Complex analysis Definition and 784 Threads

Homework Statement Evaluate the integral ∫sin(z)/(z^2-4) dz about the contour C such that it is a circle of radius 2 centered at z = 2. Homework Equations All theorems of complex analysis except residue theorem. The Attempt at a Solution There is a singularity at z = 2, so we...

Homework Statement Show ln(az) where a is a real number and z = x + iy is harmonic everywhere except z = 0. Homework Equations z = x + iy = rcos(θ) + irsin(θ) = re^iθ z = u(x,y) + iv(x,y) Cauchy Riemann test for analyticity: ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x The Attempt at a...

Say you want to evaluate an integral over some domain, so one option is to write the integral as a contour integral in the complex plane. However, there can sometimes be several different contours that all cover the same domain, but may lead to different values in the event of singularities...

Hey everyone, I'm a math major and I am having a lot of trouble in my Complex Analysis class right now. I studied for 3 hours today for our first quiz on topology of the complex plane and I got somewhere between 40-50%. I memorized all the definitions but screwed up drawing the regions...

Homework Statement 1)Show that (1-i)^{2}=-2i then evaluate (1-i)^{2004}+(1-i)^{2005} 2)Prove that every complex number with moduli 1, except z=1, can be put in the form \frac{a+i}{a-i} 3)Let m and n be positive integers without a common factor. Define z^{m/n}=(z^{m})^{1/n}, and show...

Having just gone through a section of complex analysis in a math course, I'm curious when you would actually use things like contour integration and residue theory in EE. I've been told complex analysis has all these applications in z and laplace transforms, but it seems like you only ever...

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...

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...

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}...

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...

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...

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...

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...

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...

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...

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...

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

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)?

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...

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

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...

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...

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...

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...

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...

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...

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

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)...

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...

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...

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

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...

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...

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

$$ \int_{0}^{\infty}\frac{\sin^2 x}{x^2}dx = \frac{\pi}{2} $$. [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} =...

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...

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}...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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...

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...

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.

The problem is attached regards, chwala ken