Complex Definition and 1000 Threads

I need recommendation about complex analysis book. As I'm electrical eng. student, it should cover everything one engineer need to know about that mathematical field, but without strict mathematical formalism :)

When I solve a quadratic equation I need to find a Discriminant. If D>0 I have no problem. I can find x1 and x2. And when I draw a parabola I can see the x1 and x2 on a X-line. But when D<0 I don't understand where I can find x1 and x2 on a plot of function. For example for 5x2+2x+1=0 I...

Calculate ( minus ( 2 over 3 ) + ( 2 over 3 ) i ) to the power minus 4, simplifing your answer and giving it in the form a + i b, with a and b given exactly.I found the modulus by: sqrt((-2/3)^2 + (2/3)^2) = (2*sqrt(2))/3 the argument is: pi - 1 (from a sketch in the complex plane) hence...

Hello, I'm trying to derive the perfectly matched layer for the TM mode Maxwell's equations using a complex coordinate stretching. As seen in http://math.mit.edu/~stevenj/18.369/pml.pdf . But I'm running in a bit of trouble somehow. \partial_t H_x =-\mu^{-1} \partial_y E_z\\ \partial_t H_y...

Homework Statement Show that if P(z)=a_0+a_1z+\cdots+a_nz^n is a polynomial of degree n where n\geq1 then there exists some positive number R such that |P(z)|>\frac{|a_n||z|^n}{2} for each value of z such that |z|>R Homework Equations Not sure. The Attempt at a Solution I've tried dividing...

Why do Complex Numbers arise in Quantum Mechanics' computations? What kind of physical significance do they carry? Someone told me to read this paper: W E Baylis, J Hushilt, and Jiansu Wei, Why i?, American Journal of Physics 60 (1992), no. 9, 788–797. But I found it difficult for me to...

If QM is a statistical model to approximate something underlying space time we don't quite understand yet, and there is a complex geometry underlying space time, is it possible to find other ways to simplify molecular optimizations and electron interactions in computational chemistry using...

Here's a link to a professor's notes on a contour integration example. https://math.nyu.edu/faculty/childres/lec12.pdf I don't understand where the ##e^{i\pi /2} I## comes from in the first problem. It seems like it should be ##e^{i\pi}## instead since ##-C_3## and ##C_1## are both on the real...

The way I understand it, they both have rectangular forms which are easy for addition/subtraction. Now I realize that the polar form of a complex vector can be simplified into an exponential, which is ideal for multiplication/division. But this is what confuses me; vectors don't multiply/divide...

I'm going to ask a very general question where I just would want to hear different possible methods that can be thought of in this kind of problem. I am trying to solve a very specific problem with this but I won't talk about that because I don't want someone to give me the answer but ideas for...

Homework Statement The largest value of r for which the region represented by the set { ω ε C / |ω - 4 - i| ≤ r} is contained in the region represented by the set { z ε C / |z - 1| ≤ |z + i|}, is equal to : √17 2√2 3/2 √2 5/2 √2 Homework Equations complex number = a + ib where a,b ε R The...

Hi, so I need to write a fortran code with 2, 2x2 matrices. These matrices are in the form of B=(1 exp(i)(theta) 0 0) and D=(0 0 exp(i)(theta) 1) where i is sqrt of -1 and theta is an angle between 0 and 2pi. I've expanded the exponential so it reads cos(theta)+isin(theta) and let theta=pi/2...

Homework Statement Homework Equations phasor forms voltage division current division The Attempt at a Solution Using superposition, considering only the varying voltage source. Z (L) = 4j Z (C) = 5j Total impedance: 4 is parallel with 5 = 2.44 + 1.95j series with 1 + 4j Total...

Hello, I am trying to understand how to get the residue as given by wolfram : http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2 The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...

Homework Statement In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample. (a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...

Homework Statement Good day, I've been have having difficulties finding the roots of this: Find the roots of 3ix^2 + 6x - i = 0 where i = complex number i = sqrt(-1) Homework Equations quadratic formula (apologies for the large image) The Attempt at a Solution...

Dear all, I am new to Wavelet field and I wanted to ask you for a help for an idea. I am supposed to create QRS complex (certain part of ECG signal wave) wave morphology classifier based on Wavelets in other words, I am supposed to create classifier which will separate waves with similar wave...

Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation. This is incorrect, but I think it is close: X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2] I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?

Edit: Sorry about the vague title, it was intended to be complex beam system boundary conditions but somehow it turned out like this. Hello, I am trying to learn complex beam system designs and I sometimes struggle to assign boundary conditions. For example I am trying to design the lifting...

(@mfb posted an article about this here, I think it deserves an own thread, and I did not find one, so I start one :smile:) The comet-like composition of a protoplanetary disk as revealed by complex cyanides Karin I. Öberg, Viviana V. Guzmán, Kenji Furuya, Chunhua Qi, Yuri Aikawa, Sean M...

Homework Statement How would Re(z)<0 be graphed? Homework Equations Re(z) is the real part of z The Attempt at a Solution It looks similar to y>x, but only shaded in the third quadrant, how can this be explained? not relevant anymore

Hi all, I'm preparing for a deferred exam this semester after falling ill last year. Just looking over my course notes and have a question. I understand how this works in the big picture scheme. What I don't understand however is how my instructor simplified the original equation. 1. Homework...

Hello, I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the resulting vectors are page long expressions and the reason for it is because I have four...

