Complex Definition and 1000 Threads

Homework Statement Sketch the loci, find centre point and the radius of the circle. args((z-3i)/((z+4))=π/6[/B] Homework Equations args(x/y)=args(x)-args(y) Circle theorem - inclined angle theoremThe Attempt at a Solution I sketched the circle with major arc. Radius= using Pythagorus I got...

Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with f(t) = t: $$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$ $$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...

Is there a formulation of any of the relativity theories in terms of complex analysis? As in - I imagine - every event would be a complex number in a complex field.. or something as such..

Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...

Homework Statement Prove that each subfield of the field of complex numbers contains every rational number. ' From Hoffman and Kunze's Linear Algebra Chapter 1 Section 2 Homework EquationsThe Attempt at a Solution Suppose there was a subfield of the complex numbers that did not contain every...

I have a simple complex exponential signal of the form x(t)=ejωt. To find period of the signal I tested if x(t)=x(t+nT) for all n: ejωt=ejω(t+nT) ⇒ ejωnT=1=ej2πk where n and k are integers. Then I find a general period expression as T=2πk/ωn Period T means it is the least time a signal...

Hi folks, When you have a differential equation and the unknown function is complex, like in the Schrodinger equation, What methods should you use to solve it? I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to...

Homework Statement A voltage source E_0 cos ωt is connected in series with a resistor R and a capacitor C. Write down the differential equation expressing Kirchhoff’s law. Then guess an exponential form for the current, and take the real part of your solution to find the actual current...

Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...

Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...

Hello everyone, I'have implemented a Maximum-Likelihood-Expectation-Maximization Algorithm in order to reconstruct a bild. let say, we have such a system Ax=b, where A is a complex matrix, b is a complex vector. A and b are known and we will iterately try to find the best x (which should be...

Hi guys, I am learning about investment casting at the moment and have a question I would love someone to answer. In investment casting a wax pattern needs to be created. How would that be made for a complex shape like an impeller? I've heard that lots of impellers are created using...

so i am starting with the equation x3 = √(3) - i first : change to a vector magnitude = √[ (√(3))2 + 12] = 2 and angle = tan-1( 1/√(3) ) = 30 degrees (in fourth quadrant) so i have a vector of 2 ∠ - 30 so i plot the vector on the graph and consider that : 1. the fundamental theorum of...

Homework Statement [/B] Find and classify the isolated singularities of the following: $$ f(z) = \frac {1}{e^z - 1}$$ Homework EquationsThe Attempt at a Solution I have the solution for the positions of the singularities, which is: ## z = 2n\pi i## (for ##n = 0, \pm 1, \pm 2, ...##) and this...

I felt upon a mistake I made but cannot understand. I consider the following rotation transformation inspired from special relativity : $$\left(\begin{array}{c} x'\\ict'\end{array}\right)=\left (\begin {array} {cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end...

Homework Statement consider ##f## a meromorphic function with a finite pole at ##z=a## of order ##m##. Thus ##f(z)## has a laurent expansion: ##f(z)=\sum\limits_{n=-m}^{\infty} a_{n} (z-a)^{n} ## I want to show that ##f'(z)'/f(z)= \frac{m}{z-a} + holomorphic function ## And so where a...

What is the difference between apparent power and complex power? How to differentiate them?

I am studying special relativity, and I found that you have to work with a four dimensional space, where time is a complex variable. If you do so, you end up with the Minkowski metric, were the time component is negative and space components are positive (or vice versa). My questions are, why do...

Homework Statement I'm having some trouble evaluating the integral $$\int^\infty_{-\infty} \frac{\sqrt{2a}}{\sqrt{\pi}}e^{-2ax^2}e^{-ikx}dx$$ Where a and k are positive constants Homework Equations I've been given the following integral results which may be of help $$\int^\infty_{-\infty}...

I'm having trouble figuring out to get the answers from the 2 equations. The phasors and complex numbers confuse me. Do I need to change the phasor form? How do I go about doing this thanks! (Not homework question I am trying to figure this for my exam!)

I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. So in the equation ε is a symmetric matrix and δx(t) is just the difference between two points. After D.7...

Homework Statement Homework Equations V=IR and kirchhoffs laws The Attempt at a Solution Number three is my attempt at finding a solution but I got stuck when I had to find an expression for loop one on the diagram. Thanks in advance for your help...

Hello everyone! I am not actually posting this discussion regarding a homework problem. But I wanted to get some ideas from you all about how to approach the analysis of a structure I am looking at for a Statics Project. The structure is very complex, and my project partner and I are trying...

Homework Statement Evaluate the following line integrals in the complex plane by direct integration. Homework Equations Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ The Attempt at a Solution I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the...

Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic. I tried using the Laplace Equation of Uxx+Uyy=0 I have: du/dx=Ux d^2u/dx^2=Uxx du/dy=Uy d^2u/dy^2=Uyy dv/dx=cVx d^2v/dx^2=cVxx dv/dy=cVy d^2v/dy^2=cVyy I'm not really sure how to...

