Complex Definition and 1000 Threads

Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##. However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for...

Hi. I have looked through an example of working out a trig integral using the residue theorem. The integral is converted into an integral over the unit circle centred at the origin. The singularities are found. One of them is z1 = (-1+(1-a2)1/2)/a It then states that for |a| < 1 , z1 lies...

Dear Everybody, I am having troubles figuring out why the plus or minus sign in this problem. The question is: Solve the equation $\sin\left({z}\right)=2$ for $z$ by using $\arcsin\left({z}\right)$ The work for this problem is the following: $\sin\left({z}\right)=2$...

Hi. I would like to ask regarding this function that keeps on cropping up on my study (see picture below). What I did is simply substitute values for A and b and I noticed that it ALWAYS results to a real number. If possible, I would like to obtain the "non imaginary" function that is...

Hi. I have 2 questions regarding removable singular points. 1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is...

Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...

Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?

Dear Everybody, I am wanting to check the solution to this question: Sketch the set of points determined by the given conditions: a.) $\left| z-1+i \right|=1$ b.)$\left| z+i \right|\le3$ c.)$\left| z-4i \right|\ge4$ work: I know (a.) is a circle with radius 1 and its center at (-1,1) on the...

Hi everyone. I have followed with interest in the work of physicists who have conducted research in the area of complex systems and nonlinear dynamics. For example, consider researchers such as Mark Newman from the University of Michigan, Jim Crutchfield of USC, or Doyne Farmer and David Wolpert...

Homework Statement $$\frac{1}{z}+\frac{1}{2-z}=1$$ Homework Equations Quadratic-formula and algebra The Attempt at a Solution Been struggling with this one.. I keep getting the wrong answer, but that isn't the worst part, I can live with a wrong answer as long as the math behind it is...

Homework Statement So the problem I have is this silly little equation.. $$\frac {z - 7}{z + 3} = i $$Homework Equations This is the thing, I don't think you need anything more advanced than basic algebra to solve this problem. The Attempt at a Solution And I've tried solving it doing the...

We see complex animals such as crabs living near deep sea volcanic vents. (Reference: https://ocean.si.edu/ocean-life/invertebrates/hydrothermal-vent-creatures) This is causing speculation that similar life may be living near deep sea volcanic vents on other world such as Europa. Did these...

Griffith says in problem 1.15 the potential energy has an imaginary part. my question is that any real case exists where the part of the potential energy is imaginary?

Homework Statement Prove that ## \sin(z_1+z_2) = \sin z_1\cos z_2+\sin z_2\cos z_1## such that ##z_1,z_2\in\mathbb{C}## Homework Equations ##\sin z = \sum\limits_{n=1, \mathrm{ odd}}^\infty (-1)^{(n-1)/2}\dfrac{z^n}{n!} = \sum\limits_{s=0}^\infty (-1)^s\dfrac{z^{2s+1}}{(2s+1)!}## ##\cos z =...

Homework Statement A monochromatic plane wave with wavelength 500µm is propagating through a dissipative medium with refractive index 1-0.0002i. It approaching the edge of the medium, and will pass out into free space. If the angle of incidence is not 90°, how much will the wave deflect as it...

The integral I'm looking at is of the form \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right) Where K \in \mathbb{R} and J \in \mathbb{C} The book I am following (Kardar's Statistical Physics of Fields, Chapter 3 Problem 1) asserts that by completing the square this...

Hello, I need to read a fortran data with complex numbers and real numbers, the first column is the real numbers, the second and third complex numbers (real, imaginary). I need to read the first 64 lines and then the next 64 lines in separate ways and save in a variable. for example read from...

This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...

Hello, I am a rising sophomore in Astronomy and Physics. I am taking complex variables next semester and was wondering the effort required to succeed in the class. There are some other classes I'd like to take, however I don't want to overload myself. I have taken up through multivariable calc...

1. Homework Statement . Figure 1 shows a 50 Ω load being fed from two voltage sources via their associated reactances. Determine the current i flowing in the load by: (a) Thevenin's theorem (b) Superposition (c) Transforming the two voltage sources and their associated reactances into current...

Hey, I tried to construct the derivation of the integral C with respect to Y: $$ \frac{\partial C}{\partial Y} = ? $$ $$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$ with $$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...

I'm self-studying QFT and attempting exercise 2.2 on Peskin & Schroeder. First off, I'm a bit confused on the logic the authors use in the quantization process. They first expand the fields in terms of these ##a_{\vec{p}},a_{\vec{p}}^\dagger## operators which, if I understand correctly, is...

Hey, first off, I'm not sure if this is the right section. If another section is better, please let me know and I'll be more careful next time. So, my problem is with a degree 3 complex polynomial. I'm given one zero of the equation, but since it is a complex zero, I can use the conjugate too...

Hi particular solution only. As an example of what I am talking about, this method works for this DE: $$ 4y' + 2y = 10\cos(x) \\ \\ 10 \cos(x) = \Re( 10 e^{j(x)} ) = \Re(e^{j(x)} \cdot e^{j(0)} ) \rightarrow \text{complex number that captures the amplitude and phase of 10 cos x is} \\ 10...

