What is Complex analysis: Definition and 778 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. I

    Question about complex analysis

    If we know f(x)^2 and f(x)^3 are both holomorphic, can we say that f(x) itself is also holomorphic? And how to prove that?
  2. D

    Complex Analysis: prove the function is entire

    Hey guys, i just started a complex analysis course this semester and we just went over CR-equations, and various ways to show that a function is holomorphic. I'm a bit stuck on this one homework question where we have to prove the function is entire. Homework Statement so we have...
  3. A

    SOS : Textbook for Thermodynamics & complex analysis

    Hi all, I would like some recommendations for thermodynamics. It's my first course in thermo. I'm currently using : Classical and Statistical Thermodynamics by Ashley Carter. I like the book, however it lacks examples! I am someone who learns by example...so this book isn't doing me much...
  4. R

    Complex analysis question: can't find similar question on internet

    this question doesn't seem tough but i can't find anything like it. \int\frac{e^{ax}}{1+e^{x}}dx along the real line (a is between 1 and 0). I know this is a complex analysis question, so i took the complex integral (along a semicircle where the diameter is the real numbers). by residue...
  5. M

    Finding the residue of a pole of order 2 (complex analysis)

    The problem Find Res(f,z1) With: f(z)=\frac{z}{(z^2+2aiz-1)^2} The attempt at a solution The singularities are at A=i(-a+\sqrt{a^2-1}) and at B=i(-a-\sqrt{a^2-1}) With the normal equation (take limit z->A of \frac{d}{dz}((z-A)^2 f(z)) for finding the residue of a pole of order 2, my attempt...
  6. F

    Solving F(z): Complex Analysis Explained

    I am trying to decipher what this means: F(z) = \overline{f(\bar{z})} Thanks for the help.
  7. L

    What is more important for theoretical physics: Complex analysis or EM II?

    Hello all, I'm curious as to the opinion of some people here about what is more important: Complex analysis or EM II for someone interested in going into theoretical physics (mainly particle theory). I have a hectic workload for next semester. I'm taking particle physics, EM II, grad...
  8. I

    Complex Analysis: Using polar form to show arg(z1) - arg(z2) = 2n*pi

    Homework Statement Given that z_{1}z_{2} ≠ 0, use the polar form to prove that Re(z_{1}\bar{z}_{2}) = norm (z_{1}) * norm (z_{2}) \Leftrightarrow θ_{1} - θ_{2} = 2n∏, where n is an integer, θ_{1} = arg(z_{1}), and θ_{2} = arg(z_{2}). Also, \bar{z}_{2} is the conjugate of z_{2}. Homework...
  9. B

    Complex Analysis: countour integral

    Homework Statement Compute the contour integral I around the following curve $\Gamma$: $ I = \int_\Gamma \dfraq{dz}{z^2 +1} $ see picture: http://dl.dropbox.com/u/26643017/Screen%20Shot%202012-01-07%20at%2010.39.58.png Homework EquationsThe Attempt at a Solution $\Gamma$ is an open curve...
  10. I

    Complex Analysis proof question (from Markushevich text))

    There is a proof offered in the text "Theory of Functions of a Complex Variable" by Markushevich that I have a question about. Some of the definitions are a bit esoteric since it is an older book. Here "domain" is an open connected set (in \mathbb{C}, in this case.) The proof that...
  11. gluons

    What is the name of this theorem? (complex analysis)

    I am working on a problem to evaluate integrals with simple poles offset by ε above/below the real axis. So something like this ∫ [ f(x) / (x-x0-iε) ] The answer is the sum of two integrals: the principal value of the integral with ε=0 plus the integral of iπδ(x-x0). I have done the...
  12. B

    Complex Analysis Residues at Poles

    Homework Statement Find the residue at each pole of zsin(pi*z)/(4z^2 - 1)Homework Equations An isolated singular point z0 of f is a pole of order m if and only if f(z) can be written in the form: f(z) = phi(z)/(z-z0)^m where phi(z) is analytic and nonzero at z0. Moreover, Res(z=z0) f(z) =...
  13. B

    [Complex Analysis] Transform a column into a line

    [Complex Analysis] Conformal mapping of a column into a line Homework Statement I'm having a problem with this problem (:rofl:) where I have to transform an area of 2 circles, the one contained within the other, into a straight line. I've managed to transform the circles using a Möbius...
  14. N

    [complex analysis] differentiation w.r.t. complex conjugate; does it make sense?

