# 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. ### POTW Integration Over a Line in the Complex Plane

For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$
2. ### POTW Solution to a Matrix Quadratic Equation

Let ##A## be a complex nilpotent ##n\times n##-matrix. Show that there is a unique nilpotent solution to the quadratic equation ##X^2 + 2X = A## in ##M_n(\mathbb{C})##, and write the solution explicitly (that is, in terms of ##A##).
3. ### A Even with a whimsical mathematical usage, solutions are obtained!

Hello everyone, Here, we observe that the familiar properties of the real logarithm hold true for the complex logarithm in these examples. So why does a whimsical mathematical use of real logarithm properties yield coherent solutions even in the case of complex logarithm?
4. ### I Continuity of Quotient of Complex Values

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...
5. ### Polarities of capacitor plates in a complex circuit

(a) I think the top plate of C5 could end up with either + or - charge, and not necessarily + charge as shown. This is because the connected plates of C1, C5 and C3 form an isolated system to which we can apply the law of conservation of charge i.e. Total charge just before transient currents...
6. ### Find in the form, ##x+iy## in the given complex number problem

This is the question as it appears on the pdf. copy; ##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]## My approach; ##\dfrac{3π}{4}=135^0## ##\tan 135^0=-\tan 45^0=\dfrac{-\sqrt{2}}{\sqrt{2}}## therefore, ##z=-\sqrt{2}+\sqrt{2}i## There may be a better approach.
7. ### I Numerical Solution of Complex Systems in GR

Please help me confirm that I understand this correctly. Imagine a system comprised of two black holes orbiting each other, which will eventually merge. At any point in time we describe the stress-energy tensor of the system. Assume that we could solve the EFE's for every point (t,x,y,z). This...
8. ### B Is this a complex number at the second quadrant?

Hi, PF, so long, I have a naive question: is ##\pi+\arctan{(2)}## a complex number at the second quadrant? To define a single-valued function, the principal argument of ##w## (denoted ##\mbox{Arg (w)}## is unique. This is because it is sometimes convenient to restric ##\theta=\arg{(w)}## to an...
9. ### Solve the problem involving complex numbers

Hello guys, I am refreshing on complex numbers today; kindly see attached. ok for part (a) this is a circle with centre ##(\sqrt{3}, -1)## with radius =##1## thus, we shall have,
10. ### 3x3 matrix with complex numbers

The attempt at a solution: I tried the normal method to find the determinant equal to 2j. I ended up with: 2j = -4yj -2xj -2j -x +y then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options: 1) 0= y(-4j-j^2) -x(2j-1) -2j 2)...
11. ### Complex fraction question

My first method to simplify the fraction is to to I flip ##\frac{5}{3}## up I get ##2 \times \frac{3}{5} = \frac{6}{5}## Method 2: if I flip 3 up I get ##\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}##. Method 3: I could use it multiply ##\frac{3}{3}## since this is the same as mutlipying by...
12. ### I A question about Young's inequality and complex numbers

Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
13. ### Insights An Overview of Complex Differentiation and Integration

I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks. Analysis is about differentiation. Hence, complex differentiation will be my starting point...
14. ### Find the GCD of the given complex numbers (Gaussian Integers)

Hello guys, I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean? ##z=1.2-1.4i## is rounded down to ##z=1-i##? I can see from here they came up with simultaneous equation i.e ##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
15. ### I Noether currents for a complex scalar field and a Fermion field

For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$\mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi$$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right]$$...
16. ### Find the roots of the complex number ##(-1+i)^\frac {1}{3}##

Kindly see attached...I just want to understand why for the case; ##(-1+i)^\frac {1}{3}## they divided by ##3## when working out the angles... Am assuming they used; ##(\cos x + i \sin x)^n = \cos nx + i \sin nx## and here, we require ##n## to be positive integers...unless I am not getting...
17. ### Using complex numbers to solve for a current in this circuit

First I solved 4+j3, which I squared 4 and 3 to equal 16 and 9 then I added them to get 25 and then I got the square root of 25 = 5. Then I plugged it back in to the equation. [50/(5)(50)+100] x 150 to get 50/350x 150= 1/7(150)= 21.42. I've attached the correct answer.
18. ### Symbolic integration of a Bessel function with a complex argument

Hello all I am trying to solve the following integral with Mathematica and I'm having some issues with it. where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by Where delta is a coefficient. Due to the complex arguments I'm integrating the...
19. ### Complex Matrix in vector norm

TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y} Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix I think we need to use (A*B)^T= (B^T) * (A^T) and Can you help...
20. ### How to find z^n of a complex number

Hello! (Not sure if this is pre or post calc,if I am in the wrong forum feel free to move it) So im given this complex number ## z = \frac{6}{1-i} ## and I am susposed to get it in polar form as well as z = a+bi I did that; z = 3+3i and polar form ##z =\sqrt{18} *e^{\pi/4 i} ## Now Im...
21. ### I Newton-Raphson Method With Complex Numbers

I'm trying to code Newton Raphson's method for finding zeros. I realize that even if the solution is real, it's possible for guesses to be complex. For example: $$y=\sqrt{x-6}-2$$ While 10 is a valid real root, for any guess less than 6, the result is complex. I tried to run the code allowing...
22. ### POTW Limit of Complex Sums

Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
23. ### How to Calculate the Complex Op Amp in Circuit Diagram

TL;DR Summary: How to calculate the operational amplifiers in the circuit diagram Hello Everyone, I am trying to learn the circuit diagram of one of a device in which I will be doing modifications as a part of my Masters's Research to make it performance better. My background is in Mechanical...
24. ### Complex numbers problem |z| - iz = 1-2i

Here is my attempt(photo below), but somehow the solution in the textbook is z= 2 - (3/2)i, and mine is z=(-3/2) +2i. Can someone please tell me where I am making a mistake? I suppose it's something with x being part of the real part of the 1st complex number and x being part of an imaginary...
25. ### How Can I Solve a System of Equations With Complex Numbers?

