What is Complex analysis: Definition and 781 Discussions
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).
I need help please! So I'm reading 'Complex made simple' by David C. Ullrich. I made all the requirements for this proof but the author don't give the proof of this final theorem, instead it gives a similar proof for another set of theorems.
Let ##\mathbb{D}' = \mathbb{D} \setminus \{0\}##...
##\textbf{Theorem}##
If ##u: \mathbb{D'} = \mathbb{D} \setminus \{0\} \to \mathbb{R}## is harmonic and bounded, then ##u## extends to a function harmonic in ##\mathbb{D}##.
In the next proof ##\Pi^+## is the upper half-plane.
##\textbf{Proof}##: Define a function ##U: \Pi^{+} \rightarrow...
I have a question about Daniel Fischer's answer here
Why the function ##g(w)## is well-defined on ##\mathbb{D} \setminus \{0\}##? I don't understand how ##\log## function works here and how a branch of ##\log## function can be defined on whole ##\mathbb{D} \setminus \{0\}##. For example...
The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265)
Proposition 14.5. Suppose ##D## is a bounded simply connected open set in the plane, and let ##\phi: D \rightarrow \mathbb{D}## be a conformal equivalence.
(i) If ##\zeta## is a simple boundary point of...
In the maximum modulus therem we have two sets $E1={z∈E:|f(z)|<M}E2={z∈E:|f(z)|=M}}$ I know that the set E1 is open because pre image of an open set. But should't be also E2 closed because pre image of only a point ?
May I ask when we should use Cauchy’s integral theorem and when to use residue theorem? It seems for integral 1/z, we can use both of them. What are the conditions for each of them?
Thanks in advance!
I’ve attached my attempt. I’ve tried to use triangle inequality formula to attempt, but it seems I got the value which is larger than 1. Which step am I wrong? Also, it seems I cannot neglect the minus sign in front of e^(N+1/2)*2pi. How can I deal with that?
What confuses me is that my solution differs from that given in the answers at the back of the book.
Solving the ODEs is fairly simple. They are both separable. After rearrangement and simplification, you arrive at ##x(z)=Cz^{1/2}## for a) and ##x(z)=De^{1/z}## for b). In both solutions, ##C##...
It is claimed that the modulus of ##(\log(z))^jz^\lambda##, where ##j## is a positive integer (or ##0##) and ##\lambda## a complex number, can be bounded above by ##c|z|^l## for some integer ##l## and constant ##c##. Assume we are on the branch ##0\leq \mathrm{arg}(z)<2\pi## (yes, ##0##...
The way the formula is stated in my noname lecture notes is as follows:
Then they remark that:
The last sentence puzzles me deeply, specifically the part "When ##|z_0|<R##...". Why can we expand ##\frac1{z-z_0}## when ##|z_0|<R##? This makes little sense to me. I have noticed several typos...
Firstly, the exercise itself is not difficult:
On one hand, $$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$
On the other hand, ##(a + ib)(c + id) = p+ iq## for some integers p and q, and so $$|(a + ib)(c + id)|^2 = |p + iq|^2 = p^2 + q^2.$$
Thus, ##MN = p^2 +...
Hi everyone in the following expression
##f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega ##
the book says I can't swap integrals bacause the function
##f(u) e^{i \omega(t-u)}## is not ## L^1(\mathbb{R} \times \mathbb{R})##
why ? complex...
Dear Everybody, I am wondering how to compute the partial fraction decomposition of the following rational function: ##f(z)=\frac{z+2}{(z+1)^2(z^2+1)}.##
I understand how to do the simple poles of the function and how it is related to the decomposition's constants, i.e...
Hi! I have exam in couple of weeks , and now am looking for sources to learn about dynamic systems , chaotic systems and etc.
My main goal is to learn characteristics of such systems , learn about special points in dynamic plane.
For the most part we used to create dynamic system simulations...
