Complex Definition and 1000 Threads

I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero. On one hand, as we can in principle choose whatever values we like for ##m##...

Hi everyone, I want to post this exercise and my attempt to a solution since there are a couple of points I am not entirely sure of and I might need your help. I'll address them while posting my solution 1) ##Z_N = (Z_1)^N##. I can evaluate ##Z_1## integrating with respect of both parts of the...

I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0. Thank you

I find this interesting. A pretty detailed description, of a complex geological series of events, that can't be directly seen. Here's my summary: In 2018 an usual humming was picked up by seismic equipment an island off Africa, a magma pool drained, flowed up a dyke, when horizontal, and then...

##e^{-0.6x}\sin{(5x)}-0.1=0## I have posted my graphical solution to this problem. But, how do I solve this numerically/mathematically without graphing it?

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...

I am struggling to figure out how to calculate the expectation value because I am finding it hard to do something with the exponential. I tried using Euler's formula and some commutator relations, but I am always left with some term like ##\exp(L_z)## that I am not sure how to get rid of.

Z can be any point on the argand diagram so if z molous is less than 2 , is that somehow giving us the distance from origin? But how i assumed mod sign only makes things positive therefore its not sqrt( (x+yi)^2 ) = distance ??

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

Greetings, I'd like to simulate a complex mechanical automaton with lots of gears, cams, levers and springs. Most of the parts are going to be 3D printed except for some metal springs, rods and bearings. I want to make sure everything fits together and works as expected. Here is just a small...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ... Examples 1.1 and 1.2, Section 1.2...

I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ... I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows: At the start of the above proof by Markushevich we read the following: "If...

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I am getting a complex number for my transmission angle in part (c) but I do not know what that means. Am I even doing this correctly? Any help will be greatly appreciated. Thanks!

Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...

Dear all. I can't understand how to derive Eq.(2.3a). Fourier coefficients, ##A_j## and ##B_j## are described by summation in this paper as (2.2).　I think this is weird. Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms". and I understand...

I'm looking for research projects in the study of complex system, meanly applied in social and biology problems. I would be so grateful with the indication of researchers in the theme. Thanks!

All i was able to think was that i have to find a point (x,y) such that sum of its distances from points (0,0),(1,0),(0,1) and (3,4) is minimum.I tried by assuming the point to be centre of circle passing through any of the above 3 points,But it didn't helped me.

I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ... I need help in order to fully understand a remark of Apostol in Section 16.1 ... The particular remark reads as follows: Could someone...

I can do question (a). For question (b), I can not see the relation to question (a). Can we really do question (b) using result from (a)? Please give me little hint to relate them Thanks

Homework Statement: Hi there, I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an...

Summary: Trouble with infinity and complex numbers, just curious. I'm not too familiar with set theory ... but <-∞, ∞> contains just real numbers? Does something similar to <-∞, ∞> exist in Complex numbers? My question, is it "wrong"?

Summary: Which properties of ##\mathbb{C}## are actually necessary? The following is speculative as well as a honestly meant question about the way QM is modeled. I don't want to create a new theory, just understand the necessities of the old one. Physicists use complex numbers for QM. But...

--Continued-- 7) Let ##\sum_{k=0}^9 x^k = 0## Find smallest positive argument. Same thing as previous question, but I guess I can expand to ##z+z_{2}+z_{3}+...+z_{9}=0## ##z=re^{iθ}## ##re^{iθ}+re^{2iθ}+re^{3iθ}+...## What do I do to proceed on? Cheers

Hello all! Thanks for helping me out so far :) Really appreciate it. I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :) 4) Take ##3+7i## is a solution of ##3x^2+Ax+B=0## Since ##3+7i## is...

Hello, here with some complex number questions which I need some assistance in checking :) 1) z=3+5i1+3iz=3+5i1+3i Find Re(z) and Im(z) My answer is 9595 and −25−25 respectively. Checked by Wolfram 2) Find principal argument of the complex numberz=−5+3iz=−5+3i and express it in radians up to 2...

I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ... The two books are as follows: "Functions of a Complex Variable...

Hi. If I look at the function ## (z^2+z-2)/(z-1)^2## it appears to have a double pole at z=1 but if I factorise the numerator I get ##z^2+z-2 = (z+2)(z-1)## and it is a simple pole. Is it wrong to say it is a double pole ? If I overestimate the order of the pole in this case as 2 and...

I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ... I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ... The statement of Theorem 3.1.1 (A1) reads...

I'm kind of stuck over here at one part, but something is telling me that it might be wrong too :( Do assist, thanks.

Hi. I have been trying to calculate the real definite integral with limits 2π and 0 of ## 1/(k+sin2θ) ## To avoid the denominator becoming zero I know this means |k|> 1 Making the substitution ##z= e^{iθ}## eventually ends up giving me a quadratic equation in ##z^2## with 2 pairs of roots...

## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ## ## u_y = -6xy## and ## v_x = -6xy## To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x## Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0 However...

Hi I know that division of a real number by zero is not defined. I just came across the following in a textbook on Complex Analysis by Priestley , " we are allowed to divide a complex number by zero as long as the complex number ≠ 0 " Is this correct ? What happens if the complex number is...

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So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...

Problem Statement: Computing AC voltage Relevant Equations: cut off frequency, ac voltage. Hey guys, Amazing to be in this group ! Please find attached the diagram of the problem. E = 10kV at 60Hz, C1 = 60pF and R1 = 1k. I have to compute the ac voltage in reference to ground at point [1]...

If I had to guess what the complex matrix would look like, it would be: ##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix} a+bi & 0 \\ 0 & -b+ai\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}## I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover...

Why do snowflakes freeze into complex geometric patterns?

Problem Statement: What is the correct way of computing the argument of the following equation? Relevant Equations: I am trying to compute the argument ##\Phi## of the equation $$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$ which using Euler's equation...

I've attached my work below. The numbers seem odd to me though. Are my equations correct? Is the phase angle really (0/12)? If so, what are the implications of that?

As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?

The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...

Background of problem comes from Drude model of a metal (not necessary to answer my problem but for the curious): Consider a uniform, time-dependent electric field acting on a metal. It can be shown that the conductivity is $$\sigma = \frac{\sigma_0}{1-i\omega t}$$ where $$\sigma_0 =...

By applying the Fourier transform equation, and expanding the dot product, I get a sum of terms of the form: $$V(k)=\sigma_1^x\nabla_1^x\sigma_2^y\nabla_2^y\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|}e^{-ik(r_2-r_1)} =...

sz+tz*+r=0=say w so w* = s*z* + t*z + r*=0 Now , w+w* = (s+t*)z + (t+s*)z* + r+r* = 0 = p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w) which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number. Now, again...