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. C

    Can't find total resistance in a complex star circuit

    [Thread moved from the technical forums to the schoolwork forums by the Mentors] Hi i have this assignment for homework: There is only one battery for the circuit, E=10V, R=4 Ohms and L=1H it asks me to find the time constant of the circuit. i know that a time constant in a RL circuit is t=L/R...
  2. Tertius

    A Local phase invariance of complex scalar field in curved spacetime

    I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...
  3. H

    I Sum of the dot product of complex vectors

    Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...
  4. C

    Need help with a question about powers of complex numbers

    (z-3)3=-8, solve for z. I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.
  5. B

    A Help needed with derivation: solving a complex double integral

    I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...
  6. chwala

    Find two possible values of ##z## in the complex number problem

    ok here i have, ##x^2+y^2-5x=0## ##-y= 2## I end up with the quadratic equation, ##x^2-5x+4=0## Finally giving us, ##z=4-2i## and ##z=1-2i##
  7. chwala

    Find ##z## in the form ##a+bi## under Complex Numbers

    For part (a), ##z##=##\dfrac {3+i}{3-i}## ⋅##\dfrac {3+i}{3+i}## ##z##=##\dfrac {4}{5}##+##\dfrac {3}{5}i## part (b) no problem as long as one understands the argand plane... For part (c) Modulus of ##z=1## and Modulus of ##z-z^*##=##\frac{6}{5}i##
  8. chwala

    Find the complex number which satisfies the given equation

    Find the problem here; ( i do not have the solutions...i seek alternative ways of doing the problems) ok, i let ##z=x+iy## and ##z^*= x-iy##... i ended up with the simultaneous equation; ##2x+y=4## ##x+2y=-1## ##x=1## and ##y=2## therefore our complex number is ##z=1+2i##
  9. V

    Solving for z in the Equation tan z = 1 + 2i

    Find the values of tan-1(1+2i). We can use the fact: tan-1z = (i/2)log((i+z)/(i-z)). Then with substitutions we have (i/2)log((1+3i)/(-i-1)). Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)). Do we then just proceed to solve log(1+3i) and log(-1-i)? I'm just a little confused...
  10. V

    Finding Values of Complex Equation

    I took the equation and rewrote it as: e(i+1)(log(1-i)-log(√2)). So I worked on it in sections meaning e(i+1) and then log(1-i). For e(i+1) I got eie1 and used Euler's formula for ei to get: e1(cos(1)+isin(1)). And then for log(1-i) I got ln√2 + i(-(π/4)+2kπ). Do I just bring them...
  11. BWV

    Worth learning complex exponential trig derivations in precalc?

    This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...
  12. rylest

    Stuck on complex pipe system bending moment hand calcs

    Some more details on the system are that L1 is very long (close to 100ft) and L2 is close to 30ft (the vertical pipes). The piping is all schedule 40 1/2" OD. Moment of inertia is roughly 10^-8. Components are about 2kg each. The distance of the pipes horizontally is small (around 2ft). Pressure...
  13. MrHappyTree

    Resistance in complex geometries

    For the electrical resistance ##R## of an ideal wire, we all know the formula ##R=\rho * \frac{l}{A}##. However this is only valid for a cylinder with constant cross sectional area ##A##. In a cone the cross section area is reduced over its height (or length ##l##). What is a good general...
  14. L

    Can Cauchy's Residue Theorem be Used for Functions with Branch Cuts?

    First of all I am not sure which type of singularity is ##z=0##? \ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1} It looks like that ##Res[f(z),z=0]=0##
  15. chwala

    Prove that ##12≤OP≤13## in the problem involving complex numbers

    Find the question below; note that no solution is provided for this question. My approach; Find part of my sketch here; * My diagram may not be accurate..i just noted that, ##OP## takes smallest value of ##12## when ##|z+5|=|z-5|## i.e at the end of its minor axis and greatest value ##13##...
  16. chwala

    Use binomial theorem to find the complex number

    This is also pretty easy, ##z^5=(a+bi)^5## ##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}## ##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i## ##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4## ##\bigl(\Im (z))=...
  17. chwala

    Prove the given complex number problem

    This is pretty straightforward, Let ##z=a+bi## ## \bigl(\Re (z))=a, \bigl(\Im (z))=b## ##zz^*=(a+bi)(a-bi)=a^2+b^2 =\bigl(\Re (z))^2+\bigl(\Im (z))^2## Any other approach? this are pretty simple questions ...all the same its good to explore different perspective on the same...
  18. chwala

