Complex Definition and 1000 Threads

This really isn't a homework question but I wasn't sure where to post it. I was watching a video by numberphile about complex numbers and the professer being interviewed said the most important thing about complex numbers is that they help bring algebra and geometry together. What did he mean by...

Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...

Homework Statement Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex]. Homework Equations R is the radius of convergence and the derived series is the term by term...

Homework Statement Prove that \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 Homework Equations It is specifically mentioned in the text that the epsilon-delta relation should be used, |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta . Where...

Possibly a difficult question, but I've never found a discussion on the topic. Thanks

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding the proof of Tapp's Proposition 2.2. Proposition 2.2 and its proof read as...

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding what Tapp is saying in this section regarding the function \rho_n \ : \ M_n...

Homework Statement y=[(1+y^2)^1.5]/[2(y+sqrt 3)^2]; solve for y Homework Equations see above The Attempt at a Solution I tried to use algebra to solve it, but I can't. The textbook says it can be solved numerically or by iteration. By numberically I think it means algebraically. But I don't...

Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...

Homework Statement [moderator note: Image inserted to be visible in problem statement] Homework EquationsThe Attempt at a Solution My friend and I are trying to figure out how this works. This is the questions and the solution manual. We understand where the real power comes from but not at...

Homework Statement Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1 Homework Equations |f(z)| yields the maximum value The Attempt at a Solution f(z) = exp(x) ( cosy + i siny) Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric...

Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!

Homework Statement What is the amplitude and phase of the complex function? f(t) = (1-2i)e^(iwt) Homework Equations None/unknown Normal Polar Form = Real*e^imaginary i = e^pi/2*i The Attempt at a Solution [/B]I am trying to bring this into a normal polar form to easily see the phase and...

*How can I finish off the "only if" direction? I am just unable to prove the only if direction! Using the induction hypothesis and the triangle inequality is confusing me for some reason.* Show that \begin{equation} |z_1+z_2+\dots+z_n| = |z_1| + |z_2| + \dots + |z_n| \end{equation} if and only...

1)If a= cosα + i sinα and the equation az2 + az +1 =0 has a pure imaginary root, then tanα=? 2) cosα+isinα=eiα , quadratic formula 3) What i tried to do was,i put a constant real number and tried to solve it and used demoivres theorem, although the answer is getting weirder and weirder.

Recommend a self study book for linear algebra with complex numbers

Homework Statement Find lim_{x->- \infty} \; \frac{(x^6+8)^{1/3}}{4x^2+(3x^4+1)^{1/2}} Homework Equations N/A The Attempt at a Solution Factoring out \frac {(-x^6)^{1/3}}{-x^2} leaves me with \frac{(-1-8x^{-6})}{-4+(3+x^{-4})^{1/2 }} Taking the limit at infinity gives me...

I've been trying to figure out a way to get an approximation to a complex DiffEQ. dx/dt = c1 / (c2 + c3*x*t) Does anyone have any input on wether this problem can be approximated? Thank you.

Homework Statement This isn't a standard homework problem. We were asked to do research and to find a theorem of the form: If something about the partial derivatives of u and v is true then the implication is ##D(u,v)## at ##(x_0,y_0)## exists from ##R^2## to ##R^2##Homework EquationsThe...

Homework Statement Find the principle argument Arg z when z = (sqrt(3) - i)^6 Homework EquationsThe Attempt at a Solution I'm sorry to say that I'm not sure how to solve this problem. It's my understanding that what this question is basically asking me to do is find theta such that...

Homework Statement show that the following functions are differentiable everywhere and then also find f'(z) and f''(z). (a) f(z) = iz + 2 so f(z) = ix -y +2 then u(x,y) = 2-y, v(x,y) = x Homework Equations z=x+iy z=u(x,y) +iv(x,y) Cauchy-Riemann conditions says is differentiable everywhere...

Homework Statement Homework EquationsThe Attempt at a Solution I assumed there to be two supernodes. The first one was between V1 and V2. The second supernode was between V2 and V3. This is what i tried and i am not getting the right answers. Please can someone help me out! Apologies...

I'm just starting this, but what would the complex conjugate of Ψ(x,t) in the equation : |Ψ(x,t)|^2= Ψ(x,t)* Ψ(x,t) be.. Let's just say, for example, that x is 4 and t is 9... Please help if you can.. Could you please help me out with the steps to completing this, because I really want to...

Homework Statement ((1-i)/(sqrt2))^42 express in x+iy form Homework Equations z1/z1=(r1/r2)e^(i(theta1-theta2)) The Attempt at a Solution Ive found that (1-i) has r=sqrt2 so since r is sqrt2 and x=1 y=-1 so the angle is 7pi/4 so then I have (sqrt2e^(-i7pi/4)/sqrt2)^42 now from here is where I...

Homework Statement Homework EquationsThe Attempt at a Solution I tried to attempt the question but I am not sure how to start it, at least for part (i). My biggest question, I think, is how does the multiplication of a random complex number to a Fourier-Transformed signal (V(f)) have an...

Are a field and its complex conjugate independent? It seems like they're not, as one is the complex conjugate of the other, so if you have one, you know the other. However, it seems in path integrals, you integrate over the field and its conjugate, so they can take on values that are not the...

