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. P

    Complex Analysis or Complex Variables?

    Hi everyone, I'm a Physics student going into my Junior year and I'm currently registering for my courses for the following semester and I have two options for my "complex" course, namely: --------------------------------------------------- Complex Variables Theory of functions of one complex...
  2. I

    Complex Analysis or Solid State Electronics?

    First post. Great to be here. :) So, I'm stuck in deciding which of these courses to take next semester. I'm a current rising sophomore at UT Austin who has just switched from EE to physics-still nervous about that decision, but that's a separate topic. I've already got Waves(the first...
  3. E

    Harmonic Function Values (Complex Analysis)

    Homework Statement Let u be a continuous real-valued function in the closure of the unit disk \mathbb{D} that is harmonic in \mathbb{D}. Assume that the boundary values of u are given by u(e^{it}) = 5- 4 \cos t. Furthermore, let v be a harmonic conjugate of u in \mathbb{D}...
  4. E

    Complex Analysis - Normal Families

    Homework Statement Let U be a domain in \mathbb{C} with z_0 \in U. Let \mathcal{F} be the family of analytic functions f in U such that f(z_0) = -1 and f(U) \cap \mathbb{Q}_{\geq 0} = \emptyset, where \mathbb{Q}_{\geq0} denotes the set of non-negative rational numbers. Is \mathcal{F} a normal...
  5. N

    A simple Complex Analysis Mapping

    Homework Statement http://img684.imageshack.us/img684/779/334sn.jpg The Attempt at a Solution The first part was fairly straightforward, solve for z + 1, and then get w in terms of u + iv, rationalise the denominator, and then we get (x,y) in terms of u and v, which we substitute back...
  6. E

    Complex Analysis - Contour integral

    Homework Statement I have the following problem: Compute \operatorname{Re} \int _\gamma \frac{\sqrt{z}}{z+1} dz, where \gamma is the quarter-circle \{ z: |z|=1, \operatorname{Re}z \geq 0 , \operatorname{Im} z \geq 0 \} oriented from 1 to i, and \sqrt{z} denotes the principal...
  7. N

    Complex Analysis - Solving Complex Trig functions

    Homework Statement Now, I know there's two ways to go about this and it seems everywhere I look around on the web people are solving it in a way I think that seems longer, harder and more prone to mistakes in exams. It involves using the exponential identities and taking logs. I was shown...
  8. J

    [Complex Analysis] Determining order of a pole.

    I've been studying the residue theorem and I've been having some difficulty with classifying singularities. For example, let's use the function f(z) = \frac{1}{z sinz} I know it has two singularities, one at z=0 and the other at z=2kπ for k ={0,1,2,..}, I don't know what kind of singularities...
  9. P

    Complex Analysis - Radius of convergence of a Taylor series

    Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...
  10. J

    [Complex Analysis] Help with Cauchy Integral Problem

    Homework Statement Evaluate the following integral, I = \int_{0}^{2\pi} \frac{d \theta}{(1-2acos \theta + a^2)^2}, \ 0 < a < 1 For such, transform the integral above into a complex integral of the form ∫Rₐ(z)dz, where Rₐ(z) is a rational function of z. This will be obtained through the...
  11. K

    Complex analysis to evaluate integral

    Use complex analysis to evaluate the integral [from 0 to 2∏]∫dt/(b + cost) with b < -1.
  12. K

    Complex Analysis: What is f(1-4i)?

    Suppose f is an entire function and, for every z in the complex plane, |f'(z) - (2 + 3i)| ≥ 0.00007. Suppose also that f(0) = 10 + 3i and f'(7+ 9i) = 1 + i. What is f(1 - 4i)?
  13. P

    Complex Analysis - Branch Definition

    Homework Statement Hi everyone, This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and...
  14. B

    Complex Analysis Residue Query

    Folks, I am trying to understand calculating residues. http://www.wolframalpha.com/input/?i=residue+of+1%2F%28z%5E2%2B4%29%5E2+at+z%3D2i How is that answer determined? I mean (2i)^2=-4 and hence denominator is 0...? Thanks
  15. A

    What is a Pole of Order n in Complex Analysis?

