Complex Definition and 1000 Threads

Homework Statement This is a CIE A'level maths P3 question out of an exam from 2013 in October/November. As there is no markscheme ( I at least can't find one), I would be grateful if someone could look at my solution to the problem and correct me if I made a mistake. The problem is 8.(b)...

I have a very large expression: j - Sqrt[q^2 + qp^2 - 2 q qp Cos[\[Theta]]] - \[Sqrt](qp^2 + 1/2 (16 m5^2 + ma^2 + mp^2 - Sqrt[(-(16 m5^2) - ma^2 - mp^2)^2 - 4 (ma^2 mp^2 - 16 m5^2 qp^2)])) == 0 where \[Theta] = Pi/6; ma = 980; mp = 139; j...

Homework Statement Prove that any simplicial complex is Hausdorff. Homework EquationsThe Attempt at a Solution I have proved that for any finite simplicial complex, it is metrizable and hence Hausdorff. How to show the statement for infinite case?

Homework Statement I am going over examples in my textbook and I came across this: I don't understand how they converted 18.265 at angle of 39.9 to 14.02+j11.71 Homework Equations I know how to convert from the imaginary numbers into the angle form, usually I use: Is there another equation...

Homework Statement let \epsilon_1 and \epsilon_2 be unit vectors in R3. Define two complex unit vectors as follows: \epsilon_{\pm} = \frac{1}{\sqrt{2}}(\epsilon_1 \pm i \epsilon_2) verify that epsilon plus minus constitutes a set of complex orthonormal unit vectors. That is, show that...

As we know that √-5×√-5=5 i.e multiplication with it self My question is that according to this √-1×√-1=1.but it does not hold good in case of i(complex number). I.e i^2 =-1. Why?

Can someone please explain what's going on at 47:40 Thanks in advance.

Homework Statement well this is not exactly a homework, i had an argument whith my teacher about my grade in a test, because i put a complex number in the form of R,theta and she claims that the form was costheta+isentheta, and i know that but i need to prove in a book that...

Is the fact that QM uses complex numbers should be considered as a math artefact (as it is the case when complex numbers are used for alternate current circuit analysis), or, alternatively, it has some deep and important relation to the nature (or at least to the nature of the quantum theory)...

Homework Statement Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less. Homework Equations The Attempt at a Solution I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am...

Can we order Complex Numbers ? I searched a bit most places says it can but not like the real numbers. I am confused a bit.And I am not sure abouth the truth of those sources. Thanks

<Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...

Homework Statement http://prntscr.com/eqhh2p http://prntscr.com/eqhhcg Homework EquationsThe Attempt at a Solution I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even? As far as I'm concerned I am has to do wtih...

Homework Statement Let ##f(z) = ze^z## be bounded in the region where ## |z| \leq 2##, ##Im(z) \geq 0##, and ##Re(z) \geq 0## Where does it achieve it's maximum modulus and what is that maximum modulus? Homework Equations N/A The Attempt at a Solution A theorem states that any function...

Question Describe the color changes that occur when ##NH_{3(aq)}## is gradually added, with stirring to ##[CuCl_4]^{2-}_{(aq)}## until the ##NH_{3(aq)}## is in excess. Identify the three compounds or ions responsible for the new colors. Now the marking scheme shows that somehow...

Homework Statement Calculate the following integrals on the given paths. Why does the choice of path change/not change each of the results? (a) f(z) = exp(z) on i. the upper half of the unit circle. ii. the line segment from − 1 to 1. Homework Equations ∫γf(z) = ∫f(γ(t))γ'(t)dt, with the...

Hi all, I'm working through chapter 2 of Michael Tinkham's Introduction to Superconductivity. On page 40, he asserts that the skin-depth for a general complex conductivity is (In Gaussian units) $$\delta = \frac{c}{\sqrt{2\pi\omega\left(|\sigma| + \sigma_2\right)}}$$ where $$\sigma = \sigma_1...

Homework Statement I need to evaluate the following integral using the antiderivative: $$\int log^2(z) \, dz$$ I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from ##π## to ##i##, not crossing the non-positive x-axis...

How blindly theorits trust the data comming from huge and complex experiments such as the LHC CERN? Is it possible for one person to understand the whole experiment mechanics and still be able to come up with theoretical freamworks describing the data behaviour? Is it possible even to...

1) the problem I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function (...

Homework Statement Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi> A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1| So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and...

Homework Statement Homework Equations r=sqrt(a^2+b^2) θ=arg(z) tan(θ)=b/a The Attempt at a Solution for a)[/B] finding the polar form: r=sqrt(-3^2+(-4)^2)=sqrt(7) θ=arg(z) tan(θ)=-4/-3 = 53.13 ° 300-53.13=306.87° -3-j4=sqrt(7)*(cos(306.87+j306.87) I don't know if my answer is correct...

Homework Statement i have this function \begin{equation} f(t) = e^t \end{equation} Homework Equations [/B] the Fourier seria have the form \begin{equation} f(t) = \sum C_{n} e^{int} \end{equation}The Attempt at a Solution } [/B] so i need to find the coeficients $c_{n}$ given by...

Homework Statement Let a is a complex vector given by a = 2π K - i ρ / α^2 , where ρ is a two dimensional position vector and K is the corresponding two dimensional vector in the Fourier space. In order to find magnitude of this vector, i found that it is 4π^2 K^2 + ρ^2 / α^4 . The logic...