I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...
hi guys
i found this problem in a set of lecture notes I have in complex analysis, is the following function real:
$$
f(z)=\frac{1+z}{1-z}\;\;, z=x+iy
$$
simple enough we get
$$
f=\frac{1+x+iy}{1-x-iy}=
$$
after multiplying by the complex conjugate of the denominator and simplification
$$...
I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me :
If ##g: D \rightarrow \mathbb{C}...
I came across this question on chegg for practice as I'm self learning complex analysis, but became stumped on it and without access to the solution am unable to check.
Let $$ f(z)=\frac{cos(z)} {(z-π/2)^7} $$. Then the singularity is at π/2. And on first appearance, it looks like a pole of...
Summary:: summation of the components of a complex vector
Hi,
In my textbook I have
##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}##
##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}##
For ##\hat{e_p} = \hat{x}##...
I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## .
I need to show that this is equal to ## \frac{1-...
Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).
Hi,
I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$
Since I have 2 singularities which are double poles.
I'm using this formula
$$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$
then,
$$\lim_{z\to\ \pm ia}...
Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem.
The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...
What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them.
Note- I am now studying real analysis from...
Hello everyone !
I am working on ultrasound scan and the processing of the signal received by the probe. I made the model I wanted and as I do not have an ultrasound scan machine I want to simulate the signal processing.
I will do that with the Python package Scipy and the function...
Hey! :giggle:
Question 1:
If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get...
I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof.
He is evaluating an integral along a contour which makes up the imaginary axis. He has the following:
$$ -i\int_{-iR}^{iR}...
I find the following definition in my complex analysis book :
Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one...
Hello,
I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane.
This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...
Can anyone suggest a good lecture series on Complex Analysis on YouTube? I have already searched on YouTube myself, and there are a few. But I wanted to know if any of you would recommend some particular lecture series which you consider to be good.
I don't know how to start with the factorization.
$$\frac{(-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}{(-1)^{2/9}- (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}$$
Any hints would be nice. Thank you.
My homework is on mathematical physics and I want to know the concept behind Laurent series. I want to know clearly know the process behind attaining the series representation for the expansion in sigma notation using the formula that can be found on the attached files. There are three questions...
$$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$
So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise.
The generalized complex integral would be
$$\int_{C} \frac{e^{-i \alpha...
Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
Hi there! This is my first post here - glad to be involved with what seems like a great community!
I'm trying to understand the acoustics of a finite plane-wave tube terminated by arbitrary impedances at both ends. So far all of the treatments I've managed seem only to address a different...
Hello,
My university offers a couple Complex Analysis courses, among them there is one with the following description:
Introduction to complex variables:
"substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical...
I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval.
At moment 46:46 minutes above we consider the constant function 1
$$f:[0,2\pi] \to \mathbb{C}$$
$$f(x)=1$$
The question is that:
How can we show that the...
I have reached a conclusion that no such z can be found. Are there any flaws in my argument? Or are there cases that aren't covered in this?
Attempt
##\log(\frac{1}{z})=\ln\frac{1}{|z|}+i\arg(\frac{1}{z})##
##-\log(z)=-\ln|z|-i\arg(z)##
For the real part...
Hello everyone this was a problem on one of the exams from last year and I'm having trouble with the last point ##3##
my solution for ##1##
$$\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)}{z}(1+\frac{z}{2re^{i\theta}}+\frac{re^{i\theta}}{2z})dz =$$
I divided this integral into 3 different ones and...
NOTE: Was not sure where to post this as it is a math question, but a part of my "Theoretical Physics" course.
I have no idea where to start this and am probably doing this mathematically incorrect.
given the function f(z) = cos(z+1/z) there should exist a singular point at z=0 as at z = 0...
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...
I'm kind of confused on how to evaluate the principal value as it's a topic I've never seen in complex analysis and all the literature I've read so far only deals with the formal definition, not providing an example on how to calculate it properly. Therefore, I think just understanding at least...