    If a complex function has the form: f(z) = g(z)/(z-a)n then z=a is a pole of order n. I don't really understand all this fancy terminology. Isn't a pole just like when you for a real valued function g(x)/(x-a) don't want to divide by 0 and therefore the function is defined at x=a? If so what...
  16. C

    Complex Analysis or Differential Geometry first

    I have to choose which math course I'm going to take next term. I want to take both but I'm already taking two physics courses and my college's distribution requirements require that I take an English next term... bleh... I could audit one of the physics and then take both math courses, but that...
  17. N

    Complex Analysis - Differentiability

    Homework Statement Show that the function f defined by f(z) = 3\,{x}^{2}y+{y}^{3}-6\,{y}^{2}+i \left( 2\,{y}^{3}+6\,{y}^{2}+9\,x \right) is nowhere differentiable.The Attempt at a Solution Computing the C.R equations for this, I am left with {y}^{2}+2\,y={\it xy} and x^2+(y-2)^2 = 1...
  18. D

    Complex analysis: determine whether a family of functions is a normal family

    Homework Statement Let F be the set of all analytic functions f that map the open unit disc D(0,1) into the set U = \left\{w=u+iv : -2 < u < 2 \right\} such that f(0)=0. Determine whether or not F is a normal family. Homework Equations DEF'N: A normal family on a domain (i.e. open and...
  19. D

    Complex analysis - Schwarz's Lemma

    Homework Statement Show that M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}|≤|\frac{z-z_{0}}{1-\overline{z_{0}}z}| for all z, z_{0} in {w:|w|<1| Homework Equations I think I will have to use Schwarz's Lemma: if f is analytic in disc |z|<1, f(0)=0, and |f(z)|≤1 for all z in the...
  20. W

    Complex analysis: Counting zeros using the argument principle

    Homework Statement Gamelin VIII.1.6 (8.1.6) "For a fixed number a, find the number of solutions of z^5+2z^3-z^2+z=a satisfying Re z > 0" Homework Equations The argument principle relating the change in the argument to the number of zeros and poles of the function on the domain. The...
  21. M

    MHB Complex analysis - evaluate integral

    Hi all, I need to evaluate this integral anybody could point me to a solution? I've tried to look around (google, books), but I found no clue to solve it I wrote it in latex $\displaystyle \int_0^{2\pi} \! \frac{1}{(2 + \cos \theta)^2} \mathrm{d} \theta$ Thanks for the help, matteo
  22. M

    [Complex analysis] Coefficients of Laurent series

    Homework Statement I have some past exam questions that I am confused with Homework Equations a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz The Attempt at a Solution I'm not sure how to approach this, I'm completely lost and just attempted to solve a few: a) it says f(z)...
  23. N

    Complex Analysis - Manipulating trig identities

    Homework Statement Suppose c and (1 + ic)^{5} are real, (c ≠ 0) Show that either c = ± tan 36 or c = ± tan 72The Attempt at a Solution So I considered the polar form \left( {{\rm e}^{i\theta}} \right) ^{5} and that \theta=\arctan \left( c \right) , so c = tan θ Using binomial expansion, I...
  24. D

    Complex analysis showing solutions are inside or outside R

    Homework Statement Suppose w is not in the interval [-R,R] show that the equation z+\frac{R^{2}}{z}=2w has one solution z with |z|<R and one solution z with |z|>R Homework Equations none The Attempt at a Solution the book mentions that the quadratic is left unchanged by the...
  25. A

    MHB Find Complex Root of z^5=0 | Math Solutions

    how to find the complex root of z^5 = 0 there is one real root 0
  26. C

    Complex Analysis in Electrical Engineering

    Hi, Everyone! I just want to ask about the importance of Complex numbers analysis in the discipline of Electronics and Communications Engineering. I'm taking a course called, Analytical Methods in Engineering, and it's mostly focused on how to deal with complex numbers, from applying algebraic...
  27. N

    Understanding Analyticity and Continuity in Complex Analysis

    Homework Statement Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic.The Attempt at a Solution f(x + iy) = 2sin(x) -4y + i(y^2 +4x) Through C-R equations: du/dx = 2 cos x dv/dy = 2y du/dy = -4 dv/dx = 4 So the C-R equations hold...
  28. D