The theory of a complex scalar field ##\chi## is given by $$\mathcal{L}=\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi.$$ Why is it not common to include a factor of ##\frac{1}{2}## in front of the complex ##\chi## kinetic term? What is the effect on the propagator of...

Homework Statement Consider the following matrix. A = 2 + 4i...1 + 5i 2 − 3i...2 + 3i Let B = A-1. Find b12 (i.e., find the entry in row 1, column 2 of A−1) Homework Equations A-1 = 1/(ad - cb)* [ d -b ] [ -c a ] <--imagine as 2x2 matrix with first row (d,-b) and second row...

Homework Statement Express the complex number (−3 +4i)3 in the form a + bi Homework Equations z = r(cos(θ) + isin(θ)) The Attempt at a Solution z = -3 + 4i z3 = r3(cos(3θ) + isin(3θ)) r = sqrt ((-3)2 + 42) = 5 θ = arcsin(4/5) = 0.9273 ∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273)) a = -117 b...

I am under the impression that the following cannot be stated, a < b, if the a term is a complex number and the b term is either a natural number or a complex number, or any other type of number for that matter. Firstly am I correct? Secondly, if I am, does there exist a theorem of some sort...

The only thing which makes complex numbers different from 2-dimensional vectors or any other two-component mathematical object is their multiplication, right? Complex multiplication has uses in rotations but we can easily achieve that using polar co-ordinates. And, their other applications in...

Homework Statement Homework Equations S= 3VaIa* The Attempt at a Solution After transformation: Ia = 120<0 / (6+8j) = 12<-53.13 A Total complex power = 3 * Va * Ia* = 3*120<0 * 12<53.13 = 2592W + j3456 VAR This is the power supplied from source. What would be the power consumed by load?

I've just had my first batch of lectures on complex numbers (a very new idea to me). Algebraic operations and the idea behind conjugates are straightforward enough, as these seem to boil down to vectors. My problem is sketching. I have trouble defining the real and imaginary parts, and I don't...

Homework Statement Calculate the complex power delivered by the source V = 12cos(wt) V Homework Equations V = IR The Attempt at a Solution 1. I combined 8ohm resistor and 8j ohm inductor in parallel to get 4+4j ohms 2. I combined that with 4ohm resistor in series to get a Zth of 8+4j ohms 3...

I'm working out an impedance matching problem from a textbook (it is not part of any coursework) and I am trying to figure out how they get the 315 term in the polar coordinates below. Z = (XC*RL)/(XC+RL) = (-j333*(1000))/(-j333+1000) = 315 , -71.58* = 100 -j300 ohms I calculated that...

Hey everyone, I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...

I found the above while going through my textbook, where the textbook was trying to explain that the principal value of the product of two complex numbers raised to an exponent is not necessarily equivalent to the product of the two complex number each raised to the same exponent first. Based...

I would like to prove that ##\displaystyle{\int dx'\ \frac{1}{\sqrt{AB}}\exp\bigg[i\frac{(x''-x')^{2}}{A}\bigg]\exp\bigg[i\frac{(x'-x)^{2}}{B}\bigg]=\frac{1}{\sqrt{A+B}}\exp\bigg[i\frac{(x''-x)^{2}}{A+B}\bigg]}## Is there an easy way to do this integration that does not involve squaring the...

Hello, I've heard about "H.M." and similar patients that lost the ability to create declarative memories, but can still create non-declarative memories. I was wondering, since one in this condition can for example, develop a habit of drawing complex shapes, and not know they did it, can they...

Given that we can satisfy the wave equation with a simple sine & cosine wave function (the real part of the complex wave function) in classical mechanics, why do we use the complex wave function in EM theories? In QM it is obvious that it gets more mathematically more consistent. Out of curiosity.

Let's say you have a good personal computer with these characteristics: i7-6700K, 32GB RAM, 256GB SSD, 2 x GTX 970 4GB And let's assume you can run your algorithms, software... for 24 hours. I wonder, if I would be calculating the orbital structure of an hydrogen atom with many electrons...

When working in the complex domain (##z = x + iy##), how does one write the equation of a line? I have attached a problem I was working on (and have the solution), but am curious as to why the definition of a line is given by ##ax + by = c##. Are not ##x## and ##y## also variables that take on...

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or...

Hello all, I am trying to bring this: (p \iff q ) \implies r into a CNF form. I have started with the logical equivalences: (p \implies q) = \lnot p\lor q (p \iff q) = (p \land q)\lor (\lnot p \land \lnot q) and then I have applied De Morgan's rules and the distribution rules, but...

I am trying to find the limit of ## \frac {z^2 + i}{z^4 - 1} ## as ## z ## approaches ##i##. I've broken the solution down to: ##\frac {(z + \sqrt{i})(z - \sqrt{i})}{(z+1)(z-1)(z+i)(z-i)} ## but this does not seem to get me anywhere. The solution says ## -0.5 ## but I don't quite understand how...

Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...

Homework Statement Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time. Homework Equations ##\pi=\dot{\phi}^{*}## The Attempt at a Solution...

I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...

Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...

Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework EquationsThe Attempt at a Solution I have...