Topological Manifolds":If K is a simplicial complex whose geometric realization is a 1-manifold, each vertex of K lies one exactly two edges. This proposition confuses me. If we look at the geometric realization of a simplex with two vertices, then...
  37. K

    Differentiation with respect to a complex expression

    Homework Statement Homework Equations $$(x-a)(x+a)=x^2-a^2$$ The Attempt at a Solution I have to express ##~\displaystyle x^2+16=f\left( \frac{x}{x-1} \right)## I guess it has to be ##~\displaystyle \left( \frac{x}{x-1} \right)^n-a~## or ##~\displaystyle \left( \frac{x}{x-1} \pm a...
  38. L

    B Graphical Representation of a Complex Sphere

    @fresh_42 @FactChecker After thinking, I understood that the answer for this question might make the complex numbers comprehensible for me. My question in detail is as follow Let the equation of a sphere with center at the origin be ##Z1²+Z2²+Z3² = r²## where Z1 = a+ib, Z2 = c+id, Z3 = s+it...
  39. L

    B Representation of complex of square root of negative i with unitary power.

    Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?
  40. W

    Complex Plane Homework: Mobius Transformation Advice

    Homework Statement Homework EquationsThe Attempt at a Solution I'm not sure how to even begin this problem. My notes mentioned something about a Mobius Transformation but that's not something that I've been taught, and certainly not something I'm familiar with. Any advice would be greatly...
  41. W

    Complex Numbers: Euler's formula problem

    Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...
  42. karush

    MHB S10.03.25 Write complex number in rectangular form

    $\tiny{s10.03.25}$ $\textsf{Write complex number in rectangular form}$ \begin{align*}\displaystyle z&=4\left[\cos\frac{7\pi}{4} + i\sin \frac{7\pi}{4} \right]\\ \end{align*} $\textit{ok from the unit circle: $\displaystyle\cos{\left(\frac{7\pi}{4}\right)}=\frac{\sqrt{2}}{2}$}\\$ $\textit{and...
  43. D

    Abaqus modeling of a complex material

    hello all i'm trying to modal a complex material with matrix of material X and small spherical inclusion of material Y, i would like to have the ability to control the density of the inclusions and the surface properties between the material. does anyone know about a guide for the situation...
  44. L

    I How can we consider a complex number as two separate real numbers for in X and Y plane?

    How is it possible to ignore the addition sign and imaginary number without contradicting fundamental Mathematics? I find it difficult to understand.
  45. karush

    MHB Hcc8.11 change each to complex form and find product

    $\tiny{hcc8.11}$ $\textsf{Find product $(1+3i)(2-2i)$}\\$ $8 + 4i$ $\textsf{Then change each to complex form and find product. with DeMoine's Theorem}$ $\textit{ok looked at an example but ??}
  46. TeethWhitener

    I Complex scalar field commutation relations

    I'm trying to derive the commutation relations of the raising and lowering operators for a complex scalar field and I had a question. Let's start with the commutation relations: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...
  47. Ygggdrasil

    Omnigenetic model for complex traits

    Related to the recent discussions on this forum about the potential for genetically engineering humans in the future, researchers from Stanford University recently published a fascinating article in the journal Cell, looking into the genetics of complex traits, like height, as well as the...
  48. Math Amateur

    MHB What Is the Top Recommended Book on Complex Analysis for Beginners on MHB?

    What book do MHB members regard as the best for a rigorous but clear and (moderately) easily understood introduction to complex analysis? (Note - would be good if the book had hints to solutions of exercise.) Peter
  49. hideelo

    I SO(2n) representation on n complex fields

    If I have a lagrangian which has terms of the form ##\Psi^{\dagger}_\mu \Psi^\mu## then I can decompose the n complex ##\Psi## fields into 2n real fields by ##\Psi_\mu = \eta_{2\mu+1} + i\eta_{2\mu}##. When I look at the lagrangian now it seems to have SO(2n) symmetry from mixing the 2n real...
  50. T

    How can I create a unique projection of my company logo on light shades?

    I've been tasked with designing light shades for my companies new building. The current goal is to 3D print them, and include the company logo/name on them. The lights are for design purposes only, and aren't being used to illuminate the room. The hard part is, I want the logo/name clearly...
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