    MHB Trig integration complex analysis

    $$ \int_0^{\pi}\frac{ad\theta}{a^2 + \sin^2\theta} = \int_0^{2\pi}\frac{ad\theta}{1 + 2a^2 - \cos\theta} = \frac{\pi}{\sqrt{1 + a^2}} $$ Consider $a > 0$ and $a < 0$ First I don't think the second part is correct. Shouldn't it be $1 + 2a^2 - \cos 2\theta$?
  29. D

    MHB How Does Complex Analysis Explain the Integral of Sin^2(x)/x^2?

    $$ \int_{0}^{\infty}\frac{\sin^2 x}{x^2}dx = \frac{\pi}{2} $$. [Hint: Consider the integral of $(1 - e^{2ix})/x^2)$.] If we look at the complex sine, we have that $\sin z = \frac{e^{iz}-e^{-iz}}{2i}$. Then $$ \sin^2z = \frac{e^{-2iz}-e^{2iz}}{4} $$ so $$ \frac{\sin^2 z}{z^2} =...
  30. D

    Complex Analysis - Proving an analytic function f(z) is constant

    Homework Statement Let f(z) be an analytic function in the complex plane ℂ, and let \phi be amonotonic function of a real variable. Assume that U(x,y) = \phi(V(x,y)) where U(x,y) is the real part of f(z) and V(x,y) is the imaginary part of f(z). Prove that f is constant. Homework Equations...
  31. N

    Complex Analysis - Finding the equation of a circle

    Homework Statement If \frac{z}{z + 3} is purely imaginary, show that z lies on a certain circle and find the equation of that circle.The Attempt at a Solution So, \frac{z}{z + 3} = \frac{x + iy}{x + iy + 3} Multiplying by the complex conjugate (and simplifying), we get, \frac{x^{2} + y^{2}...
  32. M

    Solving Complex Analysis Questions: Are My Answers Right?

    Homework Statement I just wrote a test and was wondering if I got these questions right, I already solved them, please see the attached pictures below. Here are the questions; sorry for non-latex form 1) Let gamma be a positively oriented unit circle (|z|=1) in C solve: i) integral of...
  33. N

    Complex Analysis - Values of Real and Imaginary parts

    Homework Statement Simplify in terms of real and imaginary parts of x and y and sketch them. 1) Re \frac{z}{z-1} = 0 2) I am \frac{1}{z} ≥ 1 The Attempt at a Solution 1) \frac{x + iy}{x + iy -1} = 0 Am I allowed to just vanish the imaginary components here and have \frac{x}{x...
  34. N

    Complex Analysis - Finding the image through a mapping

    Homework Statement The point 1 + i is rotated anticlockwise through \frac{∏}{6} about the origin. Find its image. The Attempt at a Solution The point 1 + i creates an angle of arctan(1/1) = ∏/4 The rotation is by a further angle β = ∏/6. So the new point w in the w-plane from...
  35. N

    Complex Analysis - Value of imaginary part.

    Homework Statement Suppose both c and (1 + ic)^{5} are real (c \neq 0). Show that c = ± \sqrt{5 ± 2\sqrt{5}} Now use another method to show that either c = ± tan 36◦ or c = ± tan 72◦ The Attempt at a Solution I expanded it out, but I'm not entirely too sure how to solve this for...
  36. A

    Complex analysis non-constant analytic function

    There does not exist a non-constant analytic function in the unit circle which is real valued on the unit circle. I am not able to see why. I am trying to apply Louisville's Theorem, or maybe Open Mapping Th., but I fail. Is there a way of extending this function so that it entire? and even...
  37. A

    Complex Analysis: Two Questions About Non-Constant Analytic Functions

    Two questions: 1)Quote comes from a textbook: Each non-constant function analythic function with f(0)=0 is,in a small nbhd of 0, the composition of a conformal map with the nth-power map...The proof is given and I think I am comfortable with it.. My question is a lot simpler (I think)...
  38. J