Hi, just starting with complex matrices, would appreciate checking if I'm on the right track. 1) Three angular momentum matrices satisfy the commutation relation: [Jx , Jy]= i.Jz If 2 of the matrices have real components, show the elements of the 3rd must be pure imaginary. (I assume pure...

Homework Statement I'm going crazy. I've done this problem nearly 20 times and keep getting the same answer. I've read my textbook so many times too! What am I doing wrong? Homework Equations Zcapacitor = 1/(jwC) Zinductor = jwL Zresistor = R The Attempt at a Solution Z for the...

Homework Statement I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained. Homework Equations 0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...

<<Moderator note: Remember that filling in the complete homework template is mandatory in the homework forums. This thread has not been deleted due to containing relevant replies.>> 1. Homework Statement ((x)^(1/x))' Homework Equations This probably isn't overly dificult, but it has got me...

Hi, In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional...

Hey all. I am giving Panhellenic exams this year, and part of my preparation comes from solving previous' years questions. This problem was a complex number based one, in the 2013 Panhellenic exams. It goes as follows: Consider 3 complex numbers, α0,α1 and α2 which belong in the line given by...

Hi, I am trying to find the current of a circuit using mesh analysis so far I have; My voltages V1 = 415 ∠ 90° or 0 + j415 V2 = 415 ∠ 0° or 415 + j0 impedances Z1 = j4 Z2 = j6 My formula is; -V1+Z1*I+Z2*I+V2=0 Which equates to; -0+j415+j4*I+j6*I+415+j0=0 Could...

Why must the gauge group be in a complex representation so that chirality of the fermions is respected? thanks

This problem has been on my mind for a while. ---------- **Problem:** Show that **if** \begin{equation} |z_1+z_2+\dots+z_n| = |z_1| + |z_2| + \dots + |z_n| \end{equation} **then** $z_k/z_{\ell} \ge 0$ for any integers $k$ and $\ell$, $1 \leq k, \ell \leq n,$ for which $z_{\ell} \ne 0.$...

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

Matrix A: 0 -6 10 -2 12 -20 -1 6 -10 I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of: 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 So, how do I find the nonzero eigenvectors of the...

For example, if someone with an architectural modeling software designed a beautiful mansion or other type of intricate and ornate structure, is there some equation in the computer or some type of mathematical structure to what was designed? Are there potential mathematical blueprints for all...

Do I have the right idea? I'm not getting what is in the solutions manual but it's been wrong on past chapter and I'm beyond frustration.

Homework Statement Here is my equation that I want to find theta angle Homework EquationsThe Attempt at a Solution I try to set different value of cos (theta) to find theta but it failed , I want to know the main solving strategy Thanks

Homework Statement Find the solutions to z^{\frac{3}{4}}=\sqrt{6}+\sqrt{2}i Homework Equations de Moivre's theorem The Attempt at a Solution z^{\frac{3}{4}}=2\sqrt{2}e^{\frac{\pi i}{6}}=2\sqrt{2}e^{\frac{\pi i}{6}+2k\pi}=2\sqrt{2}e^{\frac{\pi +12k\pi}{6}i} z=4e^{\frac{4}{3}{\frac{\pi...

Hi PF! I am trying to run the following plot: k = .001; figure; hold on [X,Y]=meshgrid(-4:0.01:4); a = 5.56*10^14; b = .15/(2*.143*10^(-6)); for n = 1:8 k = k*2^(n-1); Z = a./(X.^2+Y.^2).*exp(b.*(X-sqrt(X.^2+Y.^2)))-k; contour3(X,Y,Z) end which works great if a = b = 1. But now...

I've just encountered the following theorem : If T is a linear operator in a complex vector space V then if < v , Tv > =0 for all v in V then T=0 But the theorem doesn't hold in real 2-D vector space as the operator could be the operator that rotates any vector by 90 degrees. My question...

Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...

Homework Statement In the circuit shown, R1 = 0.810 Ω, R2 = 8.10 Ω, R3 = 81.0 Ω, and R0 = 810 Ω. See attachment Calculate the equivalent resistance of the circuit when a 7.70 V power supply is connected between points A and C. Calculate the current through R0 and R1 Homework...

Geologists Climb Into Iceland Volcano, Come Out With Stunning Images http://news.yahoo.com/geologists-climb-iceland-volcano-come-stunning-images-132015005.html Talk about a hot job. I hope they get samples for isotopic analysis. There are "small vents of blue sulfur-dioxide gas rise from the...

Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...

Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...

Homework Statement Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique? (a) f(1/n)=4 for n=1,2,\dots (b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots A side question: Is corollary 1 from my...

Does anyone know of any tools/graphs that show relationships in networks in a more "smoky" form that evolves in time? Preferably that correlates with spatial location though not necessarily to a map. This is a "normal" picture and not what I am looking for, rather something for which the...

Hi, I need to plot the last function of this: But I don't know how to generate the sum. I know the for loop is totally wrong, but I can't go any further. This is what I have: Can someone fix the summation loop part for me? Thanks in advance

The configuration and dimensions of any experiment are important in determining wave amplitudes. Then why are the orientations of complex waves not considered when they are added? For example in two dimensions; To find a resulting wave at a point P1 from two paths R1 & R2 we have...

This really isn't a homework question but I wasn't sure where to post it. I was watching a video by numberphile about complex numbers and the professer being interviewed said the most important thing about complex numbers is that they help bring algebra and geometry together. What did he mean by...