Homework Statement Calculate the total number of compex multiplications required for the calculation in (b) when FFTs are used to perform the Discrete Fourier Transforms and Inverse Discrete Fourier Transforms.[/B] There were two FFT multiplied together and one inverse FFT of that product to...

When do we call a representation complex? What are examples of complex representations? Also, when we say real and complex forms of Lie algebras, is that related to real and complex representation classification? I read that spinors are complex representations of SO(3), because their...

Homework Statement The question involves finding the arc length of the parametric equation x = e^t + e^-t and y = 5 - 2t Homework Equations Arc length of a parametric equation ∫√(dy/dt)^2 + (dx/dt)^2 dt limits are from 0<t<3 The Attempt at a Solution Taking the derivative of both x and y...

Homework Statement Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B] Homework Equations 1. z=a+bi 2. re^itheta The Attempt at a Solution I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...

Suppose we have a massless complex field in 3+1 spacetime where E^2 = P^2. Suppose that the only excitations that are possible are those that in some rest frame consist of an excitation of a pair of states p1 and p2 such that p1 = -p2 and ιp1ι = ιp2ι = mc^2 = (+or-)E, and the pair of states p1...

I have started studying complex integration recently and i just can't seem to get the things in my head . the biggest problem i am facing is that : when solving real number integrals the area under the curve of the function is what integration means ... but i can't seem to find an analogy...

Homework Statement prove that sqrt2|z| greater than or equal to |Rez| + |Imz| Homework Equations |z|^2 = x^2 + y^2 Rez=x, Imz=yThe Attempt at a Solution so far I've worked it down to this. 2(x^2 + y^2) greater than or equal to x^2 + 2xy + y^2 I've used a few different values for x and y and...

Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...

1. Evaluate the limit http://www4a.wolframalpha.com/Calculate/MSP/MSP64511d2754i3f4iaefab00001fa62g875680a1ia?MSPStoreType=image/gif&s=44&w=125.&h=45. 2. No formulas 3.The answer is -1/9. I have tried multiplying the top by the conjugate but that seems wrong as there are no square roots...

Homework Statement Homework Equations Down The Attempt at a Solution As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the beginning n=1 the residue...

Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region $|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.

How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...

hello can you tell me please why we introduced complex numbers? what was the problem that we couldn't express with rest of algebra and we introduced complex numbers? I am basically interested in why we introduced complex number to describe and analyze AC circuits, like voltage, current and...

Homework Statement Ytt = 1 Yxx with initial conditions of yT(x,0) = 0 y(x,0) = \begin{cases} 1 & \text{if } x \geq 0 \ & \text{if } x \leq 1 \\ 0 & \text{if } otherwise \end{cases} Sketch the solution of this wave equation for 5 representative values of t, when the solution of the wave is...

Kc1 = (5.8*2/5)^2 / (14/5)(1.4/5) = 6.865 6.865 = [HI]^2 / (45/253.8/100)(0.5/2/100) [HI] = 5.52x10^-3 M mass no of HI = [HI] x (126.9+1) x 100 = 70.6g is it correct? And how to do 3bii I GOT 3bi Kc2 = [HCl(g)]^2/ / Kc(1) [H2(g)] [Cl2(g)]

Hi All, I'm working out a program to emulate a quantum computer (definitely in a nascent stage), and I'm struggling with a piece of the math. I looked at the math sections in these forums, but thought this might be more appropriate to post it. I'll try to conceptually outline the problem, and...

Homework Statement Solve the equation ## z^3 + 6z = 20 ## (this was considered by Cardan in Ars magna). Homework Equations Please see the 2nd attachement. The Attempt at a Solution I want to know if my solution is correct because the book (2nd attachment) says that there should only be 3...

Homework Statement Hi Guys, I am trying to solve this question, please look at the attached picture Homework Equations The general equation for a inverting amp is -Rf/R1 * Vin = Vout The Attempt at a Solution Well as the question says the two resistors, R2 and R must be treated as parallel...

Homework Statement If arg(\frac{z-ω}{z-ω^2}) = 0, \ then\ prove \ that\ Re(z) = -1/2 Homework Equations ω and ω^2 are non-real cube roots of unity. The Attempt at a Solution arg(z-ω) = arg(z-ω^2) So, z-ω = k(z-w^2) Beyond that, I'm not sure how to proceed. Using the rotation formula may also...

Hello, I am evaluating: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using the following contour: $R$ is the big radius, $\epsilon$ is small radius (of small circle) Question before: Which $\log$ branch is this? I asked else they said, $$-\pi/2 \le arg(z) \le 3\pi/2$$ But in the...

Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...

Is there in a nutshell an explanation or even a single reason why complex numbers have so many fascinating consequences and give rise to so much deep stuff like analytic functions (with all its stunning properties), Riemann surfaces, analytic continuations, modular forms, zeta function, its...

This is an interesting complex analysis problem; **The figure on the bottom left is what is being referred to,Fig7-10.** **Firstly: (1)** How is the branch point $z=0$ at $z=0$?? We have $f(0) = 0$ that is not a discontinuity is it? **Secondly:(2)** It says that: $AB$ and $GH$ are coincident...

Hello, Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ We know that because $f(x)$ is even:$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = \frac{1}{2} \cdot \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Consider a complex function, with $z = x + iy$ $$f(z) =...