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. S

    Simplifying natural log of complex number

    Homework Statement The problem is to sketch lines of constant u and v in the image plane for the function Log[(z+1)/(z-1)]. Homework Equations z=x+iy The Attempt at a Solution In order to do this I have to get the expression into u+iv form, so that I can read off and manipulate the u and v...
  2. L

    Complex numbers - I'm sure this is an easy - Argand diagram

    Homework Statement OABC is a square on an Argand diagram. O Represents 0, A represents -4 + 2i, B Represents z, C represents w and D is the point where the diagonals of the square meet. (There are two possible squares that meet this criteria) Find the complex number represented by C and D in...
  3. T

    Complex integral for z-transform causality

    This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property \lim_{|z|\to\infty} \frac{X(z)}{z} = 0 Is that sufficient to guarantee that \frac{1}{2\pi...
  4. A

    Graduate course as a UG: Complex Analysis or Topology?

    As an undergraduate, which graduate-level course will prepare me better for grad school, Complex Analysis or Topology? I probably can't fit both into my schedule, but I can definitely fit one. I have already taken undergraduate complex analysis and I'm taking now undergraduate topology. My...
  5. S

    MHB Complex numbers and conjugates

    Hi everyone, Can you please assist with the following problem? The complex numbers z and w are such that for the real variable x, (x-z)(x-w)=ax2+bx+c for real a,b and c. By letting z=p+qi and w=r+si, prove that z and w must be conjugates of one another.So far, I have determined that a=1...
  6. Summer95

    Derivative of complex exponential differs by a sign

    I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.
  7. B

    Complex Pulley/Force/Circular motion question.

    Homework Statement A rope is wrapped through an angle θ about a horizontal pole (So for ex- ample, θ = 2π would imply the rope goes around one full time). The rope and the pole have a static friction coeffecient of μ, and the pole is of radius r. From one end of the rope hangs a mass m. How...
  8. M

    Complex Direct Stress equation

    Attached image with problem. 2 Questions; 1. δn.|AB| = δx.|BC|.cosΘ becomes δn = δx.|BC|.cosΘ / |AB| This is dividing the entire "δx.|BC|.cosΘ" equation by |AB| or just the |BC| part? Is there a difference? 2. Half way down page, boxed equation. How does δx.|BC|.cosΘ / |AB| = δxcos^2Θ and...
  9. G

    What is the Complex Angle for 2√(3)-2i?

    Homework Statement I need to find the complex angle θ for: 2√(3)-2i in polar form. Homework EquationsThe Attempt at a Solution If I draw a complex plane, I can see that 2√3 on the real axis gives 0°, and -2i gives 3π/2 (270°), but it's incorrect. How can I find the complex angle of 2√(3)-2i...
  10. C

    Finding solution of equation in complex domain?

    (1+a)n = (1-a)n I tried following: (1+a)n = (1-a)n [(1+a)/(1-a)]n=1 but what can i do next?
  11. J

    Solve Complex Integral: Find Residues & Singularities

    Homework Statement Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral ## \int_C \frac{z}{1-\cos{z}}dz, ## where ##C=\{z:|z|=1\}## and positively oriented. Homework Equations [/B] Residue theorem: Let C be a simple closed...
  12. J

    How can the number of zeros of a complex function in a given domain be proven?

    Homework Statement Let ##D={z : |z| <1}##. How many zeros (counted according to multiplicty) does the function ##f(z)=2z^4-2z^3+2z^2-2z+9## have in ##D##? Prove that you answer is correct. Homework Equations 3. The Attempt at a Solution [/B] The function has no zeros in ##D##, which can be...
  13. RJLiberator

    Proof involving complex conjugates and Matrices

    Homework Statement Show that (A+B)*=A*+B* Homework Equations I think I am missing a property to prove this. The Attempt at a Solution This should be easier then I am making it out to be. But I seem to be missing one key property to do this. A*+B* is just A(ij)*+B(ij)* = Right hand side...
  14. davidbenari

    Complex Mapping - Is transforming boundaries enough?

    Say I will make the transformation from the ##z## plane to the ##w## plane. Moreover, I'll transform a region ##R## with boundary ##C## in the ##z## plane to something in the ##w## plane. Why is it that if I know the equations for ##C## then I can transform these and immediately know that ##R##...
  15. B

    Complex number and split complex number

    Is correct to afirm that a solution of a quadratic equation or is a complex number or is a split complex number?
  16. Sirsh

    Complex Number Division and Addition

    Homework Statement This is not for a mathematics unit but is part of an electrical question I'm trying to solve but I cannot solve this equation. The complex numbers Zp and Zr are both real and imaginary, whereas Xm is purely imaginary. Homework Equations Zp = (Xm*Zr)/(Xm+Zr) Zp =...
  17. davidbenari

    ##\int \frac{dz}{z} ## along a line on the complex plane

    Some time ago I stumbled upon the integration ##\int \frac{dz}{z} ## along a line on the complex plane. I was confused because ##Ln(z)## is a multivalued function but apparently the way you do it is by only considering the principal branch from ##[-\pi,\pi]##. But I don't understand this at...
  18. M

    2D Projective Complex Space, Spin

    Just reviewing some QM again and I think I'm forgetting something basic. Just consider a qubit with basis {0, 1}. On the one hand I thought 0 and -0 are NOT the same state as demonstrated in interference experiments, but on the other hand the literature seems to say the state space is...
  19. B

    Does Minimum of Complex Set Subset Exist?