    Prove that ##c^2+d^2=1## in the problem involving complex numbers

    Easy questions, just a lot of computation... $$\frac {z}{z^*}=\frac {a+bi}{a-bi} ×\frac {a+bi}{a+bi}$$ $$c+di=\frac {a^2-b^2}{a^2+b^2}+\frac {2abi}{a^2+b^2}$$ $$⇒c^2= \frac {a^4-2a^2b^2+b^4}{(a^2+b^2)^2}$$ $$⇒d^2= \frac {4a^2b^2}{(a^2+b^2)^2}$$ Therefore...
  19. chwala

    Solve this pair of simultaneous equations involving complex numbers

    $$(1+i)z+(2-i)w=3+4i$$ $$iz+(3+i)w=-1+5i$$ ok, multiplying the first equation by##(1-i)## and the second equation by ##i##, we get, $$2z+(1-3i)w=7+i$$ $$-z+(-1+3i)w=-5-i$$ adding the two equations, we get ##z=2##, We know that, $$iz+(3+i)w=-1+5i$$ $$⇒2i+(3+i)w=-1+5i$$...
  20. S

    Finding argument of complex number

    Let: ##z=x+iy## $$z+\frac 1 z =1+2i$$ $$x+iy +\frac{1}{x+iy}=1+2i$$ $$x+iy+\frac{1}{x+iy} . \frac{x-iy}{x-iy}=1+2i$$ $$x+iy+\frac{x-iy}{x^2+y^2}=1+2i$$ $$\frac{x^3+xy^2+x+i(x^2y+y^3-y)}{x^2+y^2}=1+2i$$ So: $$\frac{x^3+xy^2+x}{x^2+y^2}=1$$ $$x^3+xy^2+x=x^2+y^2$$ and...
  21. pellis

    A How to visualise complex vector spaces of dimension 2 and above

    According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...
  22. ergospherical

    I Lorentz Transf. of Complex Null Tetrads: Formula (3.14-17)

    For a complex null tetrad ##(\boldsymbol{m}, \overline{\boldsymbol{m}}, \boldsymbol{l}, \boldsymbol{k})##, how to arrive at formulae (3.14), (3.15) and (3.17)? The equation (3.16) is clear as is. (I checked already that they work i.e. that ##\boldsymbol{e}_a' \cdot \boldsymbol{e}_b' = 2m'_{(a}...
  23. LCSphysicist

    Complex integration is giving the wrong answer by a factor of two

    $$\int_{0}^{2\pi } (1+2cost)^{n}cos(nt) dt$$ $$e^{it} = z, izdt = dz$$ $$\oint (1+e^{it}+e^{-it})^{n}\frac{e^{nit}+e^{-nit}}{2} \frac{dz}{iz} = \oint (1+z+z^{-1})^{n}\frac{z^{n}+z^{-n}}{2} \frac{dz}{iz}$$ $$\oint (z+z^{2}+1)^{n}\frac{z^{2n}+1}{z^{2n+1}} \frac{dz}{2i} = \pi Res = \pi...
  24. Leo Liu

    I Power of a complex conjugate

    Can someone please tell me why this is true? This isn't exactly the De Moivre's theorem. Thank you.
  25. docnet

    Complex conjugate of a pole is a pole?

    This isn't a homework problem, but a more general question. Let ##f## be a function with two singular points ##r## and its complex conjugate ##r^*##. let $$f=\frac{g}{z-r} \quad \text{and assume} \quad g(r)\neq 0$$ so ##r## is a simple pole of ##f##. we have conjugates that are singular...
  26. C

    Proving geometric sum for complex numbers

    I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...
  27. issue

    A complex probability question

    There is a box with 2324784 bullets. Balls are divided into 5 groop and there is a definition for each groop and how many balls and what is the probability of getting each groop. The game with the return of the balls List the probability and quantity groop 1 906192 0.389796213 groop 2 1006880...
  28. WMDhamnekar

    MHB How to Prove a Complex Number Equation and Its Trajectory Forms a Circle?

