What is Complex analysis: Definition and 766 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. Ugnius

    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...
  2. R

    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...
  3. C

    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.
  4. T

    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...
  5. patric44

    Is f(z) =(1+z)/(1-z) a real function?

    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 $$...
  6. K

    Proving that a "composition" is harmonic

    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}...
  7. penroseandpaper

    Classifying singularities of a function

    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...
  8. H

    I Sum of the dot product of complex vectors

    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}##...
  9. C

    Proving geometric sum for complex numbers

    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-...
  10. A

    Engineering Book considering FEM analysis for complex eigenvalues (incl. damping)

    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?).
  11. R

    How to find the residue of a complex function

    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}...
  12. tixi

    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...
  13. Mr.Husky

    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...
  14. Dragounat

    Complex analysis problem

    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}##
  15. hugo_faurand

    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...
  16. M

    MHB Complex Analysis

    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...
  17. D

    I Help With a Proof using Contour Integration

    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}...
  18. F

    [Complex analysis] Contradiction in the definition of a branch

    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...
  19. F

    Show that the real part of a certain complex function is harmonic

    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...
  20. S

    B Is complex analysis really much easier than real analysis?

    This author seems to say so: https://blogs.scientificamerican.com/roots-of-unity/one-weird-trick-to-make-calculus-more-beautiful/
  21. murshid_islam

    I Any Good Lecture Series on Complex Analysis?

    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.
  22. Y

    How to simplify this complex expression?

    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.
  23. HansBu

    Laurent Series (Complex Analysis)

    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...
  24. Remixex

    Contour integration around a complex pole

    $$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$ 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...
  25. entropy2008

    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.
  26. qbar

    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...
  27. H

    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) =...
  28. jeremiahrose99

    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...
  29. Jonathanos

    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...
  30. V

    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...
  31. A

    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...
  32. W

    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...
  33. starstruck_

    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...
  34. Math Amateur

    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...
  35. CharlieCW

    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...
  36. Santilopez10

    I Complex analysis theorem

    Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole. It is easy to prove that if the pole is of simple order, the value of the...
  37. D

    Textbook Recommendations: Complex Analysis

    Hello, I was interested in learning more about complex analysis. Also, very interested in analytic continuation. Can anyone recommend a good text that focuses on complex analysis. Also, is there a good textbook on number theory that anyone recommends? Thanks! <mentor - edit thread title>>
  38. A

    Find radius of convergence

    Homework Statement This is from a complex analysis course: Find radius of convergence of $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ Homework Equations I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}## The Attempt at a Solution My first reaction is that this sum looks...
  39. MakVish

    Complex Analysis: Find Analytic Functions w/ |ƒ(z)-1| + |ƒ(z)+1| = 4

    Homework Statement Find all analytic functions ƒ: ℂ→ℂ such that |ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ and ƒ(0) = √3 i The Attempt at a Solution I see that the sum of the distance is constant hence it should represent an ellipse. However, I am not able to find the exact form for ƒ(z). Any help...
  40. Santilopez10

    Complex integral problem

    Homework Statement The following is a problem from "Applied Complex Variables for Scientists and Engineers" It states: The following integral occurs in the quantum theory of collisions: $$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$ where p is real. Show that $$I=\begin{cases}0 &...
  41. O

    I Showing that a function is analytic

    Say we have ##P_k(z)## a family of entire functions, and they depend analytically on ##k## in ##\Delta##. Assume ##P_k(z)## is nonzero on ##S^1## for all ##k##. How do I see that for each ##t \ge 0##, we have that$$\sum_{|z| < 1, P_k(z) = 0} z^t$$is an analytic function of ##k##? Here, the zeros...
  42. S

    How to write the complex exponential in terms of sine/cosine?

    I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...
  43. e0ne199

    Engineering Problems about Zin in complex circuit analysis

    1. Homework Statement the problem is my answer for question (a) is not the same as the answer provided by the question, i get 2.81 - j4.49 Ω while the answer demands 2.81 + j4.49 Ω Homework Equations simplifying the circuit, details can be seen below The Attempt at a Solution...
  44. A

    Online app which plots F(z) in the complex plane

    I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given. It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z. One possible...
  45. Baibhab Bose

    A Order of poles of z∕Sin(πz²)

    When The denominator is checked, the poles seem to be at Sin(πz²)=0, Which means πz²=nπ ⇒z=√n for (n=0,±1,±2...) but in the solution of this problem, it says that, for n=0 it would be simple pole since in the Laurent expansion of (z∕Sin(πz²)) about z=0 contains the highest negative power to be...
  46. A

    I Equating coefficients of complex exponentials

    I have an equation that looks like ##i\dot{\psi_n}=X~\psi_n+\frac{C~\psi_n+D~a~\psi^\ast_{n+1}+E~b~\psi_{n+1}}{1+\beta~(D~\psi^\ast_{n+1}+E~\psi_{n+1})}## where ##E,b,D,a,C,X## are constants. I have the ansatz ##\psi_n=A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast}##, ##x## and ##A_n,B_n## are complex...
  47. T

    Laurent expansion of ##ze^{1/z}##

    Homework Statement Find a Laurent series of ##f(z)=ze^{1/z}## in powers of ##z-1##. Is there an easier way to go about this as this is not a typical expansion I see on textbooks. It seems that my incomplete solution is too complicated. Please help, exam is in two days and I am working on past...
  48. T

    Evaluating a complex integral

    Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...
  49. G

    Complex analysis question

    Hey, I have been stuck on this question for a while: I have tried to follow the hint, but I am not sure where to go next to get the result. Have I started correctly? I am not sure how to show that the integral is zero. If I can show it is less than zero, I also don't see how that shows it...
  50. T

    When is an entire function a constant?

    Homework Statement Let ##f(z)## be an entire function of ##z \in \Bbb{C}##. If ##\operatorname{Im}(f(z)) \gt 0##, then ##f(z)## is a constant. Homework Equations n/a The Attempt at a Solution I don't get how the imaginary part of ##f(z)## would be greater than any number. Aren't complex...