    Homework Statement The following doesn't come from a textbook, and I am very uncertain whether it is true or false. Suppose that ##B \subseteq \mathbb{C}## is a convex set, and consider the set ##L_B := \{|b|: b \in B \}##. Homework EquationsThe Attempt at a Solution My question is, will...
  20. B

    Define the complex number Z = u^v

    If I define the complex number z = r exp(i θ) how z = uv, so, how to express u and v in terms of r and θ? u(r, θ) = ? v(r, θ) = ? And the inverse too: r(u, v) = ? θ(u, v) = ?
  21. B

    Better definition for complex number

    I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance? In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1...
  22. F

    What is the Net Resistance in this Complex Resistor Problem?

    Hey there! I'm having some real trouble deciphering this complex resistor problem. I have heard of the Kirchhoff voltage and current rules and do know how to use them to solve some problems but I'm not sure how to apply them in this context, or if they are even used to solve this. As seen the...
  23. karush

    MHB What Are the Solutions to the Equation \(x^2 + 2i = 0\)?

    $${x}^{2}+2i=0$$ $$\left(x-? \right)\left(x-? \right)=0$$ This should be easy but I couldn't get the factor
  24. B

    Complex numbers and beyond....

    If the solution of the quadratic equation \frac{-b \pm \sqrt{b^2-4ac}}{2a} produces a new kind of number, the complex numbers a \pm i b so, the solution the cubic equation should to produce a new kind of number too, and the solution of the quartic equation too, etc...
  25. S

    Branch cuts for complex powers

    I need to perform the following integration: ##I(s) = \frac{1}{2\pi i} \int_{\gamma}\text{d}z\ z^{-s} \frac{\text{d}\ln\mathcal{F(z)}}{\text{d}z}##, where ##\mathcal{F(z)}## is analytic everywhere on the complex plane except at the zeroes of the function. For the purpose of integration, the...
  26. Remixex

    Calculus Beginners Guide to Complex Analysis

    OK, so i took a course named "Oscillations and vibrations" We began the course with an "introduction" to complex numbers, basically we raced through them in like 3 classes, we talked about how to get complex roots, adding, multiplying, Cauchy-Riemann Conditions, Cauchy's integration Theorem...
  27. Remixex

    Complex algebra problem (roots)

    Homework Statement First off i wasn't sure if i should put this in precalc or here so i just tossed a coin[/B] I must find the roots of the expression z^4 +4=0 (which I've seen repeatedly on the internet) Use it to factorize z^4 +4 into quadratic factors with real coefficients The answer is...
  28. Stephanus

    PHP Learning Web Programming: MySQL & PHP for Complex Websites

    Dear PF Forum, I want to learn web programming, but there are specifics information that I need to know. What is the most famous database in web programming? My SQL? Is it true that PF Forum database is MySQL? If this is true, then the conclusion is MySQL can handle millions of post, hundreds...
  29. A

    Complex numbers and negative roots

    I was wondering if scientists or mathematicians have any use for complex numbers involving negative roots of I as in i=(-1)^(1/2). but my question is more what would be (-1)^(-1/2)?
  30. S

    Key difference between two real and single complex variable?

    Notion of differentiability (analyticity) for function of complex variable is normally introduced and illustrated by comparison with function of single real variable. It is stated that there are infinite number of ways to approach any given point of complex plane where function is defined, not...
  31. Titan97

    Analysis Books on complex analysis and algebra

    can you recommend a good book on complex analysis? I would like a book that can sharpen my skills in solving complex number problems through graphs and also improve the algebraic part like solving problems related to roots of unity etc. (I have studied calculus myself. I have done a lot of self...
  32. ognik

    MHB Finding Complex Roots: Poles of $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2}

    I think I'm a bit rusty here, started with finding poles for $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2} 1) Out of interest, is there a complex equivalent of the rational roots test? The above function is obvious, but for a poly that has both real and complex roots? 2) I am using the exponential...
  33. A

    Integration of part of a radius gives a complex number....