How can I solve a system of equations with complex numbers 2z+w=7i zi+w=-1 I have tried substituting z with a+bi and I have tried substituting w=7i-2z but didn't get anything useful. Edit: also, I've tried, multiplying lower eq. with -1 so that I can cancel w but I get stuck with 2z and zi and...
26. ### I Applications of complex gamma (or beta) functions in physics?

An example of physical applications for the gamma (or beta) function(s) is http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf (I refer to the beta function related to the gamma function, not the other functions with this name) The applications in Wikipedia...
27. ### Complex Integration Along Given Path

From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
28. ### I Closed Form for Complex Gamma Function

Hey all, I was wondering if there was an equivalent closed form expression for ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number. I came across the following answer...
29. ### Determine if the given set is Bounded- Complex Numbers

My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...
30. ### Solve ##z^2(1-z^2)=16## using Complex numbers

The problem is as shown...all steps are pretty easy to follow. I need help on the highlighted part in red. How did they come to; ##z^4+8z^2+16-9z^2=0## or is it by manipulating ##-z^2= 8z^2-9z^2?## trial and error ...
31. ### Understanding Complex Conjugates in QM (Griffiths pg. 13)

Am looking at page 13 of QM by Griffiths - have become stuck on minor point. He is proving that a normalised solution of Schrodingers eqn stays normalised. The bit I don't get is how can you just take the complex conjugate of Schrodingers eqn and assume its true. (ie how does he get from Eqn...
32. ### Linear operator in 2x2 complex vector space

Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
33. ### B Phase Difference of Current & Voltage: Capacitors, Inductors & Complex Numbers

how does capacitors and inductors cause phase difference between current and voltage? how does complex number come into play to explain the relation between phase of current and voltage?
34. ### I Analysis of converting a DE into complex DE

In Lecture 7, Prof. Arthur Mattuck (MIT OCW 18.03) taught that the following equation $$y’ +ky = k \cos(\omega t)$$ can be solved by replacing cos⁡(ωt) by ##e^{\omega t}## and, rewriting thus, $$\tilde{y’} + k\tilde{y}= ke^{i \omega t}$$ Where ##\tilde{y} = y_1 + i y_2##. And the solution of...
35. ### I How to interpret complex solutions to simple harmonic oscillator?

Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
36. ### LaTeX Complex Fractions in LaTeX

I know of two reasonable ways to represent a complex fraction: \dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) } ##\dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) }## and \dfrac{ ^a / _b }{ ^c / _d } ##\dfrac{ ^a / _b }{ ^c / _d }## What I am...
37. ### I Real ODE yields real solution through complex numbers

Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...
38. ### Prove by induction the sum of complex numbers is complex number

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
39. ### Can we use criss-cross approach with complex number equations?

I am not sure why criss-cross approach would work here, but it seems to get the answer. What would be the reason why we could use this approach? $$\frac {z-1} {z+1} = ni$$ $$\implies \frac {z-1} {z+1} = \frac {ni} {1}$$ $$\implies {(z-1)} \times 1= {ni} \times {(z+1)}$$
40. ### Relief of Complex Gamma Fcn — was this hand drawn?

This pic is from an older text called Tables of Higher Functions (interestingly both in German first then English second) that I jumped at buying from some niche bookstore for \$40. Was this hand drawn? I think I’ve seen was it that mathegraphix or something like that linked by @fresh_42...
41. ### I Complex State Functions?

Hello. I am reading about state equations from a physics textbook, Physics by Frederick J. Keller, W. Edward Gettys, Malcolm j. Skove (Volume I). I don't understand some parts but since I have the Turkish translation of the book I must translate it as good and clear as possible. "State...
42. ### MHB Check for any complex number z

I apologize in advance for my English. I want to know if my solution is correct. :) To verify that for every complex number z, the numbers z + z¯ and z × z¯ are real. My solution: z = a + bi z¯ = a - bi z + z¯ = a + bi + a - bi = 2a ∈ R z × z¯ = (a + bi) × (a - bi) = a^2 + b^2 ∈ R
43. ### I Is it valid to express a complex number as a vector?

...and is it ever useful? An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may...
44. ### Argument of complex number

Let z = x + iy $$\arg \left(\frac{1+z^2}{1 + \bar z^{2}}\right)=\arg (1+z^2) - \arg (1 + \bar z^{2})$$ $$=\arg (1+x^2+i2xy-y^2)-\arg(1+x^2-i2xy+y^2)$$ Then I stuck. I also tried: $$\frac{1+z^2}{1 + \bar z^{2}}=\frac{1+x^2+i2xy-y^2}{1+x^2-i2xy+y^2}$$ But also stuck How to do this question...
45. ### 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...
46. ### 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...
47. ### 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}##...
48. ### 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.
49. ### 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...
50. ### 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##