    Hello, Differentiability of f : \mathbb C \to \mathbb C is characterized as \frac{\partial f}{\partial z^*} = 0. More exactly: \frac{\partial f(z,z^*)}{\partial z^*} := \frac{\partial f(z[x(z,z^*),y(z,z^*)])}{\partial z^*} = 0 where z(x,y) = x+iy and x(z,z^*) = \frac{z+z^*}{2} and...
  15. N

    Complex analysis in electrical circuits

    When discussing the i (the imaginary unit) in a math class, my math teacher commented that that complex analysis is used in studying electrical circuits. I know a little about resistors and what not, but never have I seen complex analysis used this way. I've tried looking it up, but it's been...
  16. P

    Proving f is a Polynomial with Degree at Most n in Complex Analysis

    Let D ⊂ ℂ be a domain and let f be analytic on D. Show that if there is an a ∈ D such that the kth derivative of f at a is zero for k=n, n+1, n+2,..., then f is a polynomial with degree at most n. So I believe I have a proof, but the theorems are so powerful I feel like I might be...
  17. M

    Liouville-type problem (Complex analysis)

    Homework Statement If f(z) is an entire function such that f(z)/z is bounded for |z|>R, then f''(z_0) = 0 for all z_0. Homework Equations Liouville's theorem Cauchy estimates: Suppose f is analytic for |z-z_0| ≤ ρ. If |f(z)|≤ M for |z-z_0| = ρ then the mth derivative of f at z_0 is...
  18. P

    Complex Analysis Integral by Substitution

    Im trying to take the integral, using substitution, of \int_0^1\frac{2\pi i[cos(2\pi t) + isin(2\pi t)]dt}{cos(2\pi t)+isin(2\pi t)} So I set u=cos(2\pi t)+isin(2\pi t)du=2\pi i[cos(2\pi t) + isin(2\pi t)]dt Yet when I change the endpoints of the integral I get from 1 to 1, which doesn't make...
  19. K

    Solving Complex Analysis Integral Using Residues

    Homework Statement compute I=∫_2^∞ (1/(x(x-2)^.5)) dx using the calculus of residues. be sure to choose an appropriate contour and to explain what happens on each part of that contour. Homework Equations transform to a complex integral I= ∮ (1/(z(z-2)^.5)) dz The Attempt...
  20. B

    Complex Analysis Integration with Sin and Cos

    Homework Statement Compute the integral from 0 to 2∏ of: sin(i*ln(2e^(iθ)))*ie^(iθ)/(8e^(3iθ)-1) dθ (Sorry for the mess, I don't know how to use latex) Homework Equations dθ=dz/iz sinθ = (z - z^(-1))/2i The Attempt at a Solution So I tried to change it into a contour integral of a...
  21. N

    Proving Entire Function f(z) is Constant | Complex Analysis Proof

    Q:Let f be entire and suppose that I am f(z) ≥ M for all z. Prove that f must be a constant function. A: i suppose M is a constant. So I am f(z) is a constant which means the function is a constant. Am i doing this right ? but i don't think there will be such a stupid question in my...
  22. N

    Finding the Suitable Parametrization for Computing ∫Cr (z - z0)n dz

    compute the integral ∫Cr (z - z0)n dz, with an integer and Cr the circle │z - z0│= r traversed once in the counterclockwise direction Solution: A suitable parametrization for Cr is give by z(t)= z0 + reit 0≤t≤2π ... ... My question is , how to find that suitable z(t)? i have no idea...
  23. T

    Complex Analysis: brach of the square root

    Homework Statement Let f be a quadratic polynomial function of z with two different roots z_1 and z_2. Given that a branch z of the square root of f exists in a domain D, demonstrate that neither z_1 nor z_2 can belong to D. If f had a double root, would the analogous statement be true?Homework...
  24. F

    Partial Fractions but in Complex Analysis

    Homework Statement Use partial fractions to rewrite: (2z)/(z^2+3) Homework Equations noneThe Attempt at a Solution I did this: (2z)/(z^2+3) = (Az+B)/(z^2+3) 2z = Az +B A = 2, B = 0...problem is that it just recreates the original Here is their example in the book: 1/(z^2+1) =...
  25. P

    Show No Continuous Function (Complex Analysis)

    Let n ≥ 2 be a natural number. Show there is no continuous function q_n : ℂ → ℂ such that (q_n(z))^n = z for all z ∈ ℂ. The only value of this function we can deduce is q_n(0)=0. Moreover any branch cut we take in our complex plane will touch zero. These two facts would make me a bit...
  26. N

    Intro to complex analysis problems.