    I am interested to find the length shown in red in the attached figure. I want this length as a function of d (shown in blue) and the angle θ. Then I will integrate this length to dθ from 0 to π/2. Firstly, I used the law of the triangle to determine the length s which when subtracted from the...
  34. RJLiberator

    Can someone explain to me how this is not 1/4? Complex Analy

    Homework Statement Hi all, I posted the image of the solution here. My questions concerns the evaluation of the Residue at z=i. Homework Equations None needed, all giving in the question -- simple algebraic mistake made is likely to be the problem here. The Attempt at a Solution [/B] We...
  35. JR Sauerland

    I think I'm missing something in this complex number problem

    As you can see, it says that -110 (-1 to the tenth is just -1), multiplied by i, is somehow i. Everywhere I have looked, -1 times i is negative i, but this problem disagrees. Am I missing something?
  36. RJLiberator

    Complex Analysis Contour Circle Question

    Homework Statement I have uploaded necessary image(s) for the question I have successfully accomplished a, but I am not sure how to start b. Homework Equations The sum of the integral paths added up = the desired result. The Attempt at a Solution [/B] So we start with path CR And then go...
  37. RJLiberator

    Complex Analysis Integral Question

    Homework Statement Computer the integral: Integral from 0 to infinity of (d(theta)/(5+4sin(theta)) Homework Equations integral 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1) The Attempt at a Solution I've seen this integral be computed from 0 to 2pi, where the answer is 2pi/3...
  38. ognik

    MHB Path dependance of complex conjugate

    Hi, an exercise asks to show that $ \int_{0,0}^{1,1} {z}^{*}\,dz $ depends on the path, using the 2 obvious rectangular paths. So I did: $ \int_{c} {z}^{*}\,dz = \int_{c}(x-iy) \,(dx+idy) = \int_{c}(xdx + ydy) + i\int_{c}(xdy - ydx) = \frac{1}{2}({x}^{2} + {y}^{2}) |_{c} + i(xy - yx)|_{c}...
  39. O

    IMSL diagonalizing a general complex matrix (DEVCCG)

    Under some circumstances, whenever I call DEVCCG to diagonalize a general complex matrix, the program gets stuck inside and never returns. I do not even get out an error code so that I may continue with the rest of the program. I assume the iterative diagonalization inside the procedure does not...
  40. ognik

    MHB Tricky Complex number simplification

    Hi - in an example, I can't follow the working from one of the steps to the next, the 2 steps are: $... \sqrt{\frac{1}{2}\left(1-i\right)} = \sqrt{\frac{1}{\sqrt{2}}{e^{-i(\frac{\pi}{4}-2n\pi)}}}$ I can see they equate $ \frac{1-i}{\sqrt{2}} = e^{-i(\frac{\pi}{4}-2n\pi)}$, and I can see the $...
  41. ognik

    MHB Reverse direction for complex functions

    Hi An exercise asks to show $ \int_{a}^{b}f(z) \,dz = -\int_{b}^{a}f(z) \,dz $ I can remember this for real functions, something like $ G(x) = \int_{a}^{b}f(x) \,dx = G(b) - G(a), \therefore \int_{b}^{a}f(x) \,dx = G(a) - G(b) = -\int_{a}^{b}f(x) \,dx $ I have seen 2 approaches, either...
  42. RJLiberator

    Complex Analysis Clarification Question

    Homework Statement Problem and solution found here: http://homepages.math.uic.edu/~dcabrera/math417/summer2008/section57_59.pdf The question I am interested in is #1. In the solution, the instructor differentiates the series to get to: 2/(1-z)^3 = the series. If I want the Maclaurin series of...
  43. D

    MHB How can I generate a complex loan ammortization schedule with specified figures?

    How would I create a complex loan ammortization schedule for the following figures $390,000 Loan (3) payments of $10,000 each year on Jan 5th, July 5th and Oct 5th First Payment on July 5th 2015 Ammortized over 30yrs
  44. K

    Fortran [Fortran] Help Reading Complex 2D data

    Dear All, Please, I am trying to read in a data shown in array form but no luck. The data is 10 by 10 with each 10 by 10 depicted by a local array, for e.g. 0 0 0 0 0 0. A sample of the code I tried using is as shown below and the data is as attached. Please, any help or suggestion will be...
  45. Albert1

    MHB Find the Minimum Value of a Complex Equation

    $a>1,b>1$ find the minimum value of $\dfrac {a^2}{b-1}+\dfrac {b^2}{a-1}$
  46. RJLiberator

    Complex Analysis Series Question

    Homework Statement Let 0 < r < 1. Show that from n=1 to n=∞ of Σ(r^ncos(n*theta)) = (rcos(theta)-r^2)/(1-2rcos(theta)+r^2) Hint. This is an example of the statement that sometimes the fastest path to a “real” fact is via complex numbers. Let z = reiθ. Then, since r = |z|, and 0 < r < 1, the...
  47. G

    Real part of this complex quantity

    Hi everyone, I have a dispersive wave packet of the form: ##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} e^{-y^2/(D^2+2i\frac{ct}{k_0})}## The textbook says that the enlargement of the package, on the y direction, is: ##L=\frac{1}{D}\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2} ## However I have some...
  48. O

    Can a Non-Constant Holomorphic Function Equal Zero Everywhere?

    Homework Statement With . Give an example, if it exists, of a non constant holomorphic function that is zero everywhere and has the form 1/n, where n € N. Homework Equations So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more...
  49. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  50. I

    What exactly is a complex acoustic pressure?

    I have an impedance tube and it gives me the magnitude (dB) and phase (phase) of a signaa. So does that mean the complex pressure at that point is simply the [Magnitude] +i[Phase Value]? Does I need to change the units at all?
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