Complex Definition and 1000 Threads

Let z = [a b]^T be in the 2-dimensional vector space over real numbers, and T a linear transformation on the vector space. Consider $$\lim_{z'\rightarrow \mathbf{0}} \frac{T(z+z')-T(z)}{z'}$$ I argue this could be an alternative definition for complex derivative. To illustrate this, z as a...

How do simple Architectural basic computing binary code signaling create a diversity of information and software properties from information signals of existence (=1's) and non-existence (=0's)? After reading some Theoritical and Analytical Science papers recently about the debate on...

Anyone know what the simplest possible self-contained numeral system for complex numbers would be, analogous to signed ternary for integers? My guess would be quarter-imaginary base (https://en.wikipedia.org/wiki/Quater-imaginary_base.)

is this right Q) Determine this voltage in its simplest complex number form. v = (2xj6)(3-j8) 2x3=6 2x-j8=-16 j6x3=j18 j6x-j8=-j48 v=6 +(j18-j16) - J(^2)48 (j^2 = -1) v=6 +j2 +48 V=54 + j2

I've been studying quantum mechanics this semester in school and have ran into an issue I can't find an answer for. I understand why we take the complex conjugate of the wave function, such as when calculating expectation values. I'm a little confused though as to why we take the complex...

a) I think I got this one right. Please let me know otherwise We have (let's leave the ##x## dependence of the fields implicit :wink:) $$\mathscr{L} = N \Big(\partial_{\alpha} \phi \partial^{\alpha} \phi^{\dagger} - \mu^2 \phi \phi^{\dagger} \Big) = \partial_{\alpha} \phi^{\dagger}...

I try to understand (almost) complex manifolds and related stuff. Am I right that the condition for almost complexity simply is that the metric locally can be written in terms of the complex coordinates ##z##, i.e. ##g = g(z_1, ... z_m)## (complex conjugate coordinates must not appear)? These...

Solve ##Z^2\bar{Z}=8i## i am confused on how to proceed i have tried to substitute ##z=a+ib## solve the conjugate and the square, then separate the real from the imaginary and put all in a system, but becomes too complicated

I was trying to calculate an integral of form: $$\int_{-\infty}^\infty dx \frac{e^{iax}}{x^2}$$ using contour integration, with ##a>0## above. So I would calculate a contour integral with contour being a semicircle that goes along the real axis, closing it in positive direction in the upper...

Dear Everyone, I have a question about how to solve for x near the end of the problem: \[ 1+2\sinh^{2}(z)=0 \] Here is the solution and work: \[ 1+2\sinh^2(z)=0 \\ \sinh^2(z)=\frac{-1}{2}\\ \sqrt{\sinh^2(z)}=\pm \sqrt{\frac{-1}{2}}\\ \sinh(z)=\pm i\frac{1}{\sqrt{2}}\\ \] Then we can split...

I've been trying to continue my education by self-teaching during quarantine (since I can't really go to college right now) with the MIT Opencourseware courses. I landed on one section that's got me stuck for a while which is the second part of this problem (I managed to finish the first part...

$$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$ So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise. The generalized complex integral would be $$\int_{C} \frac{e^{-i \alpha...

Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a] 4e^(-j)=4 ∠-180° c=4, 4=(√a^2 +b^2) solving for a : a=(√16-b^2) θ=tan^(-1)[b/a]= -1 b/(√16-b^2)=...

Hi PF! Here's my equation 0.5 Sech[0.997091 Sqrt[ 2.4674 + \[Xi]\[Xi]^2]] (-1. ((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2) Cosh[ 0.997091 Sqrt[2.4674 + \[Xi]\[Xi]^2]] - 1. \[Sqrt](((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2)^2 Cosh[ 0.997091 Sqrt[2.4674 +...

FBD Block 1 FBD Block 2 FBD Pulley B I'm mainly concerned with the coordinate system direction in this problem, but just to show my attempt, here are the equations I got from the system. ##-T_A + m_1g = m_1a_1## ##T_B - m_2g = m_2a_2## ##T_A - 2T_B = 0## Using the fact that the lengths...

I have this idea for LED eyes. Basically its an LED screen behind a half spherical shape. I want the user only to see what's on the display if they are looking directly at it. So for that I would need to polarize the half sphere. The half sphere can be made out of anything, the problem is how do...

i have to find such domain z=x+iy , y,x∈ℝ , |y-x|<=2, |x|<=2 i'm confused with |y-x|<=2, how should i proceed ? with abs of x i am ok.

if ## \gamma (t):= i+3e^{2it } , t \in \left[0,4\pi \right] , then \int_0^{4\pi} \frac {dz} {z} ## in order to solve such integral i substitute z with ##\gamma(t)## and i multiply by ##\gamma'(t)## that is: ##\int_0^{4 \pi} \frac {6e^{2it}}{i+3e^{2it}}dt=\left.log(i+3e^{2it}) \right|_0^{4...

Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

Consider an equation, $$\tilde{x_0} = \ln(X+ i\delta),$$ where X may be positive or negative and ##0< \delta \ll 1##. Now, if ##X>0## this evaluates to ##\ln(X)## in some limiting prescription for ##\delta \rightarrow 0## while if ##X<0##, we get ##\ln(-X) + i \pi. ## Now, consider...

Hi PF! Each element of an ##n\times m## matrix is complex valued. In the following code, I call this "domain". There is also an ##n\times m## matrix that is real valued, below I call this "f". I'd like to plot a 3D image where the ##x-y## plane is the complex plain given by the coordinates...

