Read about complex analysis | 171 Discussions | Page 1

  1. 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}...
  2. 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...
  3. 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...
  4. 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.
  5. 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...
  6. 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...
  7. 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.
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. MakVish

    Complex Analysis

    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...
  17. 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 &...
  18. 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...
  19. 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)...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. T

    Is this question incomplete? Regarding entire functions...

    Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
  28. T

    Finding residues with Laurent series.

    Homework Statement Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z-6}{z(2-z)}## ; ##\operatorname{Res}(f(z),0)## Homework Equations n/a The Attempt at a Solution Computations are done such that ##0 \lt \vert z\vert \lt 2##...
  29. Measle

    Complex Analysis - sqrt(z^2 + 1) function behavior

    Homework Statement Homework Equations The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi] The Attempt at a Solution The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
  30. Measle

    I Confused by the behavior of sqrt(z^2+1)

    (mentor note: this is a homework problem with a solution that the OP would like to understand better) In Taylor's Complex Variables, Example 1.4.10 Can someone help me understand this? I don't know what they mean by (i, i inf), or how they got it and -it
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