Complex Definition and 1000 Threads

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) =...

Hello everyone, Let ##A = (\alpha_{ij})## be an $n \times n# complex matrix. Define ##\hat## acting on ##A## as producing the matrix ##\hat{A} = (\alpha_{ij} I_n)##. I don't understand what this is saying. Isn't ##I_n## the identity matrix, and therefore the product of it with any matrix...

Within the context of real numbers, the square root function is well-defined; that is, the function ##f## defined by: ##f(x) = \sqrt{x}## Refers to the principal root of any real number x. Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##...

Mod note: This thread was moved from a technical math section, so doesn't include the homework template. I know this has been asked before, but none of the other posts have helped me. I cannot for the life of me figure out how to solve a system of equations with complex numbers. Here is a very...

Is it possible to differentiate xa+bi where a and b are real ? if so what is the answer ?

Dear All, Please help me clear some doubts about Theorem 3.3.1 in the 1st attachment. The condition ## |a| = |b| ## has only 8 cases right? ## { x+iy. x - iy, -x + iy, -x - iy, y + ix, y - ix, -y + ix, -y - ix } ## so for the condition ## |a| = |b| ## and ## b \bar c = \bar a c...

Homework Statement A uniform 300-kg, 6.0 M long, freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.0 M long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at...

Homework Statement Attached Image Homework Equations this is not a simple plane curve or a close plane curve so I use the formula: ∫ F ⋅ dr/dt dt The Attempt at a Solution From the point (0,0) to (2,4) Direction Vector v(t) = <2-0, 4-0> Parametric Equation: r(t) = (2t + 0) i + (4t + 0) j...

I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...

Hello All, I would like to know why do we multiply the complex conjugate of velocity (not just the velocity) with the complex pressure to obtain the complex acoustic intensity. Could someone please help me with this? Regards, Radel...

Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...

Homework Statement ##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx## Homework EquationsThe Attempt at a Solution Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the...

Homework Statement Suppose we have the function ##f(z) = x + iy^2## and a contour given by ##z(t) = e^t + it## on ##a \le t \le b##. Find ##x(t)##, ##y(t)##, and ##f(z(t))##. Homework EquationsThe Attempt at a Solution Well, ##x(t)## and ##y(t)## are rather simple to identity. However, I am...

Homework Statement Find the Geometric image of; 1. ## | z - 2 | - | z + 2| < 2; ## 2. ## 0 < Re(iz) < 1 ## Homework EquationsThe Attempt at a Solution In both cases i really am struggling to begin these questions, complex numbers are not my best field. There are problems before this one...

Hello everyone, I have a rather simple question. I have the curve ## C(t) = \begin{cases} 1 + it & \text{if}~ 0 \le t \le 2 \\ (t-1) + 2i & \text{if }~ 2 \le t \le 3 \end{cases} ## which is obviously formed from the two curves. This curve is regarded as an arc if the functions ##x(t)## and...

Homework Statement Solve the equation ## z^3 − z^2 + z − 1 = 0 ## first by inspection, and then by the method described above. where Z is a complex number. (Alan F. Beardon, Algebra and Geometry) The method described above is shown in the attachment. Homework Equations The method is shown in...

Hi Guys, please help me. how can i derive the primitive vector of copper oxide (I)? basically this is cuprous oxide having a cubic crystal structure but since it has oxygen in it the directions and magnitude of primitive vector are far different compare to basic cubic structure. also please help...

Homework Statement let z' = (a,b), find z in C such that z^2 = z' Homework EquationsThe Attempt at a Solution let z = (x,y) then z^2 = (x^2-y^2, 2xy) since z^2 = z', we have, (x^2-y^2, 2xy) = (a,b) comparing real and imaginary components we have; x^2-y^2 = a, 2xy = b. Now, this...

Homework Statement Hello, I have been tasked with the next problem, I have to prove that the next two integrals are complex numbers; but I have no idea of how to attack this problem. Homework Equations ∫dx f*(x) x (-ih) (∂/∂x) f(x) integrating between -∞ and ∞ ∫dx f*(x) (-ih) (∂/∂x) (x f(x))...

How to calculate the force of wind pressure, for example for the surface tilted at 45 degrees? I need to find some serious work related to the wind pressure force calculated for different 3D shapes affected by wind from different angles. Can somebody recommend me the good names, keywords...

Homework Statement The Attempt at a Solution -1 < (z-w) /(1-z*w) < 1 [/B] Hi can give clue. I am clueless

Homework Statement 1) 2w+iz = 3; 2) (3-i)w - z = 1 +3i 2wi - z = 3i; 3w - iw - z = 1 + 3i Substract (2) from (1): 2wi - z - (3w-iw-z) = 3i - (1+3i) 2wi -3w +iw = -1 3iw - 3w = -1 3w(i-1) = -1 3w = -1/(i-1) = -0.5i - 0.5 w = -i/6 - 1/6 But the answer is i/6 +...

Homework Statement What are the region of validity of the following? 1/[z2(z3+2)] = 1/z3 - 1/(6z) +4/z10 Homework EquationsThe Attempt at a Solution Knowing that this is the expansion around z=0, I am trying to find the singularities of the complex function. Which is when z2(z3+2) = 0 I...

A theorem in my textbook states the following: For every n=0,±1, ±2, --- the formula ln z=Ln z ± 2nπi defines a function, which is analytic, except at 0 and on the negative real axis, and has the derivative (ln z)'=1/z. I don't understand why the logarithm isn't analytic for negative real...

Homework Statement Calculate the complex integral along the closed path indicated: $$ \oint_C\frac{\sin{z}}{z^2+\pi^2}dz,\,\,|z-2i|=2.$$ Homework Equations $$ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} $$ $$ e^{iz}=e^{i(x+iy)}=e^{-y+ix}=e^{-y}(\cos{x}+i\sin{x}) $$ The Attempt at a Solution I really...