Complex Definition and 1000 Threads

<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...

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...

Homework Statement Solve each equation for z=a+ib z^{*2}=4z where z* is the complex conjugate The Attempt at a Solution I wrote z and z* in terms of x and iy , and tried solving for x and y, but I get quartic terms for y, it doesn't look like it will boil down, It was like over 2 pages of...

Homework Statement If Z1+Z2+Z3=0 and Z1*Z2 + Z2*Z3 + Z3*Z1=0 and Z1, Z2, Z3 are all complex, what is the value of (|z1|+|z2|+|z3|)/(|z1*z2|+|z2*z3|+|z3*z1|) Homework EquationsThe Attempt at a Solution I tried to multiply the equations by the product of all conjugates and reach some...

Humans, other animals, plants, fungi and almost all other forms of complex, multi-cellular life are known as eukaryotes. How eukaryotes evolved from simpler prokaryotic organisms is a major question in evolutionary biology. The current view is that eukaryotes evolved from the fusion between a...

Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...

I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...

Evaluate $$\sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}$$.

I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex...

Homework Statement y = 27 Homework Equations The Attempt at a Solution - I calculated the total impedance. - Divide it with the voltage to get the current. - Then I use the load impedance to find the voltage load. - And I calculated the complex power for the load. I am not comfortable...

Hello folks, 1.- In geometry we study for example the conic sections, their exentricity and properties. I was wondering what part of the mathematical science studies the different properties of complex valued distributions. One example are the spherical armonics. I guess mathematicians have...

Homework Statement I am having trouble solving systems of equations when they contain complex numbers. The context is circuit theory and phasors. For example, I am given this And the goal is to find I2 and Voc, which you can see the answers for. I just don't know how to manipulate the numbers...

I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j. What would be the effect on a vector, like 3i + 4j...

I was taught a scalar is a quantity that consists of a number (positive or negative) and it might include a measuring unit, e.g. 6, 5 kg, -900 J, etc. I was wondering if complex numbers like 3 + 7j (where j is the square root of minus 1) can be considered scalar quantities too, or is it that...