    Application of Liouville's Theorem Complex Analysis

    Homework Statement Given: f is an entire function, Re f(z) ≤ n for all z. Show f is constant. Homework Equations The Attempt at a Solution So I thought I'd use Liouville's Theorem which states that, if f(z) is entire and there is a constant m such that |f(z)| ≤ m for all z...
  39. R

    Complex analysis: Sketch the region in the complex plane

    Homework Statement Sketch: {z: \pi?4 < Arg z ≤ \pi} Homework Equations The Attempt at a Solution Is it right to assume z0 = 0 ; a = a (radius = a) ; and taking \alpha = \pi/4 ; \beta = \pi And now in order to sketch the problem after setting up the complex plane is it correct...
  40. N

    Complex Analysis - Proving an inequality

    Homework Statement Show that if |z| = 10 then 497 ≤ |z^{3} + 5iz^{2} − 3| ≤ 1503. The Attempt at a Solution I'm not an entirely sure how to begin this one, or if what I'm doing is correct. If I sub in |z| = 10 into the equation; |1000 + 500i - 3| = 997 +500i Then the modulus of...
  41. L

    Analytic functions on simple connected region (complex analysis)

    Here's the problem: Let f and g be analytic functions on a simply connected domain Ω such that f2(z) + g2(z) = 1 for all z in Ω. Show that there exists an analytic function h such that f(z) = cos (h(z)) and g(z) = sin(h(z)) for all z in Ω. Here's my attempt at a solution: f2 + g2 = 1 on Ω...
  42. Petek

    Volume 2 of Burckel's Book on Complex Analysis

    I have a copy of Robert Burckel's An Introduction to Classical Complex Analysis, Volume 1. What happened to volume 2? The introduction to volume 1 contains a description of the contents of volume 2. It also contains the table of contents of volume 2. The beginning of volume 1 lists some of the...
  43. N

    Complex Analysis - Sketching regions in a complex plane

    Homework Statement |2z -1|\geq|z + i| The Attempt at a Solution The problem I have with this one is the 2z, I just need a clue on how to go about centering this one. If it were just |z - 1|; z_{0} would be 1.
  44. chwala

    Solve the complex analysis problem

    The problem is attached regards, chwala ken
  45. P

    MHB Uniform Convergence of f{_n} in Complex Analysis on S=[0,infinity)

    Let S=[0,infinity) and let f{_n}(z)=n^2ze^-(nz) Show that f{_n} -> 0. Is the function uniformly convergent? Sorry about it being unclear but TEX tags don't see to work. f{_n} means f subscript n. Thanks
  46. D

    MHB Exploring Problem 167 in Serge Lang's Complex Analysis

    I am trying to understand a problem in my book (for reference pr 167 Serge Lang Complex Analysis). $$ f(z) = \frac{1}{z} + \sum_{n = 1}^{\infty}\frac{z}{z^2-n^2} $$ Let R>0 (is this R representing the radius of convergence?) and let N>2R (where did this come from and why?). Write $f(z) =...
  47. G

    Complex Analysis - Transcendental Solutions Help

    This isn't really homework help. I'm working through a complex analysis textbook myself, and am stumped on the complex transcendentals, but I figured this was the best place for it. I would greatly appreciate any guidance here, I'm getting very frustrated! Homework Statement The problem is to...
  48. T

    Complex Analysis: Entire Function Series

    Homework Statement I need to prove that \sum_{n=1}^{∞}[1−Cos(n−1z)] is entire. Homework Equations The Attempt at a Solution I know that I need to show that the series is differentiable for its whole domain, but I am not sure how to do that. Should I try to use the ratio test?
  49. T

    Proving Domain D in C is Simply Connected

    Homework Statement Demonstrate that a domain D\in\mathbb{C} is simply connected if and only if, for every function f which is analytic and free of zeroes in D, a branch of the square root of f exists in D. Homework Equations The Attempt at a Solution I know that by definition...
  50. M

    Can complex analysis be used to solve PDEs other than the Laplacian?

    Hey all, I was reading up on Harmonic functions and how every solution to the laplace equation can be represented in the complex plane, so a mapping in the complex domain is actually a way to solve the equation for a desired boundary. This got me wondering: is this possible for other PDEs...
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