Homework Statement Identify the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}##. My pre-caculus is very rusty, so I am not sure if I am doing this correctly. Homework Equations ##x^2 +y^2= r^2## ##\forall z,z'\in\Bbb{C}, \vert zz'\vert =\vert z\vert\vert z'\vert##...

This is not a homework problem, I just am confused a little about the differences between a Nyquist plot and the plot of a complex function. I believe they are the same given the domain of the plot of a complex function is for all real numbers equal to or greater than zero. However, I am having...

Homework Statement (1+2i+3i2)/(1-2i+3i2) answer options : a : 1 b: -i c: i d: 0 Homework Equations what is the most easy method to solve it , The Attempt at a Solution are they conjugate to each other ? if they are than z/zconjugate =1 , but how can...

Homework Statement Find roots of $$ -\lambda ^3 +(2+2i)\lambda^2-3i\lambda-(1-i) = 0 $$ Homework EquationsThe Attempt at a Solution I tried my old trick I tried to separating the 4 terms into 2 pairs and try to find a common factor in the form of ##\lambda + z## between them, $$ -\lambda ^2...

Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$

Hi, I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form... 1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity? 2. But then...

Homework Statement [/B] Homework EquationsThe Attempt at a Solution I had no problems with part a and was able to form the equation of the circle and get its centre/radius. It's part b that I'm stuck on. My notes show that for Z < 3 I would shade inside the circle but the mark scheme for...

Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.

Hi. If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1. 1. Is this "another...

Hi I wonder if anyone can help. I am not even sure I am on the right forum. I cannot solve this equation for t. It is the final sequence of a number of equations in a book about modelling athletic performance using bioenergetics. I had a high school maths education 40 years ago and I’m stumped...

Mentor note: Thread moved from technical section, so missing the homework template. Hi all, I have a homework problem which asks me to compute the complex number cos(π/4 + π/4 i). I've been playing around with it for a while now and just can't seem to get the answer I get when using Wolfram...

Reading Chandrasekhar's The mathematical theory of black holes, I reached the point in which the Newman-Penrose GR formalism is explained. Actually I'm able to grasp and understand the usage of null tetrads in GR, but The null tetrads used in this formalism, are very special, since are made by...

Homework Statement I have a simple problem relating to the superposition of plane EM waves that I'd to try out using complex notation. Could anyone run through the work to see if my understanding is right? Many thanks in advance! The incident E bit of the wave is $$\vec{E}_I = E_0 \sin(ky -...

Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...

Homework Statement Question attached: Hi I am pretty stuck on part d. I've broken the fields into real and imaginary parts as asked to and tried to compare where they previously canceled to the situation now- see below. However I can't really see this giving me a hint of any sort unless...

Hi PF! I am integrating the following sin\[Theta][x_, \[Alpha]_] := Sqrt[(2 - 2 x^2)/( 3 - 4 x Cos[\[Alpha]] + Cos[2 \[Alpha]])] cos\[Theta][x_, \[Alpha]_] := Sqrt[(Cot[\[Alpha]] - x Csc[\[Alpha]])/( 1 + 2 x Cot[\[Alpha]] Csc[\[Alpha]] - 2 Csc[\[Alpha]]^2)] \[Rho][x_, \[Alpha]_] :=...

I am sure I am overlooking something elementary, but playing around with exponentiation (this is not an assignment), I seem to be making a mistake somewhere. Please don't send me a link for a more compact way of getting the correct result; I wish to know what my particular mistake is. Suppose...

DSP Guide .com has the highly rated textbook for digital signal processing. Chapter 30 pg 561 on Complex Numbers http://www.dspguide.com/ch30.htm (chapters are free to download) Hes talking about representing sinusoids with a complex number. Author states "Multiplying complex numbers A and...

The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...

Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...

The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...

Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...

Homework Statement If Z= (1)/(z conjugate) then Z : ? Homework EquationsThe Attempt at a Solution let z= a+bi the z conjugate= a-bi (a+bi)=(1)/(a-bi) (a+bi)(a-bi)=1 a2+b2=1 does it tell from this expresssion that the complex number is a pure real ?

Homework Statement if z=(x-iy)/(x+iy) then modulus of z is : Homework EquationsThe Attempt at a Solution (x-iy)/(x+iy)= (x2-y2-2x(iy))/(x2+y2) i can't get the real part and the imaginary part to take the modulus : but the answer in any way could be = 1 ? the answer in the book is 1 .

Homework Statement Value of x and y , when (x+yi)2= 5+4i Homework EquationsThe Attempt at a Solution x2+2x(iy)-y2=5+4i x2-y2=5 -------> (1) 2x(iy)=4i (imaginary part) xy=2 --------> (2) solving the two equations x=2.388 and y=0.838 or x=-2.388 or y=-0.838 is this the right way to solve...

Homework Statement Two equal line sources of strength k are located at x = 3a and x = −3a, near a circular cylinder of radius a with axis normal to the x, y plane and passing through the origin. The fluid is incompressible and the flow is irrotational (and inviscid). Use the Milne-Thomson...