This is a question from a competitive entrance exam ...I just want to check whether my approach is correct as i don't have the answer keys . here is the question : How many complex numbers z are there such that |z+ 1| = |z+i| and |z| = 5? (A) 0 (B) 1 (C) 2 (D) 3 My approach : let z = x+iy...

I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...

Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...

Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t)Homework EquationsThe Attempt at a Solution f(t) = e(i*w0*t)) g(t) =e(i*w0*t...

Homework Statement f(z)=2x^3+3iy^2 then it wants f '(x+ix^2) The Attempt at a Solution So I take the partial with respect to x and i get 6x^2 then partial with respect to y and I get 6iy, then I plug in x for the real part and x-squared for the imaginary part, then I get f '...

Homework Statement Find the modulus and argument of z=((1+2i)^2 * (4-3i)^3) / ((3+4i)^4 * (2-i)^3 Homework Equations mod(z)=sqrt(a^2+b^2) The Attempt at a Solution In order to find the modulus, I have to use the formula below. But I'm struggling with finding out how to put the equation in...

Homework Statement Solve for ##\theta##: ##\cot \theta \sin \beta + \rho \csc^2 \theta = \cos \beta## where ##0^\circ<\beta<90^\circ, \ 0^\circ<\theta<90^\circ##, and ##0<\rho<1##. Homework Equations ##\cot^2 x +1 = \csc^2x##, the quadratic formula. The Attempt at a Solution ##\cot \theta...

Homework Statement Suppose z = x + iy. Where are the following functions differentiable? Where are they holomorphic? Which are entire? the function is f(z) = e-xe-iy Homework Equations ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x The Attempt at a Solution f(z) = e-xe-iy I convert it to polar form: f(z) =...

Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...

Homework Statement Prove the following form for an inner product in a complex space V: ##\langle u,v \rangle## ##=## ##\frac 1 4####\left| u+v\right|^2## ##-## ##\frac 1 4####\left| u-v\right|^2## ##+## ##\frac 1 4####\left| u+iv\right|^2## ##-## ##\frac 1 4####\left| u-iv\right|^2## Homework...

Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...

Homework Statement lim as z--> i , \frac{z^2-1}{z^2+1} The Attempt at a Solution [/B]When we plug in i we get -2/0, so we get division by 0, Does this mean the limit is infinity, I also tried approaching from z=x+i where x went to 0, you get the same answer, I also approached from...

Could you give me a hint how to attack this problem? Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t) I have begun as follows: e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b)) Re e^(z*t)= e^(a*t)*cos(b) What to do now?

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

Homework Statement What is the mapping of the circle of radius 1 centered at z=-2i under the mappinf f(z)=1/z The Attempt at a Solution I write the circle in polar form -2i+e^{ix} Now we invert it and multiply by the complex conjugate. so we get f(z)=...

Hi, I see a formula of gamma function and i have a question. (1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$ (2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$ (3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$ i understand the formula but...

Homework Statement Illustrate the mapping of f(z)=z+\frac{1}{z} for a parametric line. The Attempt at a Solution the equation for a parametric line is z(t)=z_0(1-t)+z_1(t) so I plug z(t) in for z in f(z), but I don't get an obvious expression on how to graph it, I tried manipulating it...

Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?

Homework Statement "ℝ×ℝ and ℂ are very similar in many ways. How do you realize ℂ as a Cartesian product of two sets? Consider how complex numbers are multiplied; by grouping real and imaginary parts, show how the pattern of complex multiplication can be used to define multiplication in ℝ×ℝ...

Homework Statement \frac{z-1}{z+1}=i I found the cartesian form, z = i, but how do I turn it into polar form?The Attempt at a Solution |z|=\sqrt{0^2+1^2}=1 \theta=arctan\frac{b}{a}=arctan\frac{1}{0} Is the solution then that is not possible to convert it to polar form?

There a simple math example that I am confused ##(\sqrt {-4})^2## Theres two ways to think 1-##\sqrt {-4}=2i## so ##(2i)^2=4i^2## which its ##-4## 2-##\sqrt {-4}##.##\sqrt {-4}##=##\sqrt {-4.-4}=\sqrt{16} =4## I think second one is wrong but I couldn't prove how, but I think its cause ##\sqrt...

Homework Statement Write the given complex number in polar form first using an argument where theta is not equal to Arg(z) z=-7i The Attempt at a Solution 7isin(\frac{-\pi}{2}+2\pi n) The weird part about this problem it asks me to not use the argument, The argument is the smallest angle...

Homework Statement If $$C = 1+cos\theta+...+cos(n-1)\theta,$$ $$S = sin\theta+...+sin(n-1)\theta,$$prove that $$C=\frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}} cos\frac{(n-1)\theta}{2} \enspace and \enspace S = \frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}}sin\frac{(n-1)\theta}{2}$$...

Homework Statement Same as title. Homework Equations Taylor expansion. The Attempt at a Solution Okay - what?! I don't even know where to begin. I taylor expanded the function and pretended like n was just some number and that doesn't help. I've never learned this. How? Can you point me in...

Homework Statement [/B] I had an inorganic lab this week which involved making VO(acac)2 from VOSO4⋅xH2O. In order to calculate the percentage yield, I need to work out x, that is, the number of water molecules coordinated with the vanadyl sulfate n-hydrate before the reaction. I'm stuck...

Homework Statement Consider the relation ## |\frac{z-i}{z*-i}| = \lambda ## where z = x + yi a) For ##\lambda = 1## show that the locus is a line in the complex plane and find its equation. b) What is the locus when ##\lambda = 0##? c) Show that for all other positive ##\lambda## the locus may...