Hi PF! Here's an ODE (for now let's not worry about the solutions, as A LOT of preceding work went into reducing the PDEs and BCs to this BVP): $$\lambda^2\phi-0.1 i\lambda\phi''-\phi'''=0$$ which admits analytic eigenvalues $$\lambda =-2.47433 + 0.17337 I, 2.47433 + 0.17337 I, -10.5087 +...

How did you find PF?: random Brownian motion Is randomness real or is it simply defined as such due to our inability to perceive hyper complex order? Randomness is a troublesome word. I'd feel better if I knew it was an objective phenomenon and not merely a placeholder description of...

Hi PF! I'm trying to find a 1D, linear, complex, 2nd order, eigenvalue BVP: know any that admit analytic solutions? Can't think of any off the top of my head. Thanks!

I stumbled upon the following problem on instagram: $$L = \left (\frac{-1+i\sqrt{3}}{2}\right )^6+\left (\frac{-1-i\sqrt{3}}{2}\right )^6+\left (\frac{-1+i\sqrt{3}}{2}\right )^5+\left (\frac{-1-i\sqrt{3}}{2}\right )^5$$ The idea is to compute it. Using a calculator, it is supposed to equal 1. My...

I need an equation to predict the flight path of a changing mass projectile under changing thrust. Any thoughts?

Hi to all member of the Physics Forums. I have this question: it is possible consider the analogue of the Schrodinger equation on the plane with configuration space ##(x,p)\in\mathbb{R}^4## on the complex disk ##\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}##? Ssnow

Summary:: Inner Product Spaces, Orthogonality. Hi there, This my first thread on this forum :) I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill) Chapter 7 - Inner Product Spaces, Orthogonality. Using some particular values for u and v, I...

Please critique this text. It came from a research article* I found but I'm only interested if the sentence is 100% accurate or not and not in the specifics of the article itself. Are they suggesting Hilbert space is always infinite? Thanks. Quantum mechanics is infinitely more complicated than...

I guess I will show my work for substantiating equation 1 and hopefully by doing so someone will be able to point out where I could generalize. ##\langle \vec{S}_{rad} \rangle = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{rad} \times \vec{B}^*_{rad}\right) = \frac{1}{2 \mu} \mathfrak{R} \left(...

I was expecting Ni2+ to be present also in the answer as it can give dsp2 and sp3 configuration. [Ni(CN)4]2- and [Ni(NH3)4]2+ have dsp2 and sp3 respectively, right?

The area of two lines that I need to find is 2.36, however i need this in exact form. The lines are y=-x/2e+1/e+e the other line is y=e^x/2 Since y=-x/2e+1/e+e is on top it is the first function. A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2 If you could please help!

So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then 1. Real part: ##\sin x \cosh y## 2. Imaginary part: ##\cos x \sinh y## If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }## How to...

I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...

Solution Attempt: \begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} = (x^2+y^2)^{-1} -x (x^2+y^2)^{-2} (2x) = (x^2+y^2)^{-1} - 2x^2 (x^2+y^2)^{-2} \\ \rule{0mm}{18pt} \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} = -x (x^2+y^2)^{-2} (2y) =...

Hello! :smile: I am locked in an exercise. I must find (and graph) the complex numbers that verify the equation: ##z^2=\bar z^2 ## If ##z=x+iy## then: ##(x+iy)^2=(x-iy)^2 ## and operating and simplifying, ##4.x.yi=0 ## and here I don't know how to continue... can you help me with ideas? thanks!

Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...

I've been trying this problem for a long time. By operating the lower part of the logarithm and clapping the real and imaginary part of the logarithm, I have come to the conclusion that the correct lines must be those in which it is true that: $ d \ frac {(x ^ 2 + y ^ 2-a ^ 2) ^ 2 + 4y ^ 2a ^...

Is it possible two find complex numbers ##C_n##, so that both equations are satisfied \sum^{\infty}_{n=1}nC_n=0 and \sum^{\infty}_{n=1}|C_n|^2=1 ?

Hello, My university offers a couple Complex Analysis courses, among them there is one with the following description: Introduction to complex variables: "substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical...

When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?

Is the spiral I drew here clockwise or counterclockwise ? What’s a trick to know whether it’s going CCW or CW. Thanks!

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...

reducing it to various forms: for example, the one in the title, or 2*pi*k(ln m) = a(ln(n/m)), and so forth. My gut feeling is that it is true (that no such foursome exists), but manipulations have not got me anywhere. Anyone push me in the right direction? I am probably overlooking something...

So just based on the cauchy riemann theorem, I think: Ux = 2 = Vy = 2xy, so f(z) is differentiable on xy = 1, and also that Vx = y^2 = -Uy = 0. That doesn't make sense to me because if 0 = y^2, then y = 0, yet that wouldn't satisfy xy = 1, would it? Furthermore, I'm not sure how I would...

I'm looking for material about the following approach : If one suppose a function over complex numbers ##f(x+iy)## then ##\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial...

question fig: solution manual: my solution: oxidation state of central cobalt is +6 due to 6 oxygen surrounding it,The other cobalt is +2 due to 2 oxygen surrounding it with NH3 ligand which is no count for oxidation state.

I have reached a conclusion that no such z can be found. Are there any flaws in my argument? Or are there cases that aren't covered in this? Attempt ##\log(\frac{1}{z})=\ln\frac{1}{|z|}+i\arg(\frac{1}{z})## ##-\log(z)=-\ln|z|-i\arg(z)## For the real part...

I tried saying z = x + iy, then squared both sides so that I would get something that looked like: |z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach. For that...