    My attempt: Let us put $\frac{1}{i+t} = \frac{1+e^{is}}{2i} \Rightarrow \frac{2i}{i+t} -1= e^{is}$ So, $\cos{s}- i\sin{s}= \frac{2i}{i+t} - 1,\Rightarrow \cos^2{(s)} - \sin^2{(s)} = \frac{-2}{(i+t)^2} +1 -\frac{4i}{i+t}$ After doing some more mathematical computations, I got $\cos{s}=...
  29. chwala

    Find the distance CD in the given complex variable question

    This is the problem; Note that i am conversant with the above steps shown in the solution, having said that i realized that we could also borrow from the understanding of gradient and straight lines in finding the distance ##CD##... it follows that the equation of ##BA= -1.5x-0.5##, implying...
  30. A

    Engineering Book considering FEM analysis for complex eigenvalues (incl. damping)

    Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).
  31. A

    Complex collision with masses and Velcro

    Drawing: I decided to attempt to approach this as several collisions. So we can start with this: Object 1-Object 2 This collision is elastic, so we know that ##P_i = P_f##. We also know ##K_i = K_f##. So, $$mv_{1i} = mv_{1f} + mv{2_f}$$ $$1 = v_{1f} + v_{2f}$$ $$v_{2f} = 1 - v_{1f}$$ and...
  32. AryaKimiaghalam

    Programs What are some good graduate programs in the Physics of Complex Systems?

    Title says it all. I am interested in studying the physics complex systems and nonlinear physics, however i find it very hard to find a good program as it seems this area of study is not the most mainstream of them. I found out about Max Planck but still want to know if there are other strong...
  33. B

    MHB Proof of Complex Numbers: Delta*w(z, z) Explained

    Hi, I have this problem and its solution but i know what right size is, but i don't understand what left size (delta*w(z, z)) is equal to
  34. T

    MHB Complex numbers such that modulus (absolute value) less than or equal to 1.

  35. T

    MHB Complex numbers such that modulus less than or equal to 1.

    https://www.physicsforums.com/attachments/11378
  36. M

    Engineering Solving Problems Involving Complex Vectors

    Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...
  37. J

    Evaluate the complex number.

    (e^(i*theta))^2 = (sin(theta)+i*cos(theta))^2 = cos(theta)^2 - sin(theta)^2 + 2*i*sin(theta)*cos(theta), so the real part would be: cos(theta)^2 - sin(theta)^2, and the imaginary part would be: 2*i*sin(theta)*cos(theta). But then I don't know where to start with the modulus or the argument?
  38. kofffie

    Chemistry Help with Complex Acid-Base Titration

    Help! I am confused with my assignment. it was about complex acid base titration. the analyte was citric acid and the titrant was a base. To find it, I need to search for C1V1=C2V2, however during the balanced equation, there is 3 mol of my base that will be reacting to 1 acid. Will the ratio of...
  39. R

    How to find the residue of a complex function

    Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...
  40. tixi

    Finding analyticity of a complex function involving ln(iz)

    Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem. The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...
  41. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  42. L

    Analysis 1 Homework Help with Complex Numbers

    I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...
  43. docnet

    B Can we have a complex number in the exponent?

    Does it make sense to write ##r^c## where ## r\in R/e ## and ##c\in C## ?
  44. R

    Finding roots and complex roots of a determinant

    I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0## I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0## I...
  45. Rubberduck2005

    A slightly complex pully problem missing the final step

    so I have already done most the work in finding the following things M1A1=M1g-T1, M2A2=M2g-T2 T2=2T1 M2A2=M2g-T2 and then through further working out(use of the invariance of length of L1 and L2 the strings attached to M1 and M2 then finding A1=-2A2) A1=2g(2M1-M2)/(4M1+M2) and...
  46. M

    Complex analysis problem

    Mentor note: Edited to fix LaTeX problems ##f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy## ## f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy) ## ## \Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}##
  47. jaychay

    MHB Struggling with Complex Number Function? Need Help?

    Can you help me with this two questions I am really struggle on how to do it Please help me Thank you in advance
  48. jaychay

    MHB Complex number equation graph problem

    Given (a,b) is the coordinate just like (x,y). Find equation Zo and coordinate (a,b) ?Please help me Thank you in advance.
  49. baby_1

    Complex Reflection and Transmission Coefficient in oblique incidence

    Hello, Something has made me confused after studying the Snell equations these days. Regarding the Balanis Advanced engineering electromagnetic( the pages have been attached), and based on that the reflection and transmission coefficient can be complex I need to rewrite the (5-23a) again...
  50. karush

    MHB 1.4.1 complex number by condition

    1.4.1 Miliani HS Find all complex numbers x which satisfy the given condition $\begin{array}{rl} 1+x&=\sqrt{10+2x} \\ (1+x)^2&=10+2x\\ 1+2x+x^2&=10+2x\\ x^2-9&=0\\ (x-3)(x+3)&=0 \end{array}$ ok looks these are not complex numbers unless we go back the the...
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