    1) How do you integrate 1/ [z^2] over the unit circle? After you integrate, do you put it in polar form or do you replace z with x + iy then solve it? I keep getting zero. It should exist since z=o is undefined, right? 2) How do you integrate x dz over gamma, when gamma is the...
  27. T

    Definition of Analytic Functions in Complex Analysis

    In Mathematics of Classical and Quantum Mechanics by Byron and Fuller, they state that "Some authors (never mathematicians) define an analytic function as a differentiable function with a continuous derivative." ..."But this is a mathematical fraud of cosmic proportions.. " Their main point...
  28. J

    Complex Analysis, Line Integrals and Cauchy Conceptually

    I am just trying to get the conceptual basics in my head. Can I think of things this way... If you are taking the integral of a function f(z) along a curve γ in a region A. If the curve is closed and f(z) is analytic on the entire curve as well as everywhere inside the curve, then the...
  29. N

    Complex Integration: Contour Evaluation and Estimation Lemma

    1.Evaluate ∫C Im(z − i)dz, where C is the contour consisting of the circular arc along |z| = 1 from z = 1 to z = i and the line segment from z = i to z = −1. 2. Suppose that C is the circle |z| = 4 traversed once. Show that §C (ez/(z+1)) dz ≤ 8∏e4/3 For question 1, should i let z=...
  30. S

    Complex Analysis: Solve Last Question in Section 1.3

    Hi All, I am trying to learn complex analysis on my own and for this I have chosen Fundamentals of Complex Analysis by Saff and Snider. I am stuck at the last question in section 1.3 which is as follows. For the linkage illustrated in the figure, use complex variables to outline a scheme...
  31. S

    How Can I Use Complex Variables to Solve for Arm Positions in a Linked System?

    Hi All, I am trying to learn complex analysis on my own and for this I have chosen Fundamentals of Complex Analysis by Saff and Snider. I am stuck at the last question in section 1.3 which is as follows. For the linkage illustrated in the figure, use complex variables to outline a scheme...
  32. N

    Find and P ((1-sqrt3 i)1-i) in Complex Analysis homework

    I have a homework question: Find (1 − \sqrt{3} i)1−i and P((1 − \sqrt{3} i)1−i). I don't know what does the P stand for, And i can't find it in the textbook either. Thanks
  33. N

    Derivative of a Complex Function: Finding f'(1 + i)

    [b]1. Let f(z) = (3e2z−ie-z)/(z2−1+i) . Find f′(1 + i). 3. Should I sub (1+i) to z and then diff it by i. Or i need to diff it by z first then sub (1+i) in it at last? Thanks
  34. T

    Complex Analysis: Liouville's Theorem

    Homework Statement Let f=u+iv be an entire function. Prove that if u is bounded, then f is constant. Homework Equations Liouville's Theorem states that the only bounded entire functions are the constant functions on \mathbb{C} The Attempt at a Solution I know that if u is bounded...
  35. F

    How Do You Decompose a Complex Function into Real and Imaginary Parts?

    One last simple question about complex analysis... Hi, sorry again for having made so many threads. I have one remaining question about complex analysis that I keep get confused on. Say that I have some complex function h(z). Sometimes I am really confused how to break that down into...
  36. J

    Definite integral using complex analysis

    Homework Statement Considering the appropriate complex integral along a semi-circular contour on the upper half plan of z, show that \int^{\infty}_{\infty} \frac{cos(ax)}{x^2 + b^2} dx = \frac{\pi}{b}e^{-ab} (a>0, b>0) Homework Equations \int_{C} = 0 For C is a semi-circle of...
  37. N

    Understanding Connectedness in Planar Sets: A Brief Overview

    I don't get the meaning of "connected" in the chapter of planar sets. The textbook said " An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S" So do i just randomly pick 2 points in S to check if they are both in...
  38. I

    Prerequisites for graduate complex analysis?

