# 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).

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1. ### A Question about proof of Great Picard Theorem

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\}##...
2. ### A Existence of a limit implies that a function can be harmonic extended

##\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...
3. ### A Question about branch of logarithm

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...
4. ### I Question about uniform convergence in a proof

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...
5. ### Is the Maximum Modulus Theorem Applicable if E2 is Open and Non-Constant?

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 ?
6. ### A Complex analysis -- Essential singularity

Can you give me two more examples for essential singularity except f(z)=e^{\frac{1}{z}}? And also a book where I can find those examples?
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### A Cauchy’s integral theorem and residue theorem, what’s the difference

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!
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### Complex contour integral proof

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?
9. ### Continue solutions of ODEs around the origin

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##...
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11. ### I Bounding modulus of complex logarithm times complex power function

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##...
12. ### I On a remark regarding the Cauchy integral formula

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...
13. ### What is a point of using complex numbers here?

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 +...
14. ### I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?

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...
15. ### I Using Residues (Complex Analysis) to compute partial fractions

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...
16. ### I need sources to learn about dynamic systems

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...
17. ### Expressing Feynman Green's function as a 4-momentum integral

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...
18. ### Prove by induction the sum of complex numbers is complex number

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
19. ### A Laurent series for algebraic functions

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...

27. ### Finding analyticity of a complex function involving ln(iz)

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...
28. ### Analysis Opinions on textbooks on Analysis

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...
29. ### What Function Satisfies the Derivative Equation in Complex Analysis?

Mentor note: Edited to fix LaTeX problems ##f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy## ## f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy) ## ## \Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}##
30. ### Help with the Python package Scipy and the Z-transform please

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...
31. ### MHB Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?

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...

40. ### Cauchy-Riemann Theorem Example in Physics

I was thinking of the wavefunction in QM but I'm not sure how it's used and when.
41. ### A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

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...
42. ### Complex Analysis - find v given u

Solution Attempt: \begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} = (x^2+y^2)^{-1} -x (x^2+y^2)^{-2} (2x) = (x^2+y^2)^{-1} - 2x^2 (x^2+y^2)^{-2} \\ \rule{0mm}{18pt} \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} = -x (x^2+y^2)^{-2} (2y) =...
43. ### I Solution to the 1D wave equation for a finite length plane wave tube

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...
44. ### Courses Complex Analysis Courses or Complex Variable Courses?

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...
45. ### A Closure of constant function 1 on the complex set

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...
46. ### Complex analysis: find contradiction of a relationship

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...
47. ### Show that an image of a Schlicht function contains ##\Delta(0,1/2)##

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...
48. ### Determine the singularity type of the given function (Theo. Phys)

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...
49. ### MHB Understanding Differentiability and Continuity in Complex Analysis

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...
50. ### Kramers-Kronig Relations: Principal Value

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...