    I'm wondering whether I could take graduate level complex analysis this spring. I planned on taking complex variables (undergraduate course), but unfortunately it conflicts with another course I want to take. I'm currently taking basic real analysis (not at the level of Rudin), point-set...
  39. T

    Complex Analysis: Properties of Line Integrals

    Homework Statement Demonstrate that \int_{-\gamma} f(z)|dz|=\int_{\gamma} f(z)|dz| where \gamma is a piecewise smooth path and f is a function that is continuous on |\gamma|. Homework Equations The Attempt at a Solution This proof seems like it should be very simple, but I am...
  40. T

    Complex Analysis: Line Integrals

    Homework Statement I have a problem as follows: Let \gamma=\beta+[e^2\pi,1] where \beta is given by \beta(t)=e^{t+it} for 0\leq 2 \leq \pi. Evaluate \int_\gamma z^{-1} dz . Homework Equations The Attempt at a Solution I know that I need to parameterize the path and I have...
  41. D

    Going over complex analysis past exam

    Hey, I've been going through a few past papers for an upcoming exam on complex analysis, I found this T/F question with a few parts I'm not confident on, I'll explain the whole lot of my work and show.[PLAIN]http://img404.imageshack.us/img404/2069/asdasdsu.jpg a) |2+3i|=|2-3i| so false b)...
  42. M

    Solve Complex Analysis Problem: Find 4 Points Mapping to 1

    i am trying to solve below problem but not getting start; so please help The function f(z) = e^(z+i*pi) has infinitely many points in the fiber of each point in its range. (A) Find four points that map to 1 (B) The natural inverse of f(z), say g(z) maps each point in its domain to infinitely...
  43. N

    Help with these complex analysis series problems?

    I don't even know where to start or go with this first problem. A) Assume that 'a sub n' E C and consider rearrangements of the convergent series the 'sum of 'a sub n' from n=1 to infinity'. Show that each of the following situations is possible and that this list includes all possibilities...
  44. S

    Complex Analysis - The Maximum Modulus Principle

    Homework Statement Find the maximum of \left|f\right| on the disc of radius 1 in the Complex Plane, for f(z)=3-\left|z\right|^{2} Homework Equations The maximum modulus principle? The Attempt at a Solution Since |z| is a real number, then surely the maximum must be 3 when z=0...
  45. S

    Using Liouville's Theorem to Show that Bounded Entire Functions are Polynomials

    Let f: ℂ→ ℂ be an entire function. If there is some nonnegative integer m and positive constants M,R such that |f(z)| ≤ M|z|m, for all z such that |z|≥ R, show that f is a polynomial of degree less that or equal to m. im really lost on this question. i feel like because...
  46. Z

    [Complex Analysis] Branch cuts of the logarithm

    Homework Statement Consider a branch of \log{z} analytic in the domain created with the branch cut x=−y, x≥0. If, for this branch, \log{1}=-2\pi i, find the following. \log⁡{(\sqrt{3}+i)} Homework Equations \log{z} = \ln{r} + i(\theta + 2k\pi) The Attempt at a Solution This one...
  47. N

    Complex analysis formula for an integral

    Homework Statement Find a formula for: \int1/(z-a)m(z-b)ndz around a ball of radius R, centred at z0 where |a| < R < |b| and m,n\inN. Homework Equations Not sure which equations to use, a cauchy integral formula maybe...? The Attempt at a Solution I've attempted to...
  48. N

    Laurent Series Complex Analysis question

    1. Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. 2...
  49. B

    Complex Analysis: Evaluating an Integral over a Contour

    Evaluate the integral of f over the contour C where: f(z) = 1/[z*(z+1)*(z+2)] where C = {z(t) = t+1 | 0 <= t < infinity} Over this contour, is f a real valued function? z(t) just maps t to the t+1, so it seems as if the contour is a real-valued continuous function, and f does not have any...
  50. S

    Complex analysis inequality proof

    Prove for all Z E C |ez-1| \leq e|z| - 1 \leq |z|e|z| I think this has to be proven using the triangle inequality but not sure how